C₇₉×#7062 #34118751

order	20224
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#7062

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₉	#99
commutator	C₂×C₄	C₂×C₁₆×C₇₉
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₇₉
socle	C₂²×C₇₉	#324
prime radical	C₂³×C₇₉	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×C₄⋊C₈×C₇₉ ◁ G

180 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 5056

A minimal faithful permutation representation — 2 generators on 111 points:

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

Conjugacy classes & character table

4108 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 50);