D₅₃×#2070 #27412811

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₅₃×#2070

Splittings

Fitting	C₅₃×#2070 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2070

Subgroups

	subgroup	G/H
center	C₂	#396×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2070	C₂
socle	C₂×C₅₃	C₂×#396
prime radical	C₂³×D₅₃	C₂×Q₈
hypercenter	#2070	D₅₃
hypocenter	C₅₃	C₂×#2070
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2070

950 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 207 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 198287);