#27281027

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1490 ⋊ C₂
Sylow	C₅₃ ⋊ #9329

Subgroups

	subgroup	G/H
center	C₂²	#16810500
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1490	C₂
socle	C₂²×C₅₃	#386
prime radical	#20284806	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1490	D₅₃
hypocenter	C₅₃	#9329
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1490

530 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 99 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:

(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)(56,58)(57,59)(60,61)(64,72)(68,76)(71,77)(73,74);(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)(55,58)(57,59)(60,61)(62,64)(63,68,70,76)(65,73)(66,74)(67,72)(69,71,75,77);(54,55,57,59,58,56,61,60)(62,63,66,69,67,70,65,75)(64,71,74,76,72,77,73,68)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfh,d^2=gh,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,f^b=fh,i^b=i^52,e^c=eh

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 66503);