#27275805

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8385

Subgroups

	subgroup	G/H
center	C₂²	#16810479
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#550	C₂
socle	C₂²×C₅₃	#379
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#550	D₅₃
hypocenter	C₅₃	#8385
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#550

1050 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 69
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 69 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)(55,58)(56,60)(62,65,67,66)(63,68,64,69);(55,58)(56,60)(62,64)(63,67)(65,69)(66,68);(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)(54,55,56,59,57,58,60,61)(62,63,66,68,67,64,65,69)

Polycyclic presentation:

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

Conjugacy classes & character table

1343 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 61281);