#27278135

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8865

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133187
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1302	C₂
socle	C₂²×C₅₃	#385
prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1302	D₅₃
hypocenter	C₅₃	#8865
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1302

348 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 95 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)(55,61,63,62)(56,65,67,66)(60,64,69,68)(70,72,74,79)(71,76,77,85)(73,82,80,78)(75,83,81,84);(54,56,58,66,59,67,57,65)(55,60,62,68,63,69,61,64)(70,71)(72,76)(73,81)(74,77)(75,80)(78,83)(79,85)(82,84);(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)(54,55,58,62,59,63,57,61)(56,64,66,60,67,68,65,69)(71,75)(73,80)(76,84)(77,81)(78,82)(83,85)

Polycyclic presentation:

2^8,53|a^2=f,b^2=f,c^2=g,d^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,i^a=i^52,d^b=dg,e^b=eh,d^c=dh,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, 63611);