GL(2,𝔽₃)×#595 #72242971

order	3264
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product

GL(2,𝔽₃)×#595

Splittings

Fitting	Q₈×C₁₇ ⋊ C₄×D₃
derived	C₁₇×SL(2,𝔽₃) ⋊ C₂×C₄
Sylow	C₁₇ ⋊ C₄×GL(2,𝔽₃)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	S₄×#595
commutator, hypocenter	C₁₇×SL(2,𝔽₃)	C₂×C₄
Fitting	Q₈×C₁₇	C₄×D₃
socle	C₂×C₁₇	C₄×S₄
prime radical	D₁₇×GL(2,𝔽₃)	C₂
cosocle	SL(2,𝔽₃)×D₁₇	C₂²
LCS	G ▷ C₁₇×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

176 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 136

A minimal faithful permutation representation — 2 generators on 25 points:

(1,5,3)(4,8,7)(10,11,12,14)(13,16,17,21)(15,20,18,19)(22,24,25,23);(2,5)(3,7)(6,8)(9,10,13,18,14,19,23,25,21,16,22,24,20,11,15,17,12)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#595×#5935, GL(2,𝔽₃)×#3633
maximal normal subgroup of	GL(2,𝔽₃)×#3633

GAP

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

gap> G := Group((1,5,3)(4,8,7)(10,11,12,14)(13,16,17,21)(15,20,18,19)(22,24,25,23), (2,5)(3,7)(6,8)(9,10,13,18,14,19,23,25,21,16,22,24,20,11,15,17,12));