#27274523

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8134

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂×C₄²
Fitting	C₅₃×#1273	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1273	D₅₃
hypocenter	C₅₃	#8134
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ #1273

414 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 179 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) = 13568

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

Polycyclic presentation:

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

Conjugacy classes & character table

2168 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 59999);