C₇₉×#8326 #34120015

order	20224
nilpotency class	3
isoclinism	isoclinic class of #8325
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇₉×#8326

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#379
commutator	C₂²×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
prime radical	C₂×#540×C₇₉	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

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

normal subgroups only

This group has 812 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) = 111
transitive degree	μ_t(G) = 5056

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

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

Conjugacy classes & character table

3634 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 1314);