C₆₇×#5997 #27190082

order	13132
isoclinism	isoclinic class of #5997
automorphism	?
homology	Abelianization (H₁) = C₄×C₆₇ · Schur multiplier (H₂) = C₇

Aliases

direct product

C₆₇×#5997

Splittings

Fitting, socle, Hall	C₇²×C₆₇ ⋊ C₄
derived, Sylow	C₇² ⋊ C₄×C₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₆₇	#5997
commutator, hypocenter	C₇²	C₄×C₆₇
Frattini	C₁	G
Fitting, socle	C₇²×C₆₇	C₄
prime radical	C₆₇×C₇⋊D₇	C₂
cosocle	C₇⋊D₇	C₂×C₆₇
LCS	G ▷ C₇²
UCS	C₁ ◁ C₆₇

28 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)(3,10,6,5)(4,13,9,8)(7,14,12,11)(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);(1,3,7,13,8,12,6)(2,9,5,14,11,10,4)

Conjugacy classes & character table

1072 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13132, 22);

C67×#5997 #27190082