There are 267 groups of order 64 up to isomorphism. This page lists them.
The list
| Group |
GAP ID (second part) |
Abelian?
|
| Cyclic group:Z64 |
1 |
Yes
|
| Direct product of Z8 and Z8 |
2 |
Yes
|
| Semidirect product of Z8 and Z8 of M-type |
3 |
No
|
| SmallGroup(64,4) |
4 |
No
|
| SmallGroup(64,5) |
5 |
No
|
| SmallGroup(64,6) |
6 |
No
|
| SmallGroup(64,7) |
7 |
No
|
| SmallGroup(64,8) |
8 |
No
|
| SmallGroup(64,9) |
9 |
No
|
| SmallGroup(64,10) |
10 |
No
|
| SmallGroup(64,11) |
11 |
No
|
| SmallGroup(64,12) |
12 |
No
|
| SmallGroup(64,13) |
13 |
No
|
| SmallGroup(64,14) |
14 |
No
|
| Semidirect product of Z8 and Z8 of semidihedral type |
15 |
No
|
| Semidirect product of Z8 and Z8 of dihedral type |
16 |
No
|
| SmallGroup(64,17) |
17 |
No
|
| Unitriangular matrix group:UT(3,Z4) |
18 |
No
|
| SmallGroup(64,19) |
19 |
No
|
| SmallGroup(64,20) |
20 |
No
|
| SmallGroup(64,21) |
21 |
No
|
| SmallGroup(64,22) |
22 |
No
|
| SmallGroup(64,23) |
23 |
No
|
| SmallGroup(64,24) |
24 |
No
|
| SmallGroup(64,25) |
25 |
No
|
| Direct product of Z16 and Z4 |
26 |
Yes
|
| Semidirect product of Z16 and Z4 of M-type |
27 |
No
|
| Semidirect product of Z16 and Z4 via fifth power map |
28 |
No
|
| SmallGroup(64,29) |
29 |
No
|
| SmallGroup(64,30) |
30 |
No
|
| SmallGroup(64,31) |
31 |
No
|
| SmallGroup(64,32) |
32 |
No
|
| SmallGroup(64,33) |
33 |
No
|
| SmallGroup(64,34) |
34 |
No
|
| SmallGroup(64,35) |
35 |
No
|
| SmallGroup(64,36) |
36 |
No
|
| SmallGroup(64,37) |
37 |
No
|
| SmallGroup(64,38) |
38 |
No
|
| SmallGroup(64,39) |
39 |
No
|
| SmallGroup(64,40) |
40 |
No
|
| SmallGroup(64,41) |
41 |
No
|
| SmallGroup(64,42) |
42 |
No
|
| SmallGroup(64,43) |
43 |
No
|
| Nontrivial semidirect product of Z4 and Z16 |
44 |
No
|
| SmallGroup(64,45) |
45 |
No
|
| Semidirect product of Z16 and Z4 via cube map |
46 |
No
|
| Semidirect product of Z16 and Z4 of dihedral type |
47 |
No
|
| Semidirect product of Z16 and Z4 of semidihedral type |
48 |
No
|
| SmallGroup(64,49) |
49 |
No
|
| Direct product of Z32 and Z2 |
50 |
Yes
|
| Modular maximal-cyclic group:M64 |
51 |
No
|
| Dihedral group:D64 |
52 |
No
|
| Semidihedral group:SD64 |
53 |
No
|
| Generalized quaternion group:Q64 |
54 |
No
|
| Direct product of Z4 and Z4 and Z4 |
55 |
Yes
|
| Direct product of SmallGroup(32,2) and Z2 |
56 |
No
|
| SmallGroup(64,57) |
57 |
No
|
| Direct product of SmallGroup(16,3) and Z4 |
58 |
No
|
| Direct product of SmallGroup(16,4) and Z4 |
59 |
No
|
| SmallGroup(64,60) |
60 |
No
|
| SmallGroup(64,61) |
61 |
No
|
| SmallGroup(64,62) |
62 |
No
|
| SmallGroup(64,63) |
63 |
No
|
| SmallGroup(64,64) |
64 |
No
|
| SmallGroup(64,65) |
65 |
No
|
| SmallGroup(64,66) |
66 |
No
|
| SmallGroup(64,67) |
67 |
No
|
| SmallGroup(64,68) |
68 |
No
|
| SmallGroup(64,69) |
