C₂×C₃×C₃₇×D₂₉ #27166182

order	12876
isoclinism	isoclinic class of D₂₉
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₃₇ · Schur multiplier (H₂) = C₂

Aliases

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

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₃₇	D₂₉
commutator, cosocle, hypocenter	C₂₉	C₂²×C₃×C₃₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₂₉×C₃₇	C₂
LCS	G ▷ C₂₉
UCS	C₁ ◁ C₂×C₃×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);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(70,71)

Conjugacy classes & character table

3552 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₂₉×#66524, C₂×C₉×C₃₇×D₂₉, C₂×C₃×D₂₉×C₃₇⋊C₃, …
maximal normal subgroup of	C₂×D₂₉×#66524, C₂×C₉×C₃₇×D₂₉, C₂×C₃×D₂₉×C₃₇⋊C₃, …
maximal quotient of	C₂×C₉×C₃₇×D₂₉, C₃×D₃×C₃₇×D₂₉, C₂×C₃²×C₃₇×D₂₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 51);

C2×C3×C37×D29 #27166182