C₂×#20276935 #27538983

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

Aliases

direct product

C₂×#20276935

Splittings

Sylow

C₅₃ ⋊ C₂×#2239

Subgroups

	subgroup	G/H
center	C₂²	#16810942
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#349	C₂
socle	C₂²×C₅₃	#481
prime radical	C₂⁴×C₅₃	SD₁₆
hypercenter	C₂×#349	D₅₃
hypocenter	C₅₃	C₂×#2239
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ C₂×#349

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

normal subgroups only

This group has 552 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) = 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)(56,58,60,76,61,67,73,69)(57,62,64,83,65,78,75,79)(59,72,74,77,71,86,80,68)(63,82,70,84,81,85,87,66);(59,71)(63,81)(66,84)(68,77)(70,87)(72,86)(74,80)(82,85);(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)(56,57,61,65)(58,66,67,84)(59,70,71,87)(60,75,73,64)(62,77,78,68)(63,80,81,74)(69,82,76,85)(72,79,86,83);(54,55)

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324459);