#27298926

order	13568
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12542

Subgroups

	subgroup	G/H
center	C₈	C₄²×D₅₃
commutator	C₄×C₅₃	C₄³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2110	C₂
socle	C₂×C₅₃	#1726
prime radical	C₂³×C₅₃	#131
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2110	D₅₃
hypocenter	C₅₃	#12542
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ #2110

350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 117 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)(54,57)(55,62)(56,66,68,89)(58,71)(59,73)(60,72)(61,77,79,100)(63,82)(64,84)(65,83)(67,88,90,110)(69,70,92,94)(74,96)(75,95)(76,97)(78,99,101,114)(80,81,103,105)(85,107)(86,106)(87,108)(91,93,111,112)(98,113)(102,104,115,116)(109,117);(54,56)(55,61)(57,70,72,94)(58,67)(59,69)(60,68)(62,81,83,105)(63,78)(64,80)(65,79)(66,73,89,97)(71,93,95,112)(74,91)(75,90)(76,92)(77,84,100,108)(82,104,106,116)(85,102)(86,101)(87,103)(88,96,110,113)(98,111)(99,107,114,117)(109,115);(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)(54,55,58,63,59,64,74,85,60,65,75,86,76,87,98,109)(56,61,67,78,69,80,91,102,68,79,90,101,92,103,111,115)(57,62,71,82,73,84,96,107,72,83,95,106,97,108,113,117)(66,77,88,99,70,81,93,104,89,100,110,114,94,105,112,116)

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,c^2=f,d^2=g,g^2=h,i^a=i^52,c^b=cg,f^b=fh,e^c=eh,i^c=i^52

Conjugacy classes & character table

1232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84402);