69 |
No
|
| SmallGroup(64,70) |
70 |
No
|
| SmallGroup(64,71) |
71 |
No
|
| SmallGroup(64,72) |
72 |
No
|
| SmallGroup(64,73) |
73 |
No
|
| SmallGroup(64,74) |
74 |
No
|
| SmallGroup(64,75) |
75 |
No
|
| SmallGroup(64,76) |
76 |
No
|
| SmallGroup(64,77) |
77 |
No
|
| SmallGroup(64,78) |
78 |
No
|
| SmallGroup(64,79) |
79 |
No
|
| SmallGroup(64,80) |
80 |
No
|
| SmallGroup(64,81) |
81 |
No
|
| SmallGroup(64,82) |
82 |
No
|
| Direct product of Z8 and Z4 and Z2 |
83 |
Yes
|
| Direct product of SmallGroup(32,4) and Z2 |
84 |
No
|
| Direct product of M16 and Z4 |
85 |
No
|
| Central product of M16 and Z8 over common Z2 |
86 |
No
|
| SmallGroup(64,87) |
87 |
No
|
| SmallGroup(64,88) |
88 |
No
|
| SmallGroup(64,89) |
89 |
No
|
| SmallGroup(64,90) |
90 |
No
|
| SmallGroup(64,91) |
91 |
No
|
| SmallGroup(64,92) |
92 |
No
|
| SmallGroup(64,93) |
93 |
No
|
| SmallGroup(64,94) |
94 |
No
|
| SmallGroup(64,95) |
95 |
No
|
| SmallGroup(64,96) |
96 |
No
|
| SmallGroup(64,97) |
97 |
No
|
| SmallGroup(64,98) |
98 |
No
|
| SmallGroup(64,99) |
99 |
No
|
| SmallGroup(64,100) |
100 |
No
|
| SmallGroup(64,101) |
101 |
No
|
| SmallGroup(64,102) |
102 |
No
|
| Direct product of SmallGroup(32,12) and Z2 |
103 |
No
|
| SmallGroup(64,104) |
104 |
No
|
| SmallGroup(64,105) |
105 |
No
|
| Direct product of SmallGroup(32,13) and Z2 |
106 |
No
|
| Direct product of SmallGroup(32,14) and Z2 |
107 |
No
|
| SmallGroup(64,108) |
108 |
No
|
| SmallGroup(64,109) |
109 |
No
|
| SmallGroup(64,110) |
110 |
No
|
| SmallGroup(64,111) |
111 |
No
|
| SmallGroup(64,112) |
112 |
No
|
| SmallGroup(64,113) |
113 |
No
|
| SmallGroup(64,114) |
114 |
No
|
| Direct product of Z8 and D8 |
115 |
No
|
| SmallGroup(64,116) |
116 |
No
|
| SmallGroup(64,117) |
117 |
No
|
| Direct product of D16 and Z4 |
118 |
No
|
| Direct product of SD16 and Z4 |
119 |
No
|
| Direct product of Q16 and Z4 |
120 |
No
|
| SmallGroup(64,121) |
121 |
No
|
| SmallGroup(64,122) |
122 |
No
|
| SmallGroup(64,123) |
123 |
No
|
| SmallGroup(64,124) |
124 |
No
|
| SmallGroup(64,125) |
125 |
No
|
| SmallGroup(64,126) |
126 |
No
|
| SmallGroup(64,127) |
127 |
No
|
| SmallGroup(64,128) |
128 |
No
|
| SmallGroup(64,129) |
129 |
No
|
| SmallGroup(64,130) |
130 |
No
|
| SmallGroup(64,131) |
131 |
No
|
| SmallGroup(64,132) |
132 |
No
|
| SmallGroup(64,133) |
133 |
No
|
| Holomorph of D8 |
134 |
No
|
| SmallGroup(64,135) |
135 |
No
|
| SmallGroup(64,136) |
136 |
No
|
| SmallGroup(64,137) |
137 |
No
|
| Unitriangular matrix group:UT(4,2) |
138 |
No
|
| SmallGroup(64,139) |
139 |
No
|
| SmallGroup(64,140) |
140 |
No
|
| SmallGroup(64,141) |
141 |
No
|
| SmallGroup(64,142) |
142 |
No
|
| SmallGroup(64,143) |
143 |
No
|
| SmallGroup(64,144) |
144 |
No
|
| SmallGroup(64,145) |
145 |
No
|
| SmallGroup(64,146) |
