C₃₇×C₃⋊#1181 #27166164

order	12876
isoclinism	isoclinic class of C₃⋊#1181
automorphism	?
homology	Abelianization (H₁) = C₄×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃₇×C₃⋊#1181
unambiguous semidirect product	C₃⋊(C₃₇×#1181)

Splittings

Fitting, socle, Hall	C₃×C₂₉×C₃₇ ⋊ C₄
derived, Hall	C₃×C₂₉ ⋊ C₄×C₃₇
Hall	C₂₉×C₃₇ ⋊ Dic₃
Hall	C₃×C₃₇ ⋊ #1181
Sylow	C₂₉ ⋊ Dic₃×C₃₇
Sylow	C₃ ⋊ C₃₇×#1181

Subgroups

	subgroup	G/H
center, hypercenter	C₃₇	C₃⋊#1181
commutator, hypocenter	C₃×C₂₉	C₄×C₃₇
Frattini	C₁	G
Fitting, socle	C₃×C₂₉×C₃₇	C₄
prime radical	C₃×C₃₇×D₂₉	C₂
cosocle	C₃×D₂₉	C₂×C₃₇
LCS	G ▷ C₃×C₂₉
UCS	C₁ ◁ C₃₇

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 3219

A minimal faithful permutation representation — 2 generators on 69 points:

(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,26)(14,19,21,23)(16,24,25,20)(22,27,28,29)(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);(1,2,5,10,19,8,16,26,29,28,21,13,6,11,20,24,12,3,7,14,22,27,17,25,15,23,18,9,4)(30,31,32)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×C₉⋊#1181, C₃₇⋊C₃×C₃⋊#1181, C₃×C₃₇×C₃⋊#1181, …
maximal normal subgroup of	C₃₇⋊C₃×C₃⋊#1181, C₃×C₃₇×C₃⋊#1181
maximal quotient of	C₃₇×C₉⋊#1181, C₃×C₃₇×C₃⋊#1181, C₃₇×C₃⋊(C₃⋊#1181)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 33);

C37×C3⋊#1181 #27166164