C₇₉×#10664 #34122353

order	20224
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₉×#10664

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₉	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
socle	C₂²×C₇₉	#399
prime radical	C₂³×C₇₉	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

304 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 10112

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

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

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, 3652);