C₅₃×#12779 #27220291

order	13568
nilpotency class	4
isoclinism	isoclinic class of #12778
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅₃×#12779

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1790
commutator	C₂×Q₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×#41	C₂³×C₅₃
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×#41×C₅₃ ◁ G

732 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 generators on 85 points:

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

Conjugacy classes & character table

1802 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 5767);