#27283028

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1452 ⋊ C₂
Sylow	C₅₃ ⋊ #9670

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1452	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂²×D₄₂₄	C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1452	D₅₃
hypocenter	C₅₃	#9670
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1452

498 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 123 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)(54,57,58,59)(55,60,56,61)(62,65,66,74,68,76,67,75)(63,69,71,73,70,77,64,72);(54,56,59,61,58,55,57,60)(62,64,67,70,68,71,66,63)(65,73,75,69,76,72,74,77);(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)(54,55,59,60,58,56,57,61)(62,63,67,71,68,70,66,64)(65,73,75,77,76,72,74,69)

Polycyclic presentation:

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

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 68504);