C₇₉×#12593 #34124282

order	20224
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈×C₇₉ · Schur multiplier (H₂) = C₄²

Aliases

direct product

C₇₉×#12593

Subgroups

	subgroup	G/H
center	C₂×C₈×C₇₉	C₄²
commutator	C₄	C₂×C₄×C₈×C₇₉
Frattini, cosocle	C₂²×C₈	C₂³×C₇₉
socle	C₂²×C₇₉	C₂×#97
prime radical	C₂³×C₇₉	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₇₉ ◁ G

338 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 5056

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

(1,2,3,5,4,6,7,8)(9,10,12,16,13,17,24,31,14,18,25,32,26,33,39,40)(11,19,22,35,21,30,38,28,23,36,20,34,37,15,27,29)(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);(9,11)(10,15,18,30)(12,20)(13,21)(14,23)(16,28,32,29)(17,19,33,36)(22,25)(24,27)(26,37)(31,34,40,35)(38,39);(9,12,13,24,14,25,26,39)(10,16,17,31,18,32,33,40)(11,20,21,27,23,22,37,38)(15,28,19,34,30,29,36,35)

Conjugacy classes & character table

6952 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5581);