#27294691

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1346 ⋊ C₂
Sylow	C₅₃ ⋊ #11783

Subgroups

	subgroup	G/H
center	C₂²	#99×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#1346	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	#20279676	C₂
cosocle	C₂×C₄⋊C₄×C₅₃	C₂³
hypercenter	#1346	D₅₃
hypocenter	C₅₃	#11783
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #1346

514 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 77
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 77 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)(54,56,58,57)(55,60,61,59)(62,65,66,75)(63,68,70,69)(64,72,74,73)(67,76,77,71);(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)(56,57)(60,61)(62,64)(63,67)(65,73)(66,74)(68,76)(69,71)(70,77)(72,75);(54,55)(56,60)(57,59)(58,61)(62,63,65,68,66,70,75,69)(64,71,73,77,74,76,72,67)

Polycyclic presentation:

2^8,53|a^2=e,c^2=gh,d^2=g,e^2=h,b^a=bd,d^a=df,f^a=fh,c^b=cfh,d^b=dg,e^b=efg,i^b=i^52,d^c=dh,i^c=i^52,e^d=eh

Conjugacy classes & character table

635 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 80167);