#27489558

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2033 ⋊ C₂
Sylow	C₅₃ ⋊ #29608

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2033	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂×#16811202	C₂
hypercenter	#2033	D₅₃
hypocenter	C₅₃	#29608
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #2033

1332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 81 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)(60,64)(63,65)(66,69,71,68)(67,73,75,76)(70,78,80,77)(72,79,81,74);(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)(55,57)(58,60)(59,63)(61,64)(62,65)(66,68,71,69)(67,72,75,81)(70,78,80,77)(73,79,76,74);(54,55)(56,57)(59,62)(63,65)(68,69)(72,81)(73,76)(77,78);(58,59)(60,63)(61,62)(64,65)(66,67)(68,76)(69,73)(70,79)(71,75)(72,77)(74,80)(78,81)

Polycyclic presentation:

2^8,53|c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^b=dh,d^c=de,f^c=fh,i^c=i^52,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, 275034);