Binary octahedral group

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Definition

The binary octahedral group can be defined in the following equivalent ways:

1. It is the Schur cover of symmetric group:S4 of type "-" which in symbols means it is the group $2 \cdot S_4^-$.
2. It is a binary von Dyck group with parameters $(2,3,4)$, i.e., it has the presentation:

$\! \langle a,b,c \mid a^4 = b^3 = c^2 = abc\rangle$
We denote the element $a^4 = b^3 = c^2$ as $z$. This element has order two.

Arithmetic functions

Function	Value	Similar groups	Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set)	48	groups with same order	As $2 \cdot S_n^-, n = 4$: $2(n!) = 2(4!) = 2(24) = 48$ As binary von Dyck group with parameters $(p,q,r) = (2,3,4)$: $\frac{4}{1/p + 1/q + 1/r - 1} = \frac{4}{1/2 + 1/3 + 1/4 - 1} = \frac{4}{1/12} = 48$
exponent of a group	24	groups with same order and exponent of a group | groups with same exponent of a group	As binary von Dyck group with parameters $(p,q,r) = (2,3,4)$: $2 \operatorname{lcm} \{ p,q,r \} = 2 \operatorname{lcm} \{ 2,3,4 \} = 2(12) = 24$
derived length	4	groups with same order and derived length | groups with same derived length	derived subgroup is isomorphic to SL(2,3). Derived subgroup of that is isomorphic to quaternion group. Derived subgroup of that is isomorphic to cyclic group:Z2, which is abelian.
nilpotency class	--		not a nilpotent group.
Frattini length	2	groups with same order and Frattini length | groups with same Frattini length	Intersection of Frattini subgroups is center of binary octahedral group which is cyclic of order two.
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group	2	groups with same order and subgroup rank of a group | groups with same subgroup rank of a group	--
max-length of a group	5	groups with same order and max-length of a group | groups with same max-length of a group

Arithmetic functions of a counting nature

Function	Value	Similar groups	Explanation for function value
number of conjugacy classes	8	groups with same order and number of conjugacy classes | groups with same number of conjugacy classes	As binary von Dyck group with parameters $(p,q,r) = (2,3,4)$: $p + q + r - 1 = 2 + 3 + 4 - 1 = 8$ See element structure of binary octahedral group Note also that the number of conjugacy classes equals that of the other double cover of $S_4$, namely general linear group:GL(2,3)
number of subgroups	35	groups with same order and number of subgroups | groups with same number of subgroups	See subgroup structure of binary octahedral group
number of conjugacy classes of subgroups	13	groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups	See subgroup structure of binary octahedral group

Group properties

Property	Satisfied	Explanation
Abelian group	No
Nilpotent group	No
Metacyclic group	No
Supersolvable group	No
Solvable group	Yes	Length four.
T-group	No
HN-group	No
Monolithic group	Yes	The center of order two is the unique minimal normal subgroup.
One-headed group	Yes

Elements

Further information: element structure of binary octahedral group

Element structure of binary octahedral group

Subgroups

Further information: subgroup structure of binary octahedral group

Quick summary

Item	Value
Number of subgroups	35
Number of conjugacy classes of subgroups	13
Number of automorphism classes of subgroups	13
Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems	2-Sylow: generalized quaternion group:Q16, Sylow number 3, fusion system ? 3-Sylow: cyclic group:Z3, Sylow number 4
maximal subgroups	Maximal subgroups are: derived subgroup of binary octahedral group (isomorphic to SL(2,3), order 24), 2-Sylow subgroups of binary octahedral group (isomorphic to Q16, order 16), and subgroups of order 12 isomorphic to dicyclic group:Dic12

Table classifying subgroups up to automorphisms

TABLE SORTING AND INTERPRETATION: Note that the subgroups in the table below are sorted based on the powers of the prime divisors of the order, first covering the smallest prime in ascending order of powers, then powers of the next prime, then products of powers of the first two primes, then the third prime, then products of powers of the first and third, second and third, and all three primes. The rationale is to cluster together subgroups with similar prime powers in their order. The subgroups are not sorted by the magnitude of the order. To sort that way, click the sorting button for the order column. Similarly you can sort by index or by number of subgroups of the automorphism class.

Automorphism class of subgroups	Representative subgroup (full list if small, generating set if large)	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes (=1 iff automorph-conjugate subgroup)	Size of each conjugacy class (=1 iff normal subgroup)	Total number of subgroups (=1 iff characteristic subgroup)	Quotient group (if it exists)
trivial subgroup	$\{ e \}$	trivial group	1	48	1	1	1	binary octahedral group
center of binary octahedral group	$\langle z \rangle$	cyclic group:Z2	2	24	1	1	1	symmetric group:S4
3-Sylow subgroups of binary octahedral group	$\langle b^2 \rangle$	cyclic group:Z3	3	16	1	4	4	--
cyclic subgroups of order four	$\langle c \rangle$	cyclic group:Z4	4	12	1			--
cyclic subgroups of order four	$\langle a^2 \rangle$	cyclic group:Z4	4	12	1			--
cyclic subgroups of order six	$\langle b \rangle$	cyclic group:Z6	6	8	1	4	4	--
quaternion subgroup of one type	$\langle aca^{-3},c \rangle$	quaternion group	8	6				--

(remaining items to be transcribed into table later)

1. The quaternion group of the form PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
   One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
   . Isomorphic to quaternion group.
2. The cyclic groups conjugate to $\langle a \rangle$. Isomorphic to cyclic group:Z8. (3)
3. The dicyclic group of order $12$, i.e., the binary von Dyck group $\Gamma(3,2,2)$. Isomorphic to dicyclic group:Dic12. (4)
4. The generalized quaternion group of order $16$. Isomorphic to generalized quaternion group. (3)
5. A unique subgroup of order $24$, isomorphic to special linear group:SL(2,3). (1)
6. The whole group. (1)

Linear representation theory

Further information: Linear representation theory of binary octahedral group

GAP implementation

Group ID

This finite group has order 48 and has ID 28 among the groups of order 48 in GAP's SmallGroup library. For context, there are 52 groups of order 48. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(48,28)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(48,28);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [48,28]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.