C₂×#20273342 #27352113

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#20273342

Splittings

Sylow

C₅₃ ⋊ C₂×#1534

Subgroups

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂²×#125×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂²×#125	D₅₃
hypocenter	C₅₃	C₂×#1534
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×#125

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 87 points:

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

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 137589);

C2×#20273342 #27352113