#27283142

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1831 ⋊ C₂
Sylow	C₅₃ ⋊ #9696

Subgroups

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1831	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂×#13133214	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#9696
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

602 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 6784

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfh,c^2=eg,d^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fh,i^b=i^52

Conjugacy classes & character table

1514 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 68618);