C₃×D₁₁×D₉₇ #27165385

order	12804
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×D₁₁×D₉₇

Splittings

Fitting, socle, Hall	C₃×C₁₁×C₉₇ ⋊ C₂²
derived, Hall	C₁₁×C₉₇ ⋊ C₂²×C₃
Hall	C₃×C₉₇ ⋊ C₂×D₁₁
Sylow	C₉₇ ⋊ C₂×C₃×D₁₁
Hall	C₃×C₁₁ ⋊ C₂×D₉₇
Sylow	C₁₁ ⋊ C₂×C₃×D₉₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₁₁×D₉₇
commutator, cosocle, hypocenter	C₁₁×C₉₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₁×C₉₇	C₂²
LCS	G ▷ C₁₁×C₉₇
UCS	C₁ ◁ C₃

40 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:

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

Conjugacy classes & character table

1050 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₁×D₉₇⋊C₉, C₉×D₁₁×D₉₇, C₃×D₁₁×D₉₇⋊C₃, …
maximal normal subgroup of	D₁₁×D₉₇⋊C₉, C₉×D₁₁×D₉₇, C₃×D₁₁×D₉₇⋊C₃, …
maximal quotient of	C₉×D₁₁×D₉₇, C₃²×D₁₁×D₉₇, C₃×D₃₃×D₉₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12804, 22);

C3×D11×D97 #27165385