146 |
No
|
| SmallGroup(64,147) |
147 |
No
|
| SmallGroup(64,148) |
148 |
No
|
| SmallGroup(64,149) |
149 |
No
|
| SmallGroup(64,150) |
150 |
No
|
| SmallGroup(64,151) |
151 |
No
|
| SmallGroup(64,152) |
152 |
No
|
| SmallGroup(64,153) |
153 |
No
|
| SmallGroup(64,154) |
154 |
No
|
| SmallGroup(64,155) |
155 |
No
|
| SmallGroup(64,156) |
156 |
No
|
| SmallGroup(64,157) |
157 |
No
|
| SmallGroup(64,158) |
158 |
No
|
| SmallGroup(64,159) |
159 |
No
|
| SmallGroup(64,160) |
160 |
No
|
| SmallGroup(64,161) |
161 |
No
|
| SmallGroup(64,162) |
162 |
No
|
| SmallGroup(64,163) |
163 |
No
|
| SmallGroup(64,164) |
164 |
No
|
| SmallGroup(64,165) |
165 |
No
|
| SmallGroup(64,166) |
166 |
No
|
| SmallGroup(64,167) |
167 |
No
|
| SmallGroup(64,168) |
168 |
No
|
| SmallGroup(64,169) |
169 |
No
|
| SmallGroup(64,170) |
170 |
No
|
| SmallGroup(64,171) |
171 |
No
|
| SmallGroup(64,172) |
172 |
No
|
| SmallGroup(64,173) |
173 |
No
|
| SmallGroup(64,174) |
174 |
No
|
| SmallGroup(64,175) |
175 |
No
|
| SmallGroup(64,176) |
176 |
No
|
| SmallGroup(64,177) |
177 |
No
|
| SmallGroup(64,178) |
178 |
No
|
| SmallGroup(64,179) |
179 |
No
|
| SmallGroup(64,180) |
180 |
No
|
| SmallGroup(64,181) |
181 |
No
|
| SmallGroup(64,182) |
182 |
No
|
| Direct product of Z16 and V4 |
183 |
Yes
|
| Direct product of M32 and Z2 |
184 |
No
|
| Central product of D8 and Z16 |
185 |
No
|
| Direct product of D32 and Z2 |
186 |
No
|
| Direct product of SD32 and Z2 |
187 |
No
|
| Direct product of Q32 and Z2 |
188 |
No
|
| SmallGroup(64,189) |
189 |
No
|
| Semidirect product of Z16 and V4 |
190 |
No
|
| SmallGroup(64,191) |
191 |
No
|
| Direct product of Z4 and Z4 and V4 |
192 |
Yes
|
| Direct product of SmallGroup(16,3) and V4 |
193 |
No
|
| Direct product of SmallGroup(16,4) and V4 |
194 |
No
|
| Direct product of SmallGroup(32,24) and Z2 |
195 |
No
|
| Direct product of D8 and Z4 and Z2 |
196 |
No
|
| SmallGroup(64,197) |
197 |
No
|
| Direct product of SmallGroup(16,13) and Z4 |
198 |
No
|
| SmallGroup(64,199) |
199 |
No
|
| SmallGroup(64,200) |
200 |
No
|
| SmallGroup(64,201) |
201 |
No
|
| Direct product of SmallGroup(32,27) and Z2 |
202 |
No
|
| SmallGroup(64,203) |
203 |
No
|
| SmallGroup(64,204) |
204 |
No
|
| SmallGroup(64,205) |
205 |
No
|
| SmallGroup(64,206) |
206 |
No
|
| SmallGroup(64,207) |
207 |
No
|
| SmallGroup(64,208) |
208 |
No
|
| Direct product of SmallGroup(32,33) and Z2 |
209 |
No
|
| SmallGroup(64,210) |
210 |
No
|
| SmallGroup(64,211) |
211 |
No
|
| SmallGroup(64,212) |
212 |
No
|
| SmallGroup(64,213) |
213 |
No
|
| SmallGroup(64,214) |
214 |
No
|
| Unitriangular matrix group of degree three over quotient of polynomial ring over F2 by square of indeterminate |
215 |
No
|
| SmallGroup(64,216) |
216 |
No
|
| SmallGroup(64,217) |
