#27283645

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₈×M₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #9789

Subgroups

	subgroup	G/H
center	C₂²	#16810525
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	#9789
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

392 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 3392

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=df,c^2=eg,d^2=gh,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,i^a=i^52,d^b=dh,f^b=fh

Conjugacy classes & character table

2132 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 69121);