#27293753

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11572

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133132
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#2566	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂²×D₄×C₅₃	C₂×C₄
cosocle	C₂×C₄⋊C₄×C₅₃	C₂³
hypercenter	#2566	D₅₃
hypocenter	C₅₃	#11572
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ #2566

416 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 91 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)(54,57,59,58)(55,61,63,62)(56,64,65,66)(60,67,68,69)(72,79)(75,83)(78,84)(80,85);(54,56)(55,60)(57,66)(58,64)(59,65)(61,69)(62,67)(63,68)(70,72)(71,75,77,83)(73,80)(74,79)(76,78,81,84)(82,85);(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)(54,55,58,62,59,63,57,61)(56,60,64,67,65,68,66,69)(70,71)(72,78,79,84)(73,81,82,76)(74,77)(75,80)(83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79229);