#27355367

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1579 ⋊ C₂
Sylow	C₅₃ ⋊ #18024

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1579	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#11719949	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1579	D₅₃
hypocenter	C₅₃	#18024
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 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)(70,74,76,72)(71,79,81,82)(73,83,75,84)(77,78,85,80);(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)(54,56,57,63,58,62,64,68)(55,59,60,66,61,65,67,69)(70,73,76,75)(71,78,81,80)(72,83,74,84)(77,82,85,79);(70,72,76,74)(71,77,81,85)(73,83,75,84)(78,82,80,79);(54,55,57,60,58,61,64,67)(56,59,63,66,62,65,68,69)(70,71,76,81)(72,82,74,79)(73,80,75,78)(77,84,85,83)

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cg,e^a=eg,e^b=eg,d^c=dgh,i^d=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, 140843);