#27301024

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1382 ⋊ C₂
Sylow	C₅₃ ⋊ #12990

Subgroups

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1382	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	#20284297	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1382	D₅₃
hypocenter	C₅₃	#12990
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1382

518 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfg,d^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,d^b=dg,f^b=fg,g^b=gh,i^b=i^52,i^c=i^52

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 86500);