D₅₃×#1544 #27352397

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₅₃×#1544

Splittings

Fitting	C₅₃×#1544 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1544

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1544	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂³×D₅₃	C₂²⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1544	D₅₃
hypocenter	C₅₃	C₂×#1544
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1544

994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

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

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

1064 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 137873);

D53×#1544 #27352397