#27295451

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1364 ⋊ C₂
Sylow	C₅₃ ⋊ #11954

Subgroups

	subgroup	G/H
center	C₂²	#102×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1364	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂²×D₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1364	D₅₃
hypocenter	C₅₃	#11954
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1364

306 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 6784

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

Polycyclic presentation:

2^8,53|a^2=e,b^2=h,c^2=fh,d^2=gh,e^2=fgh,g^2=h,b^a=bd,c^a=ch,d^a=df,d^b=dg,e^b=efg,g^b=gh,i^b=i^52,i^c=i^52

Conjugacy classes & character table

800 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 80927);