#27295655

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1369 ⋊ C₂
Sylow	C₅₃ ⋊ #12002

Subgroups

	subgroup	G/H
center	C₄	#106×D₅₃
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1369	C₂
socle	C₂×C₅₃	C₂×#334
prime radical	C₂²×D₅₃	#108
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1369	D₅₃
hypocenter	C₅₃	#12002
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

220 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 99 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:

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

Polycyclic presentation:

2^8,53|a^2=e,b^2=d,d^2=f,e^2=g,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,c^b=ch,e^b=eh,i^c=i^52

Conjugacy classes & character table

896 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81131);