C₄×C₂₉×C₃₇⋊C₃ #27166133

order	12876
isoclinism	isoclinic class of C₃₇⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₂₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₄×C₂₉×C₃₇⋊C₃
unambiguous semidirect product	(C₄×C₃₇)⋊(C₃×C₂₉)
unambiguous semidirect product	(C₄×C₂₉×C₃₇)⋊C₃

Splittings

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

Subgroups

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

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:

(2,3,5)(4,7,12)(6,10,9)(8,14,21)(11,16,18)(13,19,15)(17,24,28)(20,26,25)(22,29,23)(27,32,33)(30,34,31)(35,37,36)(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);(1,2,4,8,14,22,30,35,34,19,5,9,15,23,24,7,13,20,27,29,12,18,25,31,32,16,3,6,11,17,21,28,33,36,37,26,10)

Conjugacy classes & character table

1740 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄×C₂₉×#66526, Dic₃×C₂₉×C₃₇⋊C₃, C₃×C₄×C₂₉×C₃₇⋊C₃
maximal normal subgroup of	C₄×C₂₉×#66526, C₃×C₄×C₂₉×C₃₇⋊C₃
maximal quotient of	C₄×C₂₉×#66524, Dic₃×C₂₉×C₃₇⋊C₃, C₃×C₄×C₂₉×C₃₇⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 2);

C4×C29×C37⋊C3 #27166133