#27293132

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1312 ⋊ C₂
Sylow	C₅₃ ⋊ #11412

Subgroups

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#1312	C₂
prime radical	C₂²×D₅₃	C₄×C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1312	D₅₃
hypocenter	C₅₃	#11412
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ #1312

260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 101 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)(56,62)(59,65)(63,68)(66,69)(71,77)(76,86)(78,89)(79,90)(87,96)(88,97)(91,99)(98,101);(54,56)(55,59)(57,63)(58,62)(60,66)(61,65)(64,68)(67,69)(70,72,73,80,74,81,83,92,75,82,84,93,85,94,95,100)(71,76,78,87,79,88,91,98,77,86,89,96,90,97,99,101);(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)(54,55,57,60,58,61,64,67)(56,59,63,66,62,65,68,69)(70,71)(72,76)(73,78)(74,79)(75,77)(80,87)(81,88)(82,86)(83,91)(84,89)(85,90)(92,98)(93,96)(94,97)(95,99)(100,101)

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,c^a=ch,i^a=i^52,c^b=cf,i^c=i^52

Conjugacy classes & character table

2192 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78608);