C₂×#20270134 #27214360

order	13542
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₂×#20270134

Splittings

Fitting, socle, Hall	C₂×C₃₇×C₆₁ ⋊ C₃
derived, Hall	C₃₇×C₆₁ ⋊ C₂×C₃
Hall	C₂×C₆₁ ⋊ C₃₇⋊C₃
Hall	C₂×C₃₇ ⋊ C₆₁⋊C₃
Sylow	C₆₁ ⋊ C₂×C₃₇⋊C₃
Sylow	C₃₇ ⋊ C₂×C₆₁⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#20270134
commutator, cosocle, hypocenter	C₃₇×C₆₁	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃₇×C₆₁	C₃
LCS	G ▷ C₃₇×C₆₁
UCS	C₁ ◁ C₂

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 100
transitive degree	μ_t(G) = 4514

A minimal faithful permutation representation — 2 generators on 100 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,25)(18,27,32)(21,30,29)(23,33,26)(28,38,36)(31,41,43)(34,44,40)(35,39,37)(42,49,47)(45,52,51)(46,50,48)(53,57,55)(54,58,56)(59,61,60)(63,64,66)(65,68,73)(67,71,70)(69,75,82)(72,77,79)(74,80,76)(78,85,89)(81,87,86)(83,90,84)(88,93,94)(91,95,92)(96,98,97)(99,100);(1,2,4,8,15,25,36,47,55,60,57,41,17,3,6,11,18,28,39,22,32,43,51,56,49,27,7,13,21,31,42,50,33,12,19,29,40,48,38,14,23,34,45,53,58,44,20,5,9,16,26,37,24,35,46,54,59,61,52,30,10)(62,63,65,69,75,83,91,96,95,80,66,70,76,84,85,68,74,81,88,90,73,79,86,92,93,77,64,67,72,78,82,89,94,97,98,87,71)

Conjugacy classes & character table

1510 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄×#20270134, #41028917, #41028921, …
maximal normal subgroup of	C₄×#20270134, #41028917, #41028921, …
maximal quotient of	C₄×#20270134, C₂²×#20270134

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13542, 5);

C2×#20270134 #27214360