C₂×#20274044 #27365283

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#20274044

Splittings

Sylow

C₅₃ ⋊ C₂×#1663

Subgroups

	subgroup	G/H
center	C₂³	#13133187
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#330	C₂
socle, prime radical	C₂³×C₅₃	#120
hypercenter	C₂×#330	D₅₃
hypocenter	C₅₃	C₂×#1663
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#330

830 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 13568

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

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150759);