#27489997

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #29659

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#2062	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#2062	D₅₃
hypocenter	C₅₃	#29659
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2062

974 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

713 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 275473);