D8 in A6
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This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) dihedral group:D8 and the group is (up to isomorphism) alternating group:A6 (see subgroup structure of alternating group:A6).
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This article is about the subgroup in the group
, where
is alternating group:A6, i.e., the alternating group on the set
, and
is the subgroup:
is a 2-Sylow subgroup of
and is isomorphic to dihedral group:D8. There is a total of 45 conjugate subgroups to
in
(including
itself).
Contents
Arithmetic functions
Function | Value | Explanation | Comment |
---|---|---|---|
order of the group | 360 | ||
order of the subgroup | 8 | ||
index of the subgroup | 45 | ||
size of conjugacy class | 45 | ||
number of conjugacy classes in automorphism class | 1 |
Subgroup properties
Resemblance-based properties
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
order-conjugate subgroup | conjugate to any subgroup of the same order | Yes | Sylow implies order-conjugate | |
order-dominating subgroup | any subgroup of the whole group whose order divides the order of ![]() ![]() |
Yes | Sylow implies order-dominating | |
order-dominated subgroup | any subgroup of the whole group whose order is a multiple of the order of ![]() ![]() |
Yes | Sylow implies order-dominated | |
order-isomorphic subgroup | isomorphic to any subgroup of the group of the same order | Yes | (via order-conjugate) | |
isomorph-automorphic subgroup | Yes | (via order-conjugate) | ||
automorph-conjugate subgroup | Yes | (via order-conjugate) | ||
order-automorphic subgroup | Yes | (via order-conjugate) | ||
isomorph-conjugate subgroup | Yes | (via order-conjugate) |
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
normal subgroup | equals all its conjugate subgroups | No | (see other conjugate subgroups) | |
subnormal subgroup | No | |||
self-normalizing subgroup | equals its normalizer in the whole group | Yes | ||
abnormal subgroup | Yes | |||
weakly abnormal subgroup | Yes | |||
contranormal subgroup | Yes | |||
maximal subgroup | Yes | |||
pronormal subgroup | Yes | Sylow implies pronormal | ||
weakly pronormal subgroup | Yes | (via pronormal) |
Fusion system
The subgroup embedding induces the simple fusion system for dihedral group:D8.
GAP implementation
Construction of subgroup given group as a black box
Suppose we are already given a group that we know to be isomorphic to alternating group:A6. Then, the subgroup
can be constructed using SylowSubgroup as follows:
H := SylowSubgroup(G,2);
Construction of group-subgroup pair
The group and subgroup pair can be defined using GAP's AlternatingGroup and SylowSubgroup functions as follows:
G := AlternatingGroup(6); H := SylowSubgroup(G,2);