#27270769

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7077

Subgroups

	subgroup	G/H
center	C₂²	#16810358
commutator	C₂²×C₈×C₅₃	C₂×C₄
Frattini	C₂×C₂²⋊C₈	C₂×D₅₃
Fitting	C₅₃×#1314	C₂
socle	C₂²×C₅₃	#327
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂×C₂²⋊C₈×C₅₃	C₂²
hypercenter	#1314	D₅₃
hypocenter	C₅₃	#7077
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ #1314

218 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 49 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 — 2 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)(54,56,58,64)(55,59,61,67)(57,63,66,69)(60,62,65,68)(71,75)(72,81)(73,82)(76,78)(77,84)(79,80)(83,85);(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)(54,55)(56,62)(57,65,66,60)(58,61)(59,63,67,69)(64,68)(70,71)(72,80,83,75)(73,77)(74,79)(76,81,84,85)(78,82)

Polycyclic presentation:

2^8,53|a^2=d,b^2=g,c^2=fh,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,i^a=i^52,c^b=cf,d^b=defh,e^b=eh,f^b=fh,i^b=i^52,d^c=dgh

Conjugacy classes & character table

1178 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56245);