D₅₃×#2158 #27537117

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

D₅₃×#2158

Splittings

Fitting	C₅₃×#2158 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2158

Subgroups

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2158	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×D₅₃	D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2158	D₅₃
hypocenter	C₅₃	C₂×#2158
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 points:

(5,8,11,10)(6,13,15,25)(7,18,21,20)(9,22,24,17)(12,26,27,30)(14,23,28,36)(16,29,31,35)(19,33,34,32)(37,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39);(5,7,17,32,10,20,24,34,11,21,22,33,8,18,9,19)(6,12,23,29,13,26,28,31,15,27,36,35,25,30,14,16);(1,2,3,4)(5,6)(7,16)(8,13)(9,23)(10,25)(11,15)(12,19)(14,17)(18,29)(20,35)(21,31)(22,28)(24,36)(26,33)(27,34)(30,32);(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)(86,87)(88,89)

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, 322593);

D53×#2158 #27537117