C₇₉×#7649 #34119338

order	20224
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#7649

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
prime radical	C₂²×D₄×C₇₉	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₉ ◁ C₄³×C₇₉ ◁ G

468 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 10112

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

(1,2,3,6,4,5,7,8)(9,10,11,14,12,15,16,13)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31)(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);(2,5)(3,7)(17,19)(18,22)(20,28)(21,27)(23,30)(24,32)(25,31)(26,29);(9,11,12,16)(10,13,15,14)(18,23)(22,30)(24,31)(25,32)

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, 637);