GL(2,𝔽₃)×PSU(3,𝔽₂) #72244530

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃

Aliases

C₃²×Q₈ ⋊ D₃×Q₈

	subgroup	G/H
center, Frattini, hypercenter	C₂	S₄×PSU(3,𝔽₂)
commutator, cosocle	C₃⋊D₃×SL(2,𝔽₃)	C₂³
Fitting	C₃²×Q₈	D₃×Q₈
socle	C₂×C₃²	Q₈×S₄
prime radical	C₃⋊D₃×GL(2,𝔽₃)	C₂²
hypocenter	C₃²×SL(2,𝔽₃)	C₂×Q₈
LCS	G ▷ C₃⋊D₃×SL(2,𝔽₃) ▷ C₃²×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

575 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

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

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 17 points:

(1,2,8,4,7,6)(3,5)(9,12,14,15)(11,17,16,13);(10,12,14,16)(11,15,13,17);(2,5)(3,7)(6,8)

maximal subgroup of	GL(2,𝔽₃)×PSU(3,𝔽₂)⋊C₃, GL(2,𝔽₃)×AΓL(1,𝔽₉)
maximal normal subgroup of	GL(2,𝔽₃)×PSU(3,𝔽₂)⋊C₃, GL(2,𝔽₃)×AΓL(1,𝔽₉)

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

gap> G := Group((1,2,8,4,7,6)(3,5)(9,12,14,15)(11,17,16,13), (10,12,14,16)(11,15,13,17), (2,5)(3,7)(6,8));