C₃²×#11757702 #72269279

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₃⁵

Aliases

Fitting, Sylow	C₃²×C₃²⋊H(3,𝔽₃) ⋊ C₂²
derived	C₃⁴ ⋊ C₂²×C₃³

	subgroup	G/H
center, hypercenter	C₃²	#11757702
commutator, cosocle, hypocenter	C₃⁴	C₂²×C₃³
Frattini	C₃²	C₃³×D₃²
Fitting	C₃²×C₃²⋊H(3,𝔽₃)	C₂²
socle	C₃⁴	C₃×D₃²
LCS	G ▷ C₃⁴
UCS	C₁ ◁ C₃²

12098 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 12098 conjugacy classes of subgroups — too many to draw. Only its 322 normal subgroups are shown.

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

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 24 points:

(4,5,6)(8,10)(9,12)(11,13)(14,15)(16,17)(19,22)(20,21)(23,24);(7,8,14)(9,11,13)(10,12,15)(17,23,22,18,21,24)(19,20);(1,2,3)

324 conjugacy classes. The character table is too large to display.

maximal subgroup of	#72269296
maximal normal subgroup of	#72269296

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

gap> G := Group((4,5,6)(8,10)(9,12)(11,13)(14,15)(16,17)(19,22)(20,21)(23,24), (7,8,14)(9,11,13)(10,12,15)(17,23,22,18,21,24)(19,20), (1,2,3));