D₅₃×#2110 #27536747

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₅₃×#2110

Splittings

Fitting	C₅₃×#2110 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2110

Subgroups

	subgroup	G/H
center	C₈	C₄²×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2110	C₂
socle	C₂×C₅₃	C₂²×#97
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2110	D₅₃
hypocenter	C₅₃	C₂×#2110
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ #2110

738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 85 points:

(3,12)(7,20)(11,26)(13,15)(14,28)(19,29)(21,22)(27,30)(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);(1,3)(2,7,9,20)(4,15)(5,14)(6,12)(8,22,23,21)(10,11,25,26)(13,17)(16,19)(18,28)(24,30,32,27)(29,31);(1,2,4,8,5,10,16,24,6,9,17,23,18,25,31,32)(3,11,13,27,14,20,29,22,12,26,15,30,28,7,19,21);(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

1232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322223);

D53×#2110 #27536747