ΓL(1,𝔽₈)⋊D₄₉₇ #35229457

order	20874
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	ΓL(1,𝔽₈)⋊D₄₉₇
unambiguous semidirect product	C₇₁⋊(ΓL(1,𝔽₈)⋊D₇)

Splittings

Fitting, socle, derived, Hall	C₇²×C₇₁ ⋊ C₂×C₃
Hall	C₇×ΓL(1,𝔽₈)×C₇₁ ⋊ C₂
Hall	C₇⋊D₄₉₇ ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ D₇₁
Sylow	C₇₁ ⋊ ΓL(1,𝔽₈)⋊D₇
Sylow	C₇² ⋊ C₃×D₇₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₇₁	C₂×C₃
LCS	G ▷ C₇²×C₇₁
UCS	C₁

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1491

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

Polycyclic presentation:

2,3,7^2,71|c^a=c^6,d^a=d^6,e^a=e^70,c^b=c^2

Conjugacy classes & character table

1247 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20874, 36);

ΓL(1,𝔽8)⋊D497 #35229457