C₇₉×#11922 #34123611

order	20224
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#11922

Subgroups

	subgroup	G/H
center	C₄×C₇₉	#331
commutator	C₄²	C₂²×C₄×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
socle	C₂×C₇₉	C₂×#331
prime radical	C₇₉×#3147	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×C₄×C₈×C₇₉ ◁ G

400 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

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, 4910);