D₅₃×#2178 #27537488

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

Aliases

direct product

D₅₃×#2178

Splittings

Fitting	C₅₃×#2178 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2178

Subgroups

	subgroup	G/H
center	C₈	D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2178	C₂
socle	C₂×C₅₃	C₂×C₄×D₈
prime radical	#508×D₅₃	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2178	D₅₃
hypocenter	C₅₃	C₂×#2178
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2178

662 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:

(1,4,6,18,7,19,22,5)(2,9,10,27,12,28,11,20)(3,14,15,23,17,31,16,21)(8,24,25,30,26,32,29,13)(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)(2,8)(4,14)(5,21)(6,15)(7,17)(9,24)(10,25)(11,29)(12,26)(13,20)(16,22)(18,23)(19,31)(27,30)(28,32);(1,2,6,10,7,12,22,11)(3,13,15,24,17,30,16,32)(4,9,18,27,19,28,5,20)(8,23,25,31,26,21,29,14);(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, 322964);

D53×#2178 #27537488