C₃×C₁₁⋊#87604 #27165384

order	12804
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₁₁⋊#87604
unambiguous semidirect product	C₁₁⋊(C₃×#87604)

Splittings

Fitting, socle, Hall	C₃×C₁₁×C₉₇ ⋊ C₄
derived, Hall	C₁₁×C₉₇ ⋊ C₃×C₄
Hall	C₃×C₉₇ ⋊ Dic₁₁
Sylow	C₉₇ ⋊ C₃×Dic₁₁
Hall	C₃×C₁₁ ⋊ #87604
Sylow	C₁₁ ⋊ C₃×#87604

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₁₁⋊#87604
commutator, hypocenter	C₁₁×C₉₇	C₃×C₄
Frattini	C₁	G
Fitting, socle	C₃×C₁₁×C₉₇	C₄
prime radical	C₃×C₁₁×D₉₇	C₂
cosocle	C₁₁×D₉₇	C₂×C₃
LCS	G ▷ C₁₁×C₉₇
UCS	C₁ ◁ C₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 3201

A minimal faithful permutation representation — 2 generators on 111 points:

(2,6,4,3)(5,15,9,8)(7,11,13,12)(10,28,18,17)(14,24,22,21)(16,20,26,25)(19,45,31,30)(23,41,35,34)(27,37,39,38)(29,33,43,42)(32,66,48,47)(36,62,52,51)(40,58,56,55)(44,54,60,59)(46,50,64,63)(49,81,69,68)(53,86,73,72)(57,82,76,70)(61,78,80,79)(65,75,84,83)(67,71,85,87)(74,93,89,88)(77,94,91,90)(92,97,96,95)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110,111);(1,2,5,10,19,32,49,70,51,30,46,65,85,66,86,94,89,73,52,35,22,13,6,11,20,33,50,71,79,55,34,17,29,44,61,81,93,97,96,91,76,56,39,26,15,24,37,54,75,83,59,38,21,8,16,27,40,57,77,92,95,88,68,80,60,43,28,41,58,78,87,63,42,25,12,3,7,14,23,36,53,74,90,72,47,67,84,64,45,62,82,69,48,31,18,9,4)(98,99,101,103,105,107,108,106,104,102,100)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₁⋊(#87604⋊C₉), C₉×C₁₁⋊#87604, C₃×C₁₁⋊(#87604⋊C₃), …
maximal normal subgroup of	C₁₁⋊(#87604⋊C₉), C₉×C₁₁⋊#87604, C₃×C₁₁⋊(#87604⋊C₃), …
maximal quotient of	C₉×C₁₁⋊#87604, C₃²×C₁₁⋊#87604, C₃×C₃⋊(C₁₁⋊#87604)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12804, 21);

C3×C11⋊#87604 #27165384