#27275486

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1459 ⋊ C₂
Sylow	C₅₃ ⋊ #8317

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133187
commutator	C₂⁴×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1459	C₂
socle	C₂²×C₅₃	#379
prime radical	#20288124	C₂
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1459	D₅₃
hypocenter	C₅₃	#8317
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1459

864 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 77
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

2^8,53|a^2=f,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,e^b=eg,f^b=fg,d^c=dg,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, 60962);