C₂×D₃×C₂₉×C₃₇ #27166180

order	12876
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₂₉×C₃₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₃×C₂₉×C₃₇
unambiguous semidirect product	C₃⋊(C₂²×C₂₉×C₃₇)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₂₉×C₃₇)

Splittings

Fitting, socle	C₂×C₃×C₂₉×C₃₇ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂²×C₂₉×C₃₇
Hall	C₃×C₂₉×C₃₇ ⋊ C₂²
Hall	C₃×C₃₇ ⋊ C₂²×C₂₉
Hall	C₃×C₂₉ ⋊ C₂²×C₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₂₉×C₃₇	D₃
commutator, cosocle, hypocenter	C₃	C₂²×C₂₉×C₃₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₂₉×C₃₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₂₉×C₃₇

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 6438

A minimal faithful permutation representation — 2 generators on 71 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);(1,2,3)(70,71)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₉×C₂₉×C₃₇, C₂×D₃×C₂₉×C₃₇⋊C₃, D₃²×C₂₉×C₃₇, …
maximal normal subgroup of	C₂×D₃×C₂₉×C₃₇⋊C₃, C₂×C₃×D₃×C₂₉×C₃₇
maximal quotient of	C₂×D₉×C₂₉×C₃₇, D₃²×C₂₉×C₃₇, C₂×C₃×D₃×C₂₉×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 49);

C2×D3×C29×C37 #27166180