#27393660

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1986 ⋊ C₂
Sylow	C₅₃ ⋊ #21845

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1986	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂³×D₅₃	C₂×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1986	D₅₃
hypocenter	C₅₃	#21845
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1986

1008 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 85 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)(55,61)(59,62)(60,69)(67,68);(54,57,58,66)(55,60,61,69)(56,63,64,65)(59,67,62,68)(70,72)(71,76)(73,83)(74,81)(75,80)(77,85)(78,84)(79,82);(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)(54,56,58,64)(55,59,61,62)(57,65,66,63)(60,68,69,67)(72,80)(76,84)(79,85)(81,83);(54,55)(56,62)(57,60)(58,61)(59,64)(63,68)(65,67)(66,69)(70,71,74,77,75,78,73,82)(72,79,81,76,80,85,83,84)

Polycyclic presentation:

2^8,53|a^2=g,b^2=e,c^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,i^b=i^52,f^c=fh,i^d=i^52

Conjugacy classes & character table

1040 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 179136);