C₆₁×#11402 #29514835

order	15616
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈×C₆₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₁×#11402

Subgroups

	subgroup	G/H
center	C₄×C₈×C₆₁	C₂³
commutator	C₂²	C₂×C₄×C₈×C₆₁
Frattini, cosocle	C₄×C₈	C₂³×C₆₁
socle	C₂²×C₆₁	C₂×C₄×C₈
prime radical	C₂³×C₆₁	C₄×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₆₁ ◁ G

322 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 246 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) = 7808

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

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

Conjugacy classes & character table

6832 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 4390);