C₃₇×#66524 #27134459

order	12321
isoclinism	isoclinic class of C₃₇⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₉×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃₇×#66524

Splittings

derived	C₃₇ ⋊ C₉×C₃₇
Sylow	C₃₇² ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₃₇	C₃₇⋊C₃
commutator, hypocenter	C₃₇	C₉×C₃₇
Frattini	C₃	C₃₇×C₃₇⋊C₃
Fitting, socle, prime radical	C₃×C₃₇²	C₃
cosocle	C₃×C₃₇	C₃×C₃₇
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₃×C₃₇

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 333

A minimal faithful permutation representation — 2 generators on 83 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)(39,40,42)(41,44,49)(43,47,46)(45,51,58)(48,53,55)(50,56,52)(54,61,65)(57,63,62)(59,66,60)(64,69,70)(67,71,68)(72,74,73)(75,76,78,77,79,81,80,82,83);(38,39,41,45,51,59,67,72,71,56,42,46,52,60,61,44,50,57,64,66,49,55,62,68,69,53,40,43,48,54,58,65,70,73,74,63,47)

Conjugacy classes & character table

1665 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#66524⋊C₂, #66524⋊D₃₇, D₃₇×#66524, …
maximal normal subgroup of	C₃₇×#66524⋊C₂, #66524⋊D₃₇, D₃₇×#66524, …
maximal quotient of	C₂×C₃₇×#66524

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12321, 7);

C37×#66524 #27134459