C₅₃×#12632 #27220144

order	13568
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅₃×#12632

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#99
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×Q₈	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

512 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 73
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 73 points:

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

Conjugacy classes & character table

2438 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 5620);