#27299329

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12641

Subgroups

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2138	C₂
socle	C₂²×C₅₃	C₄×D₈
prime radical	C₂×#135×C₅₃	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2138	D₅₃
hypocenter	C₅₃	#12641
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2138

438 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)(58,61,62,77,65,78,64,63)(59,67,69,84,70,85,79,68)(60,72,73,80,76,88,75,74)(66,81,82,87,83,89,86,71);(54,55,56,57)(58,60,63,74,64,75,78,88,65,76,77,80,62,73,61,72)(59,66,67,81,69,82,84,87,70,83,85,89,79,86,68,71);(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)(58,59,62,69,65,70,64,79)(60,71,73,81,76,87,75,89)(61,67,77,84,78,85,63,68)(66,80,82,88,83,74,86,72)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=e,d^2=gh,e^2=h,f^2=g,g^2=h,b^a=bd,d^a=dg,f^a=fgh,g^a=gh,i^a=i^52,i^c=i^52

Conjugacy classes & character table

1232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84805);