C₂×#20276463 #27537052

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#20276463

Splittings

Fitting	C₂×C₅₃×#362 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2148

Subgroups

	subgroup	G/H
center	C₂²	#102×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#362	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂²×#13133240	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#362	D₅₃
hypocenter	C₅₃	C₂×#2148
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322528);

C2×#20276463 #27537052