C₃₇⋊C₃×Dic₂₉ #27166142

order	12876
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

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

Splittings

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

Subgroups

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

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 70
transitive degree	μ_t(G) = 4292

A minimal faithful permutation representation — 2 generators on 70 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)(31,32,34)(33,36,41)(35,39,38)(37,43,50)(40,45,47)(42,48,44)(46,53,57)(49,55,54)(51,58,52)(56,61,62)(59,63,60)(64,66,65)(67,70,69,68);(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)(30,31,33,37,43,51,59,64,63,48,34,38,44,52,53,36,42,49,56,58,41,47,54,60,61,45,32,35,40,46,50,57,62,65,66,55,39)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	Dic₂₉×#66526, C₃×C₃₇⋊C₃×Dic₂₉, C₃₇⋊C₃×Dic₈₇
maximal normal subgroup of	Dic₂₉×#66526, C₃×C₃₇⋊C₃×Dic₂₉
maximal quotient of	Dic₂₉×#66524, C₃×C₃₇⋊C₃×Dic₂₉, C₃₇⋊C₃×Dic₈₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 11);

C37⋊C3×Dic29 #27166142