C₃×Dic₁₁×C₉₇ #27165367

order	12804
isoclinism	isoclinic class of D₁₁
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₉₇ · Schur multiplier (H₂) = C₁

Aliases

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

Splittings

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

Subgroups

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

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)(12,15,14,13)(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);(1,2,4,6,8,10,11,9,7,5,3)

Conjugacy classes & character table

4074 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₉×Dic₁₁×C₉₇, Dic₁₁×C₉₇⋊C₉, C₃×Dic₁₁×C₉₇⋊C₃, …
maximal normal subgroup of	C₉×Dic₁₁×C₉₇, Dic₁₁×C₉₇⋊C₉, C₃×Dic₁₁×C₉₇⋊C₃, …
maximal quotient of	C₉×Dic₁₁×C₉₇, C₃×C₉₇×Dic₃₃, C₃²×Dic₁₁×C₉₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12804, 4);

C3×Dic11×C97 #27165367