#27540819

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2216 ⋊ C₂
Sylow	C₅₃ ⋊ D₄×Q₃₂

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2216	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×C₂²⋊D₅₃	D₈
hypercenter	#2216	D₅₃
hypocenter	C₅₃	D₄×Q₃₂
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 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)(58,61,64,76)(59,66,69,85)(60,71,74,65)(62,78,80,68)(63,82,79,67)(70,77,75,89)(72,88,83,86)(73,84,81,87);(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)(58,60,64,74)(59,65,69,71)(61,75,76,70)(62,77,80,89)(63,81,79,73)(66,72,85,83)(67,86,82,88)(68,87,78,84);(54,55)(56,57)(58,59,64,69)(60,70,74,75)(61,62,76,80)(63,78,79,68)(65,83,71,72)(66,67,85,82)(73,88,81,86)(77,87,89,84)

Polycyclic presentation:

2^8,53|a^2=h,b^2=h,d^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dgh,e^a=eg,g^a=gh,d^b=de,e^b=eg,g^b=gh,i^b=i^52,e^d=eg,g^d=gh,i^d=i^52

Conjugacy classes & character table

1043 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326295);