C₆₁×#7473 #29510906

order	15616
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#7473

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₁	D₁₆
commutator	C₁₆	C₄²×C₆₁
Frattini, cosocle	C₂²×C₁₆	C₂²×C₆₁
socle	C₂²×C₆₁	#357
prime radical	C₂³×C₆₁	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂²×C₈×C₆₁ ◁ C₂²×C₁₆×C₆₁ ◁ G

170 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 129
transitive degree	μ_t(G) = 7808

A minimal faithful permutation representation — 2 generators on 129 points:

(1,2,3,4)(5,6)(7,19)(8,25)(9,31)(10,34)(11,35)(12,18)(13,20)(14,28)(15,41)(16,26)(17,27)(21,54)(22,36)(23,57)(24,48)(29,44)(30,43)(32,61)(33,58)(37,56)(38,65)(39,53)(40,60)(42,59)(45,51)(46,49)(47,50)(52,62)(55,64)(63,67)(66,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(5,7,9,21,32,55,8,20,27,50,60,68,26,49,41,65,12,24,33,56,10,22,30,53,11,23,28,51,59,67,29,52)(6,13,15,36,42,54,40,48,35,62,25,46,34,45,31,47,18,39,44,64,16,37,14,19,17,38,43,63,61,66,58,57)

Conjugacy classes & character table

4636 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 461);