#27361969

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3131 ⋊ C₂
Sylow	C₅₃ ⋊ #18691

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3131	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20289120	C₂
hypercenter	#3131	D₅₃
hypocenter	C₅₃	#18691
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3131

996 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 163 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) = 3392

A minimal faithful permutation representation — 4 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)(54,57,59,58)(55,62,63,61)(56,65,67,66)(60,64,69,68);(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)(55,61)(57,58)(60,68)(62,63)(64,69)(65,66)(70,72)(71,75)(73,83)(74,79)(76,85)(77,84)(78,81)(80,82);(54,56)(55,60)(57,65)(58,66)(59,67)(61,68)(62,64)(63,69)(73,82)(76,81)(78,85)(80,83);(54,55,59,63)(56,64,67,68)(57,62,58,61)(60,65,69,66)(70,71,74,77)(72,78,79,85)(73,81,82,76)(75,83,84,80)

Polycyclic presentation:

2^8,53|a^2=h,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fh,i^c=i^52

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 147445);