D₅₃×#1385 #27299002

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

D₅₃×#1385

Splittings

Fitting	C₅₃×#1385 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1385

Subgroups

	subgroup	G/H
center	C₄	#106×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1385	C₂
socle	C₂×C₅₃	C₂×#340
prime radical	C₂³×D₅₃	Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1385	D₅₃
hypocenter	C₅₃	C₂×#1385
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

554 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 points:

(1,3,6,17,5,21,20,32,8,18,19,31,7,16,4,15)(2,9,12,25,23,24,10,14,13,27,28,29,11,26,22,30)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,2,5,11,8,13,7,23)(3,14,16,29,18,30,21,25)(4,12,6,22,20,28,19,10)(9,17,24,15,27,31,26,32);(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84478);

D53×#1385 #27299002