C₅₃×#7068 #27214580

order	13568
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7068

Subgroups

	subgroup	G/H
center	C₂×C₈×C₅₃	D₈
commutator	C₈	C₂×C₁₆×C₅₃
Frattini, cosocle	C₈²	C₂²×C₅₃
socle	C₂²×C₅₃	D₄⋊C₈
prime radical	C₂×D₈×C₅₃	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₈²×C₅₃ ◁ G

162 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 6784

A minimal faithful permutation representation — 2 generators on 85 points:

(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18)(19,22)(20,29)(21,25)(23,26)(24,27)(28,31)(30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30)

Conjugacy classes & character table

4664 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56);