C₇₉×#12047 #34123736

order	20224
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#12047

Subgroups

	subgroup	G/H
center	C₈×C₇₉	C₂×D₈
commutator	C₈	C₂²×C₈×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
socle	C₂×C₇₉	C₈×D₈
prime radical	#508×C₇₉	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₇₉ ◁ C₂×C₁₆×C₇₉ ◁ C₄×C₁₆×C₇₉ ◁ G

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

normal subgroups only

This group has 212 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) = 2528

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

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

Conjugacy classes & character table

5056 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5035);