C₇₉×#19008 #34130697

order	20224
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇₉×#19008

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇₉
prime radical	C₂³×C₇₉	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

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

normal subgroups only

This group has 786 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) = 127
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 4 generators on 127 points:

(1,2,6,3)(4,11,12,16)(5,10,15,8)(7,14,9,13)(17,18,22,25)(19,27,30,41)(20,24,32,39)(21,33,36,26)(23,34,38,29)(28,44,46,43)(31,42,35,40)(37,47,48,45)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(1,3,6,2)(4,9,12,7)(5,8,15,10)(11,14,16,13)(17,19,22,30)(18,23,25,38)(20,28,32,46)(21,34,36,29)(24,37,39,48)(26,41,33,27)(31,47,35,45)(40,43,42,44);(1,4)(2,7)(3,9)(5,14)(6,12)(8,16)(10,11)(13,15)(17,20,22,32)(18,24,25,39)(19,28,30,46)(21,35,36,31)(23,37,38,48)(26,42,33,40)(27,43,41,44)(29,47,34,45);(18,25)(23,38)(24,39)(26,33)(27,41)(37,48)(40,42)(43,44)

Conjugacy classes & character table

2686 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 11996);