#27302730

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13291

Subgroups

	subgroup	G/H
center	C₂	#20275688
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2110	C₂
socle	C₂×C₅₃	#1993
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2110	D₅₃
hypocenter	C₅₃	#13291
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2110

306 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 111 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:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,57,60,76)(55,62)(56,69)(58,74,78,85)(59,75,79,61)(63,72)(64,73)(65,81)(66,80)(67,68,83,84)(70,82)(71,77);(54,56,68,81,79,70,58,62,60,71,84,73,59,72,78,80)(55,61,69,85,64,76,82,83,66,75,77,74,65,57,63,67);(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)(56,67,71,83)(57,73,76,81)(58,77)(59,64)(60,66)(61,62,75,80)(63,68)(65,79)(69,78)(70,74,72,85)(82,84)

Polycyclic presentation:

2^8,53|a^2=f,b^2=def,c^2=eg,d^2=g,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^52,c^b=cg,e^b=eh,f^b=fh,f^c=fh

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, 88206);