C₇₉×#8509 #34120198

order	20224
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇₉×#8509

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₉	#120
commutator	C₄²	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

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

normal subgroups only

This group has 656 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) = 115
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 3 generators on 115 points:

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

Conjugacy classes & character table

3160 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 1497);