C₂×C₁₁×C₆₇⋊C₉ #27210689

order	13266
automorphism	?
homology	Abelianization (H₁) = C₂×C₉×C₁₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×C₁₁×C₆₇⋊C₉
unambiguous semidirect product	(C₂×C₁₁×C₆₇)⋊C₉

Splittings

derived, Sylow	C₆₇ ⋊ C₂×C₉×C₁₁
Hall	C₂×C₁₁×C₆₇ ⋊ C₉
Hall	C₁₁×C₆₇ ⋊ C₂×C₉
Hall	C₂×C₆₇ ⋊ C₉×C₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₁	C₆₇⋊C₃
commutator, hypocenter	C₆₇	C₂×C₉×C₁₁
Frattini	C₃	C₂×C₁₁×C₆₇⋊C₃
Fitting, socle, prime radical	C₂×C₃×C₁₁×C₆₇	C₃
cosocle	C₃×C₆₇	C₂×C₃×C₁₁
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₂×C₃×C₁₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 13266

A minimal faithful permutation representation — 2 generators on 89 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,43)(28,39,47)(31,42,44)(34,45,41)(36,48,38)(40,52,54)(46,56,55)(49,58,53)(50,57,51)(59,63,62)(60,64,61)(65,67,66)(68,75,74,70,69,76,73,72,71)(77,78,79,80,81,82,83,84,85,86,87)(88,89);(1,2,4,8,15,25,27,7,13,21,31,43,54,52,24,36,49,59,64,48,22,32,44,55,62,66,63,45,20,5,9,16,26,38,51,42,17,3,6,11,18,28,40,39,14,23,34,46,57,35,47,37,50,60,65,67,58,33,12,19,29,41,53,61,56,30,10)

Conjugacy classes & character table

1650 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄×C₁₁×C₆₇⋊C₉, C₁₁×Dic₆₇⋊C₉, Dic₁₁×C₆₇⋊C₉, …
maximal normal subgroup of	C₄×C₁₁×C₆₇⋊C₉, C₁₁×Dic₆₇⋊C₉, Dic₁₁×C₆₇⋊C₉, …
maximal quotient of	C₄×C₁₁×C₆₇⋊C₉, C₂²×C₁₁×C₆₇⋊C₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13266, 3);

C2×C11×C67⋊C9 #27210689