#27410685

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #23364

Subgroups

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#1916	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1916	D₅₃
hypocenter	C₅₃	#23364
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1916

636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 13568

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

Polycyclic presentation:

2^8,53|a^2=g,b^2=e,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cefh,d^b=de,i^c=i^52,i^d=i^52

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 196161);