#27425607

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1952 ⋊ C₂
Sylow	C₅₃ ⋊ #24725

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1952	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂×#11719945	C₂³
hypercenter	#1952	D₅₃
hypocenter	C₅₃	#24725
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1952

658 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)(54,56,57,61)(55,59,58,60)(63,71)(69,74)(70,87)(72,73)(78,88)(84,90)(85,89)(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)(62,65,66,80)(63,70,71,87)(64,75,76,92)(67,82,68,81)(69,84,74,90)(72,88,73,78)(77,93,83,79)(85,91,89,86);(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)(65,79)(66,76)(67,77)(68,83)(70,86)(71,74)(72,85)(73,89)(75,82)(78,90)(80,93)(81,92)(84,88)(87,91);(54,55,57,58)(56,60,61,59)(62,63,68,73,66,71,67,72)(64,74,83,85,76,69,77,89)(65,78,82,70,80,88,81,87)(75,91,93,90,92,86,79,84)

Polycyclic presentation:

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

Conjugacy classes & character table

875 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 211083);