C₇₉×#13079 #34124768

order	20224
nilpotency class	4
isoclinism	isoclinic class of #13077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇₉×#13079

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#1893
commutator	C₂×C₈	C₂²×C₄×C₇₉
Frattini, cosocle	C₂²×C₈	C₂³×C₇₉
prime radical	C₇₉×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×#131×C₇₉ ◁ G

486 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

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, 6067);