#27537731

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	Q₈×C₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #33496

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator	C₂×C₅₃	C₂³×C₁₆
Fitting	Q₈×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	Q₈×C₁₆	D₅₃
hypocenter	C₅₃	#33496
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

526 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)(71,77)(75,78)(76,85)(83,84);(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)(70,73,74,82)(71,76,77,85)(72,79,80,81)(75,83,78,84);(70,72,74,80)(71,75,77,78)(73,81,82,79)(76,84,85,83);(54,55,56,59,57,60,62,65,58,61,63,66,64,67,68,69)(70,71)(72,78)(73,76)(74,77)(75,80)(79,84)(81,83)(82,85)

Polycyclic presentation:

2^8,53|a^2=f,b^2=e,c^2=e,f^2=g,g^2=h,b^a=be,d^a=de,c^b=ce,i^d=i^52

Conjugacy classes & character table

2216 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323207);