#27389039

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1950 ⋊ C₂
Sylow	C₅₃ ⋊ #21372

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1950	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1950	D₅₃
hypocenter	C₅₃	#21372
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1950

714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 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)(54,56,57,60)(55,59,58,61)(65,77)(70,85)(74,88)(78,79)(82,92)(86,90)(87,93)(89,91);(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)(55,58)(59,61)(62,65)(63,70)(64,74)(66,78)(67,79)(68,77)(69,82)(71,86)(72,85)(73,87)(75,89)(76,88)(80,90)(81,91)(83,93)(84,92);(62,64,67,75,68,76,66,81)(63,69,71,83,72,84,80,73)(65,74,79,89,77,88,78,91)(70,82,86,93,85,92,90,87);(54,55)(56,59)(57,58)(60,61)(62,63,67,71,68,72,66,80)(64,73,75,69,76,83,81,84)(65,70,79,86,77,85,78,90)(74,87,89,82,88,93,91,92)

Polycyclic presentation:

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

Conjugacy classes & character table

1052 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 174515);