#27274623

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1281 ⋊ C₂
Sylow	C₅₃ ⋊ #8157

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1281	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂³×D₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1281	D₅₃
hypocenter	C₅₃	#8157
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

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

normal subgroups only

This group has 482 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) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 93 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)(55,58)(59,61)(62,65,66,76,67,77,78,89)(63,69,70,82,71,83,84,92)(64,73,75,88,74,87,79,90)(68,80,81,91,72,85,86,93);(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)(54,55,56,59,57,58,60,61)(62,64)(63,68,71,72)(65,73)(66,75)(67,74)(69,80,83,85)(70,81,84,86)(76,88)(77,87)(78,79)(82,91,92,93)(89,90);(62,63,66,70,67,71,78,84)(64,72,75,86,74,68,79,81)(65,69,76,82,77,83,89,92)(73,85,88,93,87,80,90,91)

Polycyclic presentation:

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

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 60099);