D₅₃×#2032 #27411528

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₅₃×#2032

Splittings

Fitting	C₅₃×#2032 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2032

Subgroups

	subgroup	G/H
center	C₂	#393×D₅₃
commutator, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#2032	C₂
socle	C₂×C₅₃	C₂×#393
prime radical	C₂²×D₄×D₅₃	C₂²
hypercenter	#2032	D₅₃
hypocenter	C₅₃	C₂×#2032
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #2032

1434 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1434 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) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 4 generators on 69 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,57)(55,60)(56,62)(58,66)(59,67)(61,64)(63,65)(68,69);(54,56)(55,59)(57,65,66,62)(58,63)(60,67,64,69)(61,68);(54,55)(56,59)(57,64,66,60)(58,61)(62,69,65,67)(63,68);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)

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, 197004);