#27284387

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1824 ⋊ C₂
Sylow	C₅₃ ⋊ #9928

Subgroups

	subgroup	G/H
center	C₂³	#13133185
commutator, cosocle	C₂³×C₄×C₅₃	C₂³
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1824	C₂
socle	C₂³×C₅₃	#120
prime radical	C₂⁴×D₅₃	D₄
hypercenter	#1824	D₅₃
hypocenter	C₅₃	#9928
LCS	G ▷ C₂³×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1824

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

normal subgroups only

This group has 778 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)(60,65)(62,63)(66,69)(67,72)(68,74)(70,76)(71,77)(73,80)(75,81)(78,79);(59,62)(60,65)(63,64)(66,68,70,75)(67,71,73,78)(69,74,76,81)(72,79,80,77);(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)(54,55)(56,57)(58,59,61,64)(60,62,65,63)(66,67)(68,71)(69,72)(70,73)(74,77)(75,78)(76,80)(79,81)

Polycyclic presentation:

2^8,53|a^2=g,b^2=gh,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fh,i^a=i^52,c^b=cf,d^b=dg,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, 69863);