C₂×C₇⋊D₄₆₉ #27190084

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

Aliases

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

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₇⋊D₄₆₉
commutator, cosocle, hypocenter	C₇²×C₆₇	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₆₇	C₂
LCS	G ▷ C₇²×C₆₇
UCS	C₁ ◁ C₂

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 6566

A minimal faithful permutation representation — 3 generators on 83 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)(69,70)(71,72)(73,74)(76,77)(78,79)(80,81);(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)(75,76,78,80,81,79,77)(82,83);(68,69,71,73,74,72,70)

Conjugacy classes & character table

3286 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13132, 24);

C2×C7⋊D469 #27190084