#27539599

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2253 ⋊ C₂
Sylow	C₅₃ ⋊ #33729

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2253	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₅₃	C₂×D₈
hypercenter	#2253	D₅₃
hypocenter	C₅₃	#33729
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 93 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)(63,69)(68,80)(70,82)(71,83)(78,89)(79,92)(81,90)(91,93);(62,64,73,85,66,74,77,86,67,75,87,88,76,84,65,72)(63,68,79,91,71,81,82,89,69,80,92,93,83,90,70,78);(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)(54,56,57,61)(55,58,59,60)(62,63)(64,68)(65,70)(66,71)(67,69)(72,78)(73,79)(74,81)(75,80)(76,83)(77,82)(84,90)(85,91)(86,89)(87,92)(88,93);(54,55,57,59)(56,60,61,58)(64,72)(65,73)(66,76)(68,78)(70,79)(71,83)(74,88)(75,86)(77,87)(80,89)(81,93)(82,92)(84,85)(90,91)

Polycyclic presentation:

2^8,53|a^2=e,b^2=e,c^2=fg,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,i^b=i^52,i^d=i^52

Conjugacy classes & character table

1040 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 325075);