#27295458

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1366 ⋊ C₂
Sylow	C₅₃ ⋊ #11956

Subgroups

	subgroup	G/H
center	C₂	#333×D₅₃
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1366	C₂
socle	C₂×C₅₃	C₂×#333
prime radical	C₂²×D₅₃	#100
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1366	D₅₃
hypocenter	C₅₃	#11956
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1366

292 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 89 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

644 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 80934);