C₃×C₁₁×Dic₉₇ #27165366

order	12804
isoclinism	isoclinic class of D₉₇
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₁₁ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×C₁₁×Dic₉₇

Splittings

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

Subgroups

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

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 12804

A minimal faithful permutation representation — 2 generators on 115 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)(102,103,104,105,106,107,108,109,110,111,112)(113,114,115);(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)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₉×C₁₁×Dic₉₇, C₁₁×Dic₉₇⋊C₉, C₃×C₁₁×Dic₉₇⋊C₃, …
maximal normal subgroup of	C₉×C₁₁×Dic₉₇, C₁₁×Dic₉₇⋊C₉, C₃×C₁₁×Dic₉₇⋊C₃, …
maximal quotient of	C₉×C₁₁×Dic₉₇, C₃×C₁₁×Dic₂₉₁, C₃²×C₁₁×Dic₉₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12804, 3);

C3×C11×Dic97 #27165366