C₂×C₆₇×C₇⋊D₇ #27190085

order	13132
isoclinism	isoclinic class of C₇⋊D₇
automorphism	?
homology	Abelianization (H₁) = C₂²×C₆₇ · Schur multiplier (H₂) = C₂×C₇

Aliases

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

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₆₇	C₇⋊D₇
commutator, cosocle, hypocenter	C₇²	C₂²×C₆₇
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₆₇	C₂
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂×C₆₇

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:

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

Conjugacy classes & character table

3484 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13132, 25);

C2×C67×C7⋊D7 #27190085