#27280225

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1299 ⋊ C₂
Sylow	C₅₃ ⋊ #9214

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133190
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1299	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂×#11719945	D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1299	D₅₃
hypocenter	C₅₃	#9214
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1299

428 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 97 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)(56,64)(60,68)(63,69)(65,66)(70,72)(71,76)(73,81)(74,79)(75,83)(77,85)(78,82)(80,84);(54,56,59,64)(55,60)(57,65,58,66)(61,63)(62,68)(67,69)(70,71,73,80,74,77,82,75)(72,76,81,84,79,85,78,83);(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)(54,55,58,61,59,62,57,67)(56,63,66,68,64,69,65,60)(71,75,77,80)(72,78,79,81)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfg,d^2=gh,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,i^a=i^52,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh,i^c=i^52

Conjugacy classes & character table

1034 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 65701);