C₂×D₂₉×C₃₇⋊C₃ #27166154

order	12876
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₂₉×C₃₇⋊C₃
unambiguous semidirect product	C₂₉⋊(C₂²×C₃₇⋊C₃)
unambiguous semidirect product	(C₂×C₂₉)⋊(C₂×C₃₇⋊C₃)
unambiguous semidirect product	(C₂×C₃₇×D₂₉)⋊C₃

Splittings

Fitting, socle	C₂×C₂₉×C₃₇ ⋊ C₂×C₃
derived, Hall	C₂₉×C₃₇ ⋊ C₂²×C₃
Hall	C₂×C₃₇×D₂₉ ⋊ C₃
Hall	C₂₉×C₃₇⋊C₃ ⋊ C₂²
Sylow	C₃₇ ⋊ C₂×C₃×D₂₉
Sylow	C₂₉ ⋊ C₂²×C₃₇⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₂₉×C₃₇⋊C₃
commutator, cosocle, hypocenter	C₂₉×C₃₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₂₉×C₃₇	C₂×C₃
LCS	G ▷ C₂₉×C₃₇
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 2146

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₂₉×#66526, D₃×D₂₉×C₃₇⋊C₃, C₂×C₃×D₂₉×C₃₇⋊C₃, …
maximal normal subgroup of	C₂×D₂₉×#66526, C₂×C₃×D₂₉×C₃₇⋊C₃
maximal quotient of	C₂×D₂₉×#66524, D₃×D₂₉×C₃₇⋊C₃, C₂×C₃×D₂₉×C₃₇⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 23);

C2×D29×C37⋊C3 #27166154