C₂×C₃₇×D₈₇ #27166178

order	12876
isoclinism	isoclinic class of D₈₇
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₃₇×D₈₇
unambiguous semidirect product	(C₂×C₃)⋊(C₃₇×D₂₉)
unambiguous semidirect product	(C₂×C₂₉)⋊(D₃×C₃₇)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃₇	D₈₇
commutator, cosocle, hypocenter	C₃×C₂₉	C₂²×C₃₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₂₉×C₃₇	C₂
LCS	G ▷ C₃×C₂₉
UCS	C₁ ◁ C₂×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)(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)(30,31,32)(70,71)

Conjugacy classes & character table

3330 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₃₇×D₂₆₁, C₂×C₃₇⋊C₃×D₈₇, D₃×C₃₇×D₈₇, …
maximal normal subgroup of	C₂×C₃₇⋊C₃×D₈₇, C₂×C₃×C₃₇×D₈₇
maximal quotient of	C₂×C₃₇×D₂₆₁, D₃×C₃₇×D₈₇, C₂×C₃×C₃₇×D₈₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 47);

C2×C37×D87 #27166178