C₃₇×Dic₈₇ #27166139

order	12876
isoclinism	isoclinic class of D₈₇
automorphism	?
homology	Abelianization (H₁) = C₄×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃₇×Dic₈₇
unambiguous semidirect product	C₃⋊(C₃₇×Dic₂₉)

Splittings

derived, Hall	C₃×C₂₉ ⋊ C₄×C₃₇
Hall	C₃×C₂₉×C₃₇ ⋊ C₄
Hall	C₂₉×C₃₇ ⋊ Dic₃
Hall	C₃×C₃₇ ⋊ Dic₂₉
Sylow	C₂₉ ⋊ Dic₃×C₃₇
Sylow	C₃ ⋊ C₃₇×Dic₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃₇	D₈₇
commutator, hypocenter	C₃×C₂₉	C₄×C₃₇
Frattini	C₂	C₃₇×D₈₇
Fitting, socle, prime radical	C₂×C₃×C₂₉×C₃₇	C₂
cosocle	C₂×C₃×C₂₉	C₂×C₃₇
LCS	G ▷ C₃×C₂₉
UCS	C₁ ◁ C₂×C₃₇

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 12876

A minimal faithful permutation representation — 2 generators on 73 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)(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);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(30,31,32)

Conjugacy classes & character table

3330 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×Dic₂₆₁, C₃₇⋊C₃×Dic₈₇, C₃₇×C₃⋊Dic₈₇, …
maximal normal subgroup of	C₃₇⋊C₃×Dic₈₇, C₃×C₃₇×Dic₈₇
maximal quotient of	C₃₇×Dic₂₆₁, C₃₇×C₃⋊Dic₈₇, C₃×C₃₇×Dic₈₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 8);

C37×Dic87 #27166139