C₇₉×#13589 #34125278

order	20224
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇₉×#13589

Subgroups

	subgroup	G/H
center	C₄×C₇₉	#353
commutator	C₄²	C₂²×C₄×C₇₉
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇₉
socle	C₂×C₇₉	C₂×#353
prime radical	C₇₉×#3147	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ #532×C₇₉ ◁ G

408 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Conjugacy classes & character table

2686 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6577);