C₃₁×C₂²⋊ΓL(1,𝔽₈) #14579644

order	2604
isoclinism	isoclinic class of C₂²⋊ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₃×C₃₁ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₃₁×C₂²⋊ΓL(1,𝔽₈)
unambiguous semidirect product	C₂²⋊(ΓL(1,𝔽₈)×C₃₁)
unambiguous semidirect product	C₇⋊(A₄×C₃₁)

Splittings

Fitting, socle, Hall	C₂²×C₇×C₃₁ ⋊ C₃
derived, Hall	C₂²×C₇ ⋊ C₃×C₃₁
Hall	C₇×C₃₁ ⋊ A₄
Hall	C₂²×C₃₁ ⋊ ΓL(1,𝔽₈)
Sylow	C₇ ⋊ A₄×C₃₁
Sylow	C₂² ⋊ ΓL(1,𝔽₈)×C₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₃₁	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃×C₃₁
Frattini	C₁	G
Fitting, socle	C₂²×C₇×C₃₁	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₃₁

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 868

A minimal faithful permutation representation — 2 generators on 42 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)(33,34,35)(37,38,39)(40,42,41);(32,35)(33,34)(36,37,38,40,39,41,42)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²⋊ΓL(1,𝔽₈)×ΓL(1,𝔽₃₂), C₅×C₃₁×C₂²⋊ΓL(1,𝔽₈), C₃₁×C₂²⋊(C₄₉⋊C₃), …
maximal normal subgroup of	C₂²⋊ΓL(1,𝔽₈)×ΓL(1,𝔽₃₂), C₅×C₃₁×C₂²⋊ΓL(1,𝔽₈), C₇×C₃₁×C₂²⋊ΓL(1,𝔽₈), …
maximal quotient of	C₅×C₃₁×C₂²⋊ΓL(1,𝔽₈), C₃₁×C₂²⋊(C₄₉⋊C₃), C₃₁×C₂²⋊#3862, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2604, 55);

C31×C22⋊ΓL(1,𝔽8) #14579644