C₂×#20272845 #27299084

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#20272845

Splittings

Sylow

C₅₃ ⋊ C₂×#1395

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133131
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×#41	C₂²×D₅₃
Fitting	C₂×C₅₃×#413	C₂
socle	C₂²×C₅₃	#342
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂×#41×C₅₃	C₂³
hypercenter	C₂×#413	D₅₃
hypocenter	C₅₃	C₂×#1395
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#413

766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(56,58,61,71,62,72,59,69)(57,63,66,79,67,80,64,68)(60,74,75,77,76,70,73,85)(65,78,82,86,83,87,81,84);(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)(56,57,60,65)(58,68,70,84)(59,66,73,82)(61,64,75,81)(62,67,76,83)(63,77,78,69)(71,80,85,87)(72,79,74,86);(54,55)

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84560);

C2×#20272845 #27299084