C₂×C₉×C₆₇⋊C₁₁ #27210688

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

Aliases

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

Splittings

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

Subgroups

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

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 78
transitive degree	μ_t(G) = 1206

A minimal faithful permutation representation — 2 generators on 78 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄×C₉×C₆₇⋊C₁₁, C₉×Dic₆₇⋊C₁₁, Dic₉×C₆₇⋊C₁₁, …
maximal normal subgroup of	C₄×C₉×C₆₇⋊C₁₁, C₉×Dic₆₇⋊C₁₁, Dic₉×C₆₇⋊C₁₁, …
maximal quotient of	C₄×C₉×C₆₇⋊C₁₁, C₂²×C₉×C₆₇⋊C₁₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13266, 2);

C2×C9×C67⋊C11 #27210688