C₅₃×#7200 #27214712

order	13568
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7200

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	#95
commutator	C₂×C₄	C₄×C₈×C₅₃
Frattini, cosocle	C₂³×C₈	C₂²×C₅₃
socle	C₂²×C₅₃	#336
prime radical	C₂⁴×C₅₃	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

292 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 2 generators on 77 points:

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

Conjugacy classes & character table

3392 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 188);