217 |
No
|
| SmallGroup(64,218) |
218 |
No
|
| SmallGroup(64,219) |
219 |
No
|
| SmallGroup(64,220) |
220 |
No
|
| SmallGroup(64,221) |
221 |
No
|
| SmallGroup(64,222) |
222 |
No
|
| SmallGroup(64,223) |
223 |
No
|
| SmallGroup(64,224) |
224 |
No
|
| SmallGroup(64,225) |
225 |
No
|
| Direct product of D8 and D8 |
226 |
No
|
| SmallGroup(64,227) |
227 |
No
|
| SmallGroup(64,228) |
228 |
No
|
| SmallGroup(64,229) |
229 |
No
|
| Direct product of D8 and Q8 |
230 |
No
|
| SmallGroup(64,231) |
231 |
No
|
| SmallGroup(64,232) |
232 |
No
|
| SmallGroup(64,233) |
233 |
No
|
| SmallGroup(64,234) |
234 |
No
|
| SmallGroup(64,235) |
235 |
No
|
| SmallGroup(64,236) |
236 |
No
|
| SmallGroup(64,237) |
237 |
No
|
| SmallGroup(64,238) |
238 |
No
|
| Direct product of Q8 and Q8 |
239 |
No
|
| SmallGroup(64,240) |
240 |
No
|
| SmallGroup(64,241) |
241 |
No
|
| Unitriangular matrix group:UT(3,4) |
242 |
No
|
| SmallGroup(64,243) |
243 |
No
|
| SmallGroup(64,244) |
244 |
No
|
| SmallGroup(64,245) |
245 |
No
|
| Direct product of Z8 and E8 |
246 |
Yes
|
| Direct product of M16 and V4 |
247 |
No
|
| SmallGroup(64,248) |
248 |
No
|
| SmallGroup(64,249) |
249 |
No
|
| Direct product of D16 and V4 |
250 |
No
|
| Direct product of SD16 and V4 |
251 |
No
|
| Direct product of Q16 and V4 |
252 |
No
|
| SmallGroup(64,253) |
253 |
No
|
| Direct product of holomorph of Z8 and Z2 |
254 |
No
|
| SmallGroup(64,255) |
255 |
No
|
| SmallGroup(64,256) |
256 |
No
|
| SmallGroup(64,257) |
257 |
No
|
| SmallGroup(64,258) |
258 |
No
|
| SmallGroup(64,259) |
259 |
No
|
| Direct product of E16 and Z4 |
260 |
Yes
|
| Direct product of D8 and E8 |
261 |
No
|
| Direct product of E8 and Q8 |
262 |
No
|
| Direct product of SmallGroup(16,13) and V4 |
263 |
No
|
| Direct product of SmallGroup(32,49) and Z2 |
264 |
No
|
| Direct product of SmallGroup(32,50) and Z2 |
265 |
No
|
| SmallGroup(64,266) |
266 |
No
|
| Elementary abelian group:E64 |
267 |
Yes
|
GAP implementation
The order 64 is part of GAP's SmallGroup library. Hence, any group of order 64 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.
Further, the collection of all groups of order 64 can be accessed as a list using GAP's AllSmallGroups function.
Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:
gap> SmallGroupsInformation(64);
There are 267 groups of order 64.
They are sorted by their ranks.
1 is cyclic.
2 - 54 have rank 2.
55 - 191 have rank 3.
192 - 259 have rank 4.
260 - 266 have rank 5.
267 is elementary abelian.
For the selection functions the values of the following attributes
are precomputed and stored:
IsAbelian, PClassPGroup, RankPGroup, FrattinifactorSize and
FrattinifactorId.
This size belongs to layer 2 of the SmallGroups library.
IdSmallGroup is available for this size.</pre.