#27296151

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1344 ⋊ C₂
Sylow	C₅₃ ⋊ #12095

Subgroups

	subgroup	G/H
center	C₂²	#16810811
commutator, cosocle	C₂×C₂²⋊C₄×C₅₃	C₂³
Frattini	C₂×C₂²⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#1344	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
prime radical	#16811185	C₂²
hypercenter	#1344	D₅₃
hypocenter	C₅₃	#12095
LCS	G ▷ C₂×C₂²⋊C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #1344

458 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 93 points:

(9,12,14,25)(10,16,18,32)(11,20,22,23)(13,26,27,24)(15,28,29,30)(17,31,33,40)(19,34,35,39)(21,37,38,36)(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)(85,86)(87,88)(89,90)(91,92);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,18,29)(12,23,25,20)(13,21,27,38)(16,30,32,28)(17,19,33,35)(24,37,26,36)(31,39,40,34);(1,2)(3,5)(4,7)(6,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,34)(22,35)(23,39)(24,31)(25,32)(26,40)(27,33)(28,36)(29,38)(30,37)(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)(86,87)(88,89)(90,91)(92,93)

Polycyclic presentation:

2^8,53|b^2=h,c^2=h,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=gh,i^a=i^52,c^b=ch,d^b=df,e^b=egh,d^c=dg,i^c=i^52,e^d=eh

Conjugacy classes & character table

623 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81627);