C₅₃×#11544 #27219056

order	13568
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#11544

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#99
commutator	C₂⁴	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

380 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 3392

A minimal faithful permutation representation — 3 generators on 85 points:

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

Conjugacy classes & character table

1961 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 4532);