ΓL(1,𝔽₈)⋊D₅₈₁ #37567378

order	24402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	ΓL(1,𝔽₈)⋊D₅₈₁
unambiguous semidirect product	C₈₃⋊(ΓL(1,𝔽₈)⋊D₇)

Splittings

Fitting, socle, derived, Hall	C₇²×C₈₃ ⋊ C₂×C₃
Hall	C₇×ΓL(1,𝔽₈)×C₈₃ ⋊ C₂
Hall	C₇⋊D₅₈₁ ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ D₈₃
Sylow	C₈₃ ⋊ ΓL(1,𝔽₈)⋊D₇
Sylow	C₇² ⋊ C₃×D₈₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₈₃	C₂×C₃
LCS	G ▷ C₇²×C₈₃
UCS	C₁

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 1743

A minimal faithful permutation representation — 2 generators on 97 points:

(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)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(85,88,89,86,87,90)(92,93)(94,95)(96,97);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(84,85,87,90,89,88,86)(91,92,94,96,97,95,93)

Polycyclic presentation:

2,3,7^2,83|c^a=c^6,d^a=d^6,e^a=e^82,c^b=c^2

Conjugacy classes & character table

1457 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24402, 20);

ΓL(1,𝔽8)⋊D581 #37567378