C₇²×Dic₆₇ #27190071

order	13132
isoclinism	isoclinic class of D₆₇
automorphism	?
homology	Abelianization (H₁) = C₄×C₇² · Schur multiplier (H₂) = C₇

Aliases

direct product	C₇²×Dic₆₇
unambiguous semidirect product	C₆₇⋊(C₄×C₇²)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇²	D₆₇
commutator, hypocenter	C₆₇	C₄×C₇²
Frattini	C₂	C₇²×D₆₇
Fitting, socle, prime radical	C₂×C₇²×C₆₇	C₂
cosocle	C₂×C₆₇	C₂×C₇²
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₂×C₇²

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 — 2 generators on 85 points:

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

Conjugacy classes & character table

3430 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13132, 11);

C72×Dic67 #27190071