C₇×D₇×D₆₇ #27190077

order	13132
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₇×D₇×D₆₇
unambiguous semidirect product	(D₇×C₆₇)⋊(C₂×C₇)
unambiguous semidirect product	(C₇×D₆₇)⋊(C₂×C₇)
unambiguous semidirect product	D₄₆₉⋊(C₂×C₇)

Splittings

Fitting, socle, Hall	C₇²×C₆₇ ⋊ C₂²
derived	C₇×C₆₇ ⋊ C₂²×C₇
Sylow	C₆₇ ⋊ C₂×C₇×D₇
Sylow	C₇² ⋊ C₂×D₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₇×D₆₇
commutator, cosocle, hypocenter	C₇×C₆₇	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₇²×C₆₇	C₂²
LCS	G ▷ C₇×C₆₇
UCS	C₁ ◁ C₇

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 938

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

Conjugacy classes & character table

1225 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13132, 17);

C7×D7×D67 #27190077