#27294083

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1331 ⋊ C₂
Sylow	C₅₃ ⋊ #11649

Subgroups

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1331	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	#20285291	C₂
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1331	D₅₃
hypocenter	C₅₃	#11649
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1331

348 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79559);