#27304426

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#356 ⋊ C₂
Sylow	C₅₃ ⋊ #13590

Subgroups

	subgroup	G/H
center	C₂	C₂×#16810415
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄⋊Q₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#356	C₂
socle	C₂×C₅₃	C₂×#353
prime radical	#16811173	C₄
cosocle	C₄⋊Q₈×C₅₃	C₂³
hypercenter	C₂×#356	D₅₃
hypocenter	C₅₃	#13590
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈ ◁ C₂×#356

398 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 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)(56,66)(60,75)(65,77)(67,82)(68,72)(69,83)(76,84)(79,85);(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)(55,60)(57,69)(58,72)(59,66)(61,77)(62,79)(63,65)(64,75)(67,73)(68,74)(70,82)(71,83)(76,78)(80,84)(81,85);(54,55,57,62,59,64,71,81)(56,65,67,79,66,77,82,85)(58,63,73,80,74,61,70,78)(60,69,76,72,75,83,84,68)

Polycyclic presentation:

2^8,53|a^2=e,d^2=gh,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch,d^b=dgh,e^b=efg,f^b=fh,i^b=i^52,e^d=egh,f^e=fh

Conjugacy classes & character table

704 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89902);