D₃₇×D₈₇ #27166169

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

Aliases

direct product	D₃₇×D₈₇
unambiguous semidirect product	(C₃×D₃₇)⋊D₂₉
unambiguous semidirect product	(C₂₉×D₃₇)⋊D₃

Splittings

Fitting, socle, derived, Hall	C₃×C₂₉×C₃₇ ⋊ C₂²
Hall	C₂₉×C₃₇ ⋊ C₂×D₃
Hall	C₃×C₃₇ ⋊ C₂×D₂₉
Hall	C₃×C₂₉ ⋊ C₂×D₃₇
Sylow	C₃₇ ⋊ C₂×D₈₇
Sylow	C₂₉ ⋊ D₃×D₃₇
Sylow	C₃ ⋊ D₂₉×D₃₇

Subgroups

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

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 3219

A minimal faithful permutation representation — 2 generators on 69 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)(31,32)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(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)(30,31,32)(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)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₃₇×D₂₆₁, D₈₇×D₃₇⋊C₃, C₃×D₃₇×D₈₇, …
maximal normal subgroup of	D₈₇×D₃₇⋊C₃, C₃×D₃₇×D₈₇
maximal quotient of	D₃₇×D₂₆₁, C₃×D₃₇×D₈₇, D₁₁₁⋊D₈₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 38);

D37×D87 #27166169