D₅₃×#2135 #27536932

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₅₃×#2135

Splittings

Fitting	C₅₃×#2135 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2135

Subgroups

	subgroup	G/H
center	C₂	C₂×#99×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×Q₈	C₂³×D₅₃
Fitting	C₅₃×#2135	C₂
socle	C₂×C₅₃	C₂²×#99
prime radical	C₂×#41×D₅₃	C₂²
cosocle	C₂×Q₈×C₅₃	C₂⁴
hypercenter	#2135	D₅₃
hypocenter	C₅₃	C₂×#2135
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2135

1198 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:

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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322408);

D53×#2135 #27536932