#27275109

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8244

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1299	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1299	D₅₃
hypocenter	C₅₃	#8244
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1299

368 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 121 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)(70,72,74,82,75,80,73,81)(71,77,78,84,79,85,76,83);(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)(54,56,58,64,57,63,65,69)(55,59,61,67,60,62,66,68)(71,76,79,78)(72,80)(77,84,85,83)(81,82);(54,55,57,60)(56,62,63,59)(58,66,65,61)(64,67,69,68)(70,71,74,78,75,79,73,76)(72,77,82,84,80,85,81,83)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=eh,d^2=g,e^2=g,f^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,d^b=dh,i^c=i^52

Conjugacy classes & character table

1040 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 60585);