#27353323

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #17827

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×#41×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₅₃	D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×#41	D₅₃
hypocenter	C₅₃	#17827
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

610 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

2150 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 138799);