#27284322

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #9919

Subgroups

	subgroup	G/H
center	C₂³	#13133187
commutator, cosocle	C₂³×C₄×C₅₃	C₂³
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#327	C₂
socle	C₂³×C₅₃	#120
prime radical	C₂×#121×C₅₃	C₂²
hypercenter	C₂×#327	D₅₃
hypocenter	C₅₃	#9919
LCS	G ▷ C₂³×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#327

716 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 113 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) = 81
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 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)(55,57)(58,60,62,61)(59,64,65,63)(66,69)(67,72)(68,74)(70,76)(71,77)(73,80)(75,81)(78,79);(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)(60,61)(64,65)(66,68,70,75)(67,71,73,78)(69,74,76,81)(72,79,80,77);(54,55)(56,57)(58,59)(60,64)(61,63)(62,65)(66,67)(68,71)(69,72)(70,73)(74,77)(75,78)(76,80)(79,81)

Polycyclic presentation:

2^8,53|b^2=h,c^2=g,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fh,c^b=cf,d^b=dg,i^b=i^52,d^c=dh,i^c=i^52

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 69798);