AGL(1,𝔽₇)×#13091120 #72269783

order	61236
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

AGL(1,𝔽₇)×#13091120

Splittings

Fitting, derived	C₇×#10624749 ⋊ C₂²×C₃
Hall	ΓL(1,𝔽₈)×#10624749 ⋊ C₂²
Sylow	C₇ ⋊ C₂×C₃×#13091120

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×#10624749	C₂²×C₃
Frattini	C₃³	AGL(1,𝔽₇)×C₃⋊(C₃⋊D₃)
socle	C₃×C₇	C₂×C₃×#92338
LCS	G ▷ C₇×#10624749
UCS	C₁

10084 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:

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

Conjugacy classes & character table

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

Relationships

holomorph of

C₇×3₋¹⁺²

GAP

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

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

AGL(1,𝔽7)×#13091120 #72269783