Dic₁₁×C₉₇⋊C₃ #27165374

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

Aliases

direct product	Dic₁₁×C₉₇⋊C₃
unambiguous semidirect product	C₁₁⋊(C₄×C₉₇⋊C₃)
unambiguous semidirect product	(Dic₁₁×C₉₇)⋊C₃

Splittings

derived, Hall	C₁₁×C₉₇ ⋊ C₃×C₄
Hall	Dic₁₁×C₉₇ ⋊ C₃
Hall	C₁₁×C₉₇⋊C₃ ⋊ C₄
Sylow	C₉₇ ⋊ C₃×Dic₁₁
Sylow	C₁₁ ⋊ C₄×C₉₇⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₁×C₉₇⋊C₃
commutator, hypocenter	C₁₁×C₉₇	C₃×C₄
Fitting, socle, cosocle	C₂×C₁₁×C₉₇	C₂×C₃
prime radical	C₂×C₁₁×C₉₇⋊C₃	C₂
LCS	G ▷ C₁₁×C₉₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 112
transitive degree	μ_t(G) = 4268

A minimal faithful permutation representation — 2 generators on 112 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)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,59)(41,55,66)(44,58,63)(47,61,60)(50,64,57)(52,67,54)(56,71,73)(62,75,77)(65,78,74)(68,80,72)(69,76,70)(79,86,85)(81,88,84)(82,87,83)(89,93,92)(90,94,91)(95,97,96)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110,111,112);(1,2,4,8,15,25,38,40,14,23,34,47,62,76,51,66,53,69,82,90,95,97,88,64,33,12,19,29,42,57,72,83,75,43,17,3,6,11,18,28,41,56,55,24,36,50,65,79,87,67,35,48,63,77,85,92,96,93,78,46,20,5,9,16,26,39,54,70,58,27,7,13,21,31,44,59,73,71,37,52,68,81,89,94,80,49,22,32,45,60,74,84,91,86,61,30,10)(98,99,101,103,105,107,108,106,104,102,100)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×Dic₁₁×C₉₇⋊C₃, Dic₃₃×C₉₇⋊C₃
maximal normal subgroup of	C₃×Dic₁₁×C₉₇⋊C₃
maximal quotient of	Dic₁₁×C₉₇⋊C₉, C₃×Dic₁₁×C₉₇⋊C₃, Dic₃₃×C₉₇⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12804, 11);

Dic11×C97⋊C3 #27165374