C₇₉×#12303 #34123992

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#12303

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₇₉
prime radical	C₇₉×#3204	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₂×M₁₆×C₇₉ ◁ G

322 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 111
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 3 generators on 111 points:

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

Conjugacy classes & character table

1975 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5291);