C₇₉×#12892 #34124581

order	20224
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₉×#12892

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇₉
Frattini, cosocle	C₂²×C₈	C₂³×C₇₉
socle	C₂²×C₇₉	C₂×#107
prime radical	C₂³×C₇₉	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ C₂×C₄×C₈×C₇₉ ◁ G

274 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 158 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)(3,5)(4,6)(7,8)(9,10,15,22,17,23,16,35)(11,19,26,39,28,40,27,36)(12,29,30,38,24,18,25,37)(13,31,14,34,33,21,32,20)(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);(1,3,4,7)(2,5,6,8)(9,11,14,25,16,27,13,24,17,28,32,30,15,26,33,12)(10,18,21,36,22,37,20,19,23,29,31,39,35,38,34,40);(2,6)(5,8)(9,12,13,27,15,30,14,11,17,24,33,26,16,25,32,28)(10,19,20,38,35,36,21,29,23,40,34,37,22,39,31,18)

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, 5880);