#27294813

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1352 ⋊ C₂
Sylow	C₅₃ ⋊ #11811

Subgroups

	subgroup	G/H
center	C₂²	#99×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₂²⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#1352	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂⁴×D₅₃	D₄
cosocle	C₂×C₂²⋊C₄×C₅₃	C₂³
hypercenter	#1352	D₅₃
hypocenter	C₅₃	#11811
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #1352

496 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 69 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)(55,60)(56,63)(61,65)(64,66)(71,76)(75,83)(77,84)(78,85);(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)(54,56)(55,59,60,68)(57,66)(58,63)(61,69,65,62)(64,67)(70,72)(71,75)(73,81)(74,80)(76,83)(77,85)(78,84)(79,82);(54,55)(56,62)(57,65,67,61)(58,60)(59,64,68,66)(63,69)(70,71)(72,78,80,85)(73,77)(74,76)(75,79,83,81)(82,84)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,d^2=gh,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,d^b=dg,e^b=efgh,i^b=i^52,i^c=i^52,e^d=eh

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, 80289);