C₂×#72210357 #72258342

order	6912
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

Fitting	C₂⁵×C₃² ⋊ C₂²×D₃
derived	C₃²×C₂²⋊A₄ ⋊ C₂⁴

	subgroup	G/H
center, hypercenter	C₂	#72210357
commutator, cosocle, hypocenter	C₃²×C₂²⋊A₄	C₂⁴
Frattini	C₂²	C₂×D₃²×S₄
Fitting	C₂⁵×C₃²	C₂²×D₃
socle	C₂³×C₃²	C₂²×S₄
LCS	G ▷ C₃²×C₂²⋊A₄
UCS	C₁ ◁ C₂

13721 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 13721 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 4 generators on 16 points:

(4,5)(6,7)(10,13,15)(11,12,16);(1,2)(3,4,5);(11,16)(13,14);(7,8)(9,10)(11,14)(12,15)(13,16)

maximal subgroup of	D₃×#72210357
maximal quotient of	D₃×#72210357

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((4,5)(6,7)(10,13,15)(11,12,16), (1,2)(3,4,5), (11,16)(13,14), (7,8)(9,10)(11,14)(12,15)(13,16));