#27300113

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1813 ⋊ C₂
Sylow	C₅₃ ⋊ #12806

Subgroups

	subgroup	G/H
center	C₄	#16810596
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×#41	C₂²×D₅₃
Fitting	C₅₃×#1813	C₂
socle	C₂×C₅₃	#1799
prime radical	#16811130	C₂²
cosocle	C₂×#41×C₅₃	C₂³
hypercenter	#1813	D₅₃
hypocenter	C₅₃	#12806
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×#131 ◁ #1813

420 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 1696

A minimal faithful permutation representation — 3 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)(55,61)(56,65)(60,73)(62,76)(63,71)(64,77)(66,80)(67,81)(68,83)(74,75)(78,79)(82,85);(54,56,57,66,59,65,70,80)(55,60,62,64,63,75,68,79)(58,67,69,82,72,81,84,85)(61,74,76,78,71,73,83,77);(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)(56,64)(57,68)(58,71,72,61)(59,63)(60,66)(62,70)(65,79)(67,77,81,78)(69,76,84,83)(73,85,74,82)(75,80)

Polycyclic presentation:

2^8,53|a^2=f,b^2=d,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,i^a=i^52,c^b=ch,f^c=fg,i^c=i^52,f^e=fh

Conjugacy classes & character table

866 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85589);