#27275528

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1479 ⋊ C₂
Sylow	C₅₃ ⋊ #8327

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1479	C₂
socle	C₂²×C₅₃	#379
prime radical	C₂³×D₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1479	D₅₃
hypocenter	C₅₃	#8327
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1479

684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 101 points:

(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)(56,62)(59,65)(63,68)(66,69)(70,73,75,86)(71,77,79,84)(72,81,83,97)(74,85,89,101)(76,90,91,98)(78,92,87,99)(80,94,93,95)(82,96,88,100);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,56)(55,59)(57,63)(58,62)(60,66)(61,65)(64,68)(67,69)(70,72,75,83)(71,76,79,91)(73,81,86,97)(74,88,89,82)(77,90,84,98)(78,93,87,80)(85,100,101,96)(92,95,99,94);(54,55,57,60,58,61,64,67)(56,59,63,66,62,65,68,69)(70,71)(72,80)(73,84,86,77)(74,87)(75,79)(76,82)(78,89)(81,95,97,94)(83,93)(85,92,101,99)(88,91)(90,96,98,100)

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,c^2=g,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ch,d^b=dg,f^c=fg,i^c=i^52

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 61004);