C₂₉×Dic₃₇⋊C₃ #27166141

order	12876
isoclinism	isoclinic class of D₃₇⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₂₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂₉×Dic₃₇⋊C₃
unambiguous semidirect product	Dic₃₇⋊(C₃×C₂₉)
unambiguous semidirect product	(C₂₉×Dic₃₇)⋊C₃

Splittings

derived, Sylow	C₃₇ ⋊ C₃×C₄×C₂₉
Hall	C₂₉×Dic₃₇ ⋊ C₃
Hall	C₂₉×C₃₇⋊C₃ ⋊ C₄
Hall	C₂₉×C₃₇ ⋊ C₃×C₄
Hall	Dic₃₇ ⋊ C₃×C₂₉
Hall	C₃₇⋊C₃ ⋊ C₄×C₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₂₉	D₃₇⋊C₃
commutator, hypocenter	C₃₇	C₃×C₄×C₂₉
Frattini	C₂	C₂₉×D₃₇⋊C₃
Fitting, socle	C₂×C₂₉×C₃₇	C₂×C₃
prime radical	C₂×C₂₉×C₃₇⋊C₃	C₂
cosocle	C₂×C₃₇	C₂×C₃×C₂₉
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₂×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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#13050625, C₃×C₂₉×Dic₃₇⋊C₃, C₂₉×C₃⋊(Dic₃₇⋊C₃)
maximal normal subgroup of	C₂₉×#13050625, C₃×C₂₉×Dic₃₇⋊C₃
maximal quotient of	C₂₉×#13050607, C₃×C₂₉×Dic₃₇⋊C₃, C₂₉×C₃⋊(Dic₃₇⋊C₃)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 10);

C29×Dic37⋊C3 #27166141