#27299293

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12634

Subgroups

	subgroup	G/H
center	C₄	#99×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×#41	C₂²×D₅₃
Fitting	C₅₃×#2126	C₂
socle	C₂×C₅₃	C₄×#99
prime radical	C₅₃×#532	C₂²
cosocle	C₂×#41×C₅₃	C₂³
hypercenter	#2126	D₅₃
hypocenter	C₅₃	#12634
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2126

452 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 121 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) = 1696

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

Polycyclic presentation:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84769);