C₇₉×#12119 #34123808

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12117
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#12119

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₇₉
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₂×M₁₆×C₇₉ ◁ G

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

normal subgroups only

This group has 404 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)(3,12)(4,7)(5,21)(6,11)(8,14)(9,17)(10,18)(13,25)(15,27)(16,26)(19,23)(20,24)(22,29)(28,30)(31,32)(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);(2,7)(4,17)(5,18)(8,22)(9,10)(11,24)(12,25)(13,29)(14,15)(19,20)(21,23)(26,28)(27,32)(30,31);(1,3,6,16)(2,8,11,27)(4,13,20,30)(5,15,18,14)(7,22,24,32)(9,28,23,25)(10,26,21,12)(17,29,19,31)

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, 5107);