#27271124

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7215

Subgroups

	subgroup	G/H
center	C₂²	#16810389
commutator	C₂²×C₄×C₅₃	C₄²
Frattini	C₂²×C₄⋊C₄	C₂×D₅₃
Fitting	C₅₃×#1463	C₂
socle	C₂²×C₅₃	#342
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂²×C₄⋊C₄×C₅₃	C₂²
hypercenter	#1463	D₅₃
hypocenter	C₅₃	#7215
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ #1463

422 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 69
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 2 generators on 69 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)(55,58)(59,60)(63,66,68,67);(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)(54,55,56,59,57,60,61,58)(62,63)(64,66)(65,68)(67,69)

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,c^2=fg,d^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,i^a=i^52,d^b=dg,i^b=i^52,d^c=dh

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56600);