#27302807

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#359 ⋊ C₂
Sylow	C₅₃ ⋊ #13307

Subgroups

	subgroup	G/H
center	C₂²	#16810758
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#359	C₂
socle	C₂²×C₅₃	#447
hypercenter	C₂×#359	D₅₃
hypocenter	C₅₃	#13307
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#359

552 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 97 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) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 89 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)(55,57)(58,60)(59,65)(61,70)(62,72)(63,73)(64,66)(67,83)(68,84)(69,79)(71,76)(74,81)(75,77)(78,87)(80,88)(82,85)(86,89);(54,55)(56,57)(58,60)(59,64)(61,75)(62,71)(63,73)(65,66)(67,80)(68,82)(69,74)(70,77)(72,76)(78,86)(79,81)(83,88)(84,85)(87,89);(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)(58,59)(60,69)(61,74)(62,79)(63,68)(64,70)(65,71)(66,77)(67,76)(72,84)(73,83)(75,80)(78,88)(81,86)(82,87)(85,89)

Polycyclic presentation:

2^8,53|d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,i^a=i^52,c^b=cf,d^b=dg,g^b=gh,e^c=eh,i^c=i^52

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, 88283);