C₂×#20276343 #27536794

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#20276343

Splittings

Sylow

C₅₃ ⋊ C₂×#2117

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×#131×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂²×#41×C₅₃	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×#131	D₅₃
hypocenter	C₅₃	C₂×#2117
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×#131

1154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3392

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)(57,64)(63,78)(65,67)(66,80)(68,81)(79,83)(82,84)(85,87);(56,58)(57,63)(59,72)(60,74)(61,69)(62,71)(64,78)(65,81)(66,83)(67,68)(70,76)(73,86)(75,77)(79,80)(82,85)(84,87);(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)(56,57,60,66,61,67,75,84,62,64,76,80,59,65,73,82)(58,68,70,85,69,78,86,83,71,81,74,87,72,63,77,79);(54,55)

Conjugacy classes & character table

2168 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322270);

C2×#20276343 #27536794