C₇₉×#12300 #34123989

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#12300

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₇₉
prime radical	C₇₉×#3197	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₂×M₁₆×C₇₉ ◁ G

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

normal subgroups only

This group has 328 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:

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

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, 5288);