#27437818

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1882 ⋊ C₂
Sylow	C₅₃ ⋊ #25619

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1882	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288632	C₂
hypercenter	#1882	D₅₃
hypocenter	C₅₃	#25619
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1882

1052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

2^8,53|c^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,i^b=i^52,d^c=dh,g^c=gh,i^d=i^52

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 223294);