C₆₇×C₇⋊Dic₇ #27190074

order	13132
isoclinism	isoclinic class of C₇⋊D₇
automorphism	?
homology	Abelianization (H₁) = C₄×C₆₇ · Schur multiplier (H₂) = C₇

Aliases

direct product	C₆₇×C₇⋊Dic₇
unambiguous semidirect product	C₇⋊(Dic₇×C₆₇)

Splittings

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

Subgroups

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

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 13132

A minimal faithful permutation representation — 3 generators on 85 points:

(1,4,3,2)(6,7)(8,9)(10,11)(13,14)(15,16)(17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(12,13,15,17,18,16,14);(5,6,8,10,11,9,7)

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, 14);

C67×C7⋊Dic7 #27190074