#27295180

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1348 ⋊ C₂
Sylow	C₅₃ ⋊ #11890

Subgroups

	subgroup	G/H
center	C₂²	#102×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1348	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	#20284298	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1348	D₅₃
hypocenter	C₅₃	#11890
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1348

416 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 3392

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)(56,61)(58,60)(62,64,73,76,66,74,65,71)(63,68,75,72,70,67,69,77);(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)(54,56)(55,58)(57,61)(59,60)(62,63)(64,72)(65,75)(66,70)(67,71)(68,76)(69,73)(74,77);(54,55)(56,60,61,58)(57,59)(63,67)(64,71)(65,73)(68,70)(69,77)(72,75)(74,76)

Polycyclic presentation:

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

Conjugacy classes & character table

794 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 80656);