ΓL(1,𝔽₈)×#13091120 #72249196

order	30618
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃

Aliases

direct product

ΓL(1,𝔽₈)×#13091120

Splittings

Fitting, derived	C₇×#10624749 ⋊ C₂×C₃
Hall	ΓL(1,𝔽₈)×#10624749 ⋊ C₂
Sylow	C₇ ⋊ C₃×#13091120

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×#10624749	C₂×C₃
Frattini	C₃³	ΓL(1,𝔽₈)×C₃⋊(C₃⋊D₃)
socle	C₃×C₇	C₃×#92338
LCS	G ▷ C₇×#10624749
UCS	C₁

4400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 189

A minimal faithful permutation representation — 4 generators on 34 points:

(2,3,4)(5,7,6)(9,13)(10,16)(11,12)(14,23)(15,26)(17,28)(18,20)(19,22)(21,27)(24,30)(25,33)(29,31)(32,34);(1,2,3,5,4,6,7)(8,17,16,12,20,18,11,10,28)(9,25,27,15,29,24,19,14,34)(13,32,23,22,30,31,26,21,33);(10,17,20)(16,18,28)(21,30,32)(24,27,34);(8,9,13)(10,14,21)(11,19,26)(12,15,22)(16,27,23)(17,25,32)(18,24,31)(20,29,30)(28,34,33)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#13091120
maximal normal subgroup of	AGL(1,𝔽₇)×#13091120

GAP

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

gap> G := Group((2,3,4)(5,7,6)(9,13)(10,16)(11,12)(14,23)(15,26)(17,28)(18,20)(19,22)(21,27)(24,30)(25,33)(29,31)(32,34), (1,2,3,5,4,6,7)(8,17,16,12,20,18,11,10,28)(9,25,27,15,29,24,19,14,34)(13,32,23,22,30,31,26,21,33), (10,17,20)(16,18,28)(21,30,32)(24,27,34), (8,9,13)(10,14,21)(11,19,26)(12,15,22)(16,27,23)(17,25,32)(18,24,31)(20,29,30)(28,34,33));

ΓL(1,𝔽8)×#13091120 #72249196