C₂×C₃×C₁₁×D₉₇ #27165403

order	12804
isoclinism	isoclinic class of D₉₇
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₁₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂×C₃×C₁₁×D₉₇

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₁	D₉₇
commutator, cosocle, hypocenter	C₉₇	C₂²×C₃×C₁₁
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₁×C₉₇	C₂
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₂×C₃×C₁₁

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 6402

A minimal faithful permutation representation — 2 generators on 113 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)(98,99,100,101,102,103,104,105,106,107,108)(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)(112,113)

Conjugacy classes & character table

3300 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₁₁×D₉₇⋊C₉, C₂×C₉×C₁₁×D₉₇, C₂×C₃×C₁₁×D₉₇⋊C₃, …
maximal normal subgroup of	C₂×C₁₁×D₉₇⋊C₉, C₂×C₉×C₁₁×D₉₇, C₂×C₃×C₁₁×D₉₇⋊C₃, …
maximal quotient of	C₂×C₉×C₁₁×D₉₇, C₃×D₃×C₁₁×D₉₇, C₂×C₃²×C₁₁×D₉₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12804, 40);

C2×C3×C11×D97 #27165403