D₅₃×#1381 #27298977

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

Aliases

direct product

D₅₃×#1381

Splittings

Fitting	C₅₃×#1381 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1381

Subgroups

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1381	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×D₅₃	Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1381	D₅₃
hypocenter	C₅₃	C₂×#1381
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 89 points:

(1,2,3,4)(5,7,10,21,9,20,24,35,11,22,25,36,23,34,8,19)(6,12,15,27,16,28,31,33,17,29,30,32,14,26,13,18)(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);(5,6,9,14,11,17,23,16)(7,18,20,32,22,33,34,27)(8,15,10,13,24,30,25,31)(12,21,26,35,29,36,28,19);(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, 84453);

D53×#1381 #27298977