D₇²×C₆₇ #27190079

order	13132
automorphism	?
homology	Abelianization (H₁) = C₂²×C₆₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₇²×C₆₇
unambiguous semidirect product	(C₇×D₇)⋊(C₂×C₆₇)
unambiguous semidirect product	(C₇⋊D₇)⋊(C₂×C₆₇)

Splittings

Fitting, socle, Hall	C₇²×C₆₇ ⋊ C₂²
derived, Sylow	C₇² ⋊ C₂²×C₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₆₇	D₇²
commutator, cosocle, hypocenter	C₇²	C₂²×C₆₇
Frattini	C₁	G
Fitting, socle	C₇²×C₆₇	C₂²
LCS	G ▷ C₇²
UCS	C₁ ◁ C₆₇

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:

(1,2,4,6,7,5,3)(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);(2,3)(4,5)(6,7)(8,9,11,13,14,12,10)

Conjugacy classes & character table

1675 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13132, 19);

D72×C67 #27190079