#27400644

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1996 ⋊ C₂
Sylow	C₅₃ ⋊ #22513

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#1996	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂²×C₂²⋊D₅₃	C₂³
cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	#1996	D₅₃
hypocenter	C₅₃	#22513
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1996

1114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 77 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);(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)(63,68)(64,70)(66,73)(67,74)(69,72)(71,77)(75,76);(54,55,57,58)(56,60,61,59)(62,64,66,71)(63,67,69,75)(65,70,73,77)(68,74,72,76);(55,58)(59,60)(62,63)(64,67)(65,72,73,68)(66,69)(70,76,77,74)(71,75)

Polycyclic presentation:

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

Conjugacy classes & character table

887 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 186120);