C₇₉×#10676 #34122365

order	20224
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₉×#10676

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₉	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

410 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

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, 3664);