#27389231

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1983 ⋊ C₂
Sylow	C₅₃ ⋊ #21397

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1983	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂×#13133188	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1983	D₅₃
hypocenter	C₅₃	#21397
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1983

1150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|a^2=g,b^2=eg,d^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,e^b=eh,g^b=gh,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, 174707);