#27275455

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#324 ⋊ C₂
Sylow	C₅₃ ⋊ #8309

Subgroups

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₅₃
commutator	C₂⁴×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#324	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×D₅₃	C₂×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#324	D₅₃
hypocenter	C₅₃	#8309
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#324

874 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 139 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) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 89 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)(58,61,62,69)(59,64,65,71)(60,67,68,73)(63,70,66,72)(76,83)(79,82)(84,85)(87,89);(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)(58,60)(59,63)(61,67)(62,68)(64,70)(65,66)(69,73)(71,72)(74,76)(75,79)(77,85)(78,83)(80,82)(81,89)(84,86)(87,88);(54,55)(56,57)(58,59)(60,66,68,63)(61,64)(62,65)(67,72,73,70)(69,71)(74,75,77,81,78,80,86,88)(76,82,84,89,83,79,85,87)

Polycyclic presentation:

2^8,53|a^2=f,c^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,f^b=fg,i^b=i^52,i^c=i^52

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 60931);