#27278501

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8919

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133187
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1300	C₂
socle	C₂²×C₅₃	#385
prime radical	C₂³×C₅₃	#123
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1300	D₅₃
hypocenter	C₅₃	#8919
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1300

336 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 89 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) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 63977);