#27303619

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2123 ⋊ C₂
Sylow	C₅₃ ⋊ #13437

Subgroups

	subgroup	G/H
center	C₂	#20275965
commutator, cosocle	C₂×#41×C₅₃	C₂³
Frattini	C₂×#41	C₂²×D₅₃
Fitting	C₅₃×#2123	C₂
socle	C₂×C₅₃	#2044
prime radical	#16811187	C₂²
hypercenter	#2123	D₅₃
hypocenter	C₅₃	#13437
LCS	G ▷ C₂×#41×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2123

422 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 71 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:

(1,3,4,15,7,17,6,14)(2,9,10,25,12,26,8,22)(5,16,18,32,20,24,21,31)(11,13,27,28,19,30,23,29)(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)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84);(2,8)(3,14)(4,6)(5,20)(10,12)(11,27)(15,17)(16,31)(19,23)(22,25)(24,32)(28,29)(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)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85);(1,2,7,12)(3,13,17,30)(4,8,6,10)(5,19,20,11)(9,24,26,16)(14,28,15,29)(18,27,21,23)(22,31,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

626 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89095);