#27275441

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#323 ⋊ C₂
Sylow	C₅₃ ⋊ #8305

Subgroups

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₅₃
commutator	C₂⁴×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#323	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂²⋊D₅₃	C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#323	D₅₃
hypocenter	C₅₃	#8305
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#323

1154 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 143 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) = 81
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 81 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)(55,57)(58,60,62,61)(59,64,63,65)(66,69,70,77)(67,72,73,79)(68,75,76,81)(71,78,74,80);(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)(59,63)(60,61)(66,68)(67,71)(69,75)(70,76)(72,78)(73,74)(77,81)(79,80);(54,55)(56,57)(58,59,61,65,62,63,60,64)(66,67)(68,74,76,71)(69,72)(70,73)(75,80,81,78)(77,79)

Polycyclic presentation:

2^8,53|a^2=f,c^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,f^b=fg,i^c=i^52

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 60917);