#27285490

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10096

Subgroups

	subgroup	G/H
center	C₂²	#16810531
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting, prime radical	C₅₃×#2657	C₂
socle	C₂²×C₅₃	#392
hypercenter	#2657	D₅₃
hypocenter	C₅₃	#10096
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2657

684 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 1696

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

Polycyclic presentation:

2^8,53|a^2=h,d^2=g,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,i^a=i^52,c^b=cf,d^b=dg,e^b=egh,d^c=dh,e^c=egh

Conjugacy classes & character table

863 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 70966);