C₅₃×#7394 #27214906

order	13568
nilpotency class	5
isoclinism	isoclinic class of #7394
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#7394

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1343
commutator	C₂×C₂²⋊C₄	C₂×C₄×C₅₃
Frattini, cosocle	#379	C₂²×C₅₃
prime radical	C₅₃×#2021	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂³×C₅₃ ◁ C₂⁴×C₅₃ ◁ C₅₃×#379 ◁ G

418 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 2 generators on 69 points:

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

Conjugacy classes & character table

1325 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 382);