C₁₆×Q₁₆×C₇₉ #34123741

order	20224
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₆×Q₁₆×C₇₉

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₇₉	D₄
commutator	C₄	C₂²×C₁₆×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
socle, prime radical	C₂²×C₇₉	C₈×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₇₉ ◁ C₄×C₁₆×C₇₉ ◁ G

218 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 111
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 3 generators on 111 points:

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

Conjugacy classes & character table

8848 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5040);