C₄×SL(2,𝔽₉) #72227897

order	2880
isoclinism	isoclinic class of SL(2,𝔽₉)
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃

non-solvable

Aliases

direct product

C₄×SL(2,𝔽₉)

Subgroups

	subgroup	G/H
center, Fitting, hypercenter	C₂×C₄	A₆
commutator, hypocenter	SL(2,𝔽₉)	C₄
Frattini, socle, cosocle	C₂²	C₂×A₆
prime radical	C₂×SL(2,𝔽₉)	C₂
LCS	G ▷ SL(2,𝔽₉)
UCS	C₁ ◁ C₂×C₄

131 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 84 points:

(1,4,2,3)(5,27,81,32,8,54,58,62)(6,57,47,28,12,38,69,42)(7,29,44,77,16,53,73,74)(9,19,66,20,18,39,33,40)(10,76,55,25,23,48,82,13)(11,63,79,24,26,60,50,51)(14,84,21,22,31,65,45,49)(15,17,83,56,35,36,72,71)(30,59,34,67,61,41,64,46)(37,75,43,52,70,78,68,80);(5,34,41,58,43)(6,62,47,83,36)(7,82,26,37,63)(8,64,59,81,68)(9,20,75,31,52)(10,46,42,61,73)(11,70,60,16,55)(12,32,69,72,17)(13,39,54,66,74)(14,80,18,40,78)(15,29,48,56,21)(19,27,33,77,25)(22,38,65,51,79)(23,67,28,30,44)(24,50,49,57,84)(35,53,76,71,45)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72227900

GAP

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

gap> G := Group((1,4,2,3)(5,27,81,32,8,54,58,62)(6,57,47,28,12,38,69,42)(7,29,44,77,16,53,73,74)(9,19,66,20,18,39,33,40)(10,76,55,25,23,48,82,13)(11,63,79,24,26,60,50,51)(14,84,21,22,31,65,45,49)(15,17,83,56,35,36,72,71)(30,59,34,67,61,41,64,46)(37,75,43,52,70,78,68,80), (5,34,41,58,43)(6,62,47,83,36)(7,82,26,37,63)(8,64,59,81,68)(9,20,75,31,52)(10,46,42,61,73)(11,70,60,16,55)(12,32,69,72,17)(13,39,54,66,74)(14,80,18,40,78)(15,29,48,56,21)(19,27,33,77,25)(22,38,65,51,79)(23,67,28,30,44)(24,50,49,57,84)(35,53,76,71,45));

C4×SL(2,𝔽9) #72227897