C₆₁×#11416 #29514849

order	15616
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₆₁ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₆₁×#11416

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₆₁
Frattini, cosocle	C₄×C₈	C₂³×C₆₁
socle	C₂²×C₆₁	C₂×#97
prime radical	C₂³×C₆₁	#97
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ G

228 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 109
transitive degree	μ_t(G) = 7808

A minimal faithful permutation representation — 3 generators on 109 points:

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

Conjugacy classes & character table

3904 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 4404);