Dic₃×C₂₉×C₃₇ #27166137

order	12876
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₂₉×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	Dic₃×C₂₉×C₃₇
unambiguous semidirect product	C₃⋊(C₄×C₂₉×C₃₇)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₂₉×C₃₇	D₃
commutator, hypocenter	C₃	C₄×C₂₉×C₃₇
Frattini	C₂	D₃×C₂₉×C₃₇
Fitting, socle, prime radical	C₂×C₃×C₂₉×C₃₇	C₂
cosocle	C₂×C₃	C₂×C₂₉×C₃₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₂₉×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)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44)(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);(1,2,3)

Conjugacy classes & character table

6438 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×Dic₉×C₃₇, Dic₃×C₂₉×C₃₇⋊C₃, C₂₉×C₃⋊Dic₃×C₃₇, …
maximal normal subgroup of	Dic₃×C₂₉×C₃₇⋊C₃, C₃×Dic₃×C₂₉×C₃₇
maximal quotient of	C₂₉×Dic₉×C₃₇, C₂₉×C₃⋊Dic₃×C₃₇, C₃×Dic₃×C₂₉×C₃₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 6);

Dic3×C29×C37 #27166137