C₂⁷×D₆₇ #31769735

order	17152
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁸

Aliases

direct product	C₂⁷×D₆₇
unambiguous semidirect product	C₆₇⋊C₂⁸
unambiguous semidirect product	(C₂×C₆₇)⋊C₂⁷

Splittings

Fitting, socle	C₂⁷×C₆₇ ⋊ C₂
derived, Sylow	C₆₇ ⋊ C₂⁸

Subgroups

	subgroup	G/H
center, hypercenter	C₂⁷	D₆₇
commutator, cosocle, hypocenter	C₆₇	C₂⁸
Frattini	C₁	G
Fitting, socle	C₂⁷×C₆₇	C₂
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₂⁷

834398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 8576

A minimal faithful permutation representation — 8 generators on 81 points:

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

Conjugacy classes & character table

4480 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 1083472);

C27×D67 #31769735