C₇₉×#11490 #34123179

order	20224
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₉×#11490

Subgroups

	subgroup	G/H
center	C₂×C₈×C₇₉	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₇₉
Frattini, cosocle	C₂²×C₈	C₂³×C₇₉
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
prime radical	#532×C₇₉	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₇₉ ◁ C₂×C₄×C₈×C₇₉ ◁ G

364 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 119
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 119 points:

(1,2,3,6,4,5,7,8)(9,10,12,17,14,19,27,35,15,20,28,36,13,18,26,34)(11,21,24,39,23,33,40,31,25,37,22,38,29,16,30,32)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(2,5)(3,7)(9,11)(10,16)(12,22)(13,29)(14,23)(15,25)(17,31)(18,37)(19,21)(20,33)(24,28)(26,40)(27,30)(32,36)(34,39)(35,38);(9,12,14,27,15,28,13,26)(10,17,19,35,20,36,18,34)(11,22,23,30,25,24,29,40)(16,31,21,38,33,32,37,39)

Conjugacy classes & character table

6004 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 4478);