C₂×C₂₉⋊(D₃₇⋊C₃) #27166153

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

Aliases

direct product	C₂×C₂₉⋊(D₃₇⋊C₃)
unambiguous semidirect product	(C₂×C₂₉)⋊(D₃₇⋊C₃)
unambiguous semidirect product	(C₂×C₃₇⋊C₃)⋊D₂₉
unambiguous semidirect product	(C₂×D₁₀₇₃)⋊C₃

Splittings

Fitting, socle	C₂×C₂₉×C₃₇ ⋊ C₂×C₃
derived, Hall	C₂₉×C₃₇ ⋊ C₂²×C₃
Hall	C₂×D₁₀₇₃ ⋊ C₃
Hall	C₂₉×C₃₇⋊C₃ ⋊ C₂²
Hall	C₃₇⋊C₃ ⋊ C₂×D₂₉
Sylow	C₃₇ ⋊ C₂×C₃×D₂₉
Sylow	C₂₉ ⋊ C₂×D₃₇⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₂₉⋊(D₃₇⋊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:

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

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

Relationships

maximal subgroup of	C₂×C₂₉⋊(#66526⋊C₂), D₃×C₂₉⋊(D₃₇⋊C₃), C₂×C₃×C₂₉⋊(D₃₇⋊C₃), …
maximal normal subgroup of	C₂×C₂₉⋊(#66526⋊C₂), C₂×C₃×C₂₉⋊(D₃₇⋊C₃)
maximal quotient of	C₂×C₂₉⋊(#66524⋊C₂), D₃×C₂₉⋊(D₃₇⋊C₃), C₂×C₃×C₂₉⋊(D₃₇⋊C₃), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 22);

C2×C29⋊(D37⋊C3) #27166153