#27401206

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1927 ⋊ C₂
Sylow	C₅₃ ⋊ #22568

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1927	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂²×#11719949	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1927	D₅₃
hypocenter	C₅₃	#22568
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1927

994 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)(67,73)(69,77)(71,74)(72,79)(75,81)(78,80);(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)(59,63)(62,65)(66,68,70,76)(67,72,74,80)(69,75,77,81)(71,78,73,79);(58,60,61,64)(59,62,63,65)(66,67,70,74)(68,72,76,80)(69,73,77,71)(75,79,81,78);(54,55,56,57)(58,59)(60,62)(61,63)(64,65)(67,71)(69,77)(72,78)(73,74)(75,81)(79,80)

Polycyclic presentation:

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

Conjugacy classes & character table

1046 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 186682);