C₂×C₇×D₇×C₆₇ #27190087

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

Aliases

direct product	C₂×C₇×D₇×C₆₇
unambiguous semidirect product	C₇⋊(C₂²×C₇×C₆₇)
unambiguous semidirect product	(C₂×C₇)⋊(C₂×C₇×C₆₇)

Splittings

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

Subgroups

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

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 1876

A minimal faithful permutation representation — 2 generators on 83 points:

(1,2,3,4,5,6,7)(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);(8,9,11,13,14,12,10)(82,83)

Conjugacy classes & character table

4690 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13132, 27);

C2×C7×D7×C67 #27190087