C₅₃×#12778 #27220290

order	13568
nilpotency class	4
isoclinism	isoclinic class of #12778
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅₃×#12778

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1790
commutator	C₂×Q₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×#41	C₂³×C₅₃
prime radical	C₅₃×#3549	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×#41×C₅₃ ◁ G

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

normal subgroups only

This group has 860 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) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,6)(4,9,12,15)(5,10,11,8)(7,14,16,13)(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);(2,6)(3,11)(7,15)(8,10)(9,16)(12,14);(2,7)(3,11)(4,13)(6,15)(8,16)(9,10)

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, 5766);