#27274630

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8158

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1458	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1458	D₅₃
hypocenter	C₅₃	#8158
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1458

400 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 60106);