#27537276

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	D₄×C₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #33434

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	D₄×C₁₆×C₅₃	C₂
prime radical	#16811170	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	D₄×C₁₆	D₅₃
hypocenter	C₅₃	#33434
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ D₄×C₁₆

690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 89 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)(58,61)(59,66)(60,69)(62,73)(63,74)(64,72)(65,77)(67,79)(68,80)(70,82)(71,83)(75,86)(76,85)(78,87)(81,88)(84,89);(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)(58,60,62,70,63,71,75,84,64,59,65,67,76,68,78,81)(61,69,73,82,74,83,86,89,72,66,77,79,85,80,87,88);(55,57)(61,72)(66,69)(73,77)(74,85)(79,82)(80,83)(86,87)(88,89);(54,55)(56,57)(58,59,62,67,63,68,75,81,64,60,65,70,76,71,78,84)(61,69,73,82,74,83,86,89,72,66,77,79,85,80,87,88)

Polycyclic presentation:

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

Conjugacy classes & character table

2192 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322752);