#27304681

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄×Q₃₂×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #13652

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133158
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₄×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₁₆
prime radical	C₂×D₈₄₈	C₂²
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₄×Q₃₂	D₅₃
hypocenter	C₅₃	#13652
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

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

normal subgroups only

This group has 200 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) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 89 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)(54,56,57,55)(59,65)(64,77)(66,80)(67,81)(68,76)(73,89)(78,82)(79,83);(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)(59,64)(60,69)(61,70)(62,74)(65,77)(66,67)(68,83)(71,88)(72,86)(73,78)(75,87)(76,79)(80,81)(82,89)(84,85);(54,55,57,56)(58,59)(60,64)(61,73)(62,76)(63,65)(66,69)(67,70)(68,74)(71,83)(72,77)(75,89)(78,85)(79,84)(80,86)(81,87)(82,88)

Polycyclic presentation:

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

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90157);