C₇₉×#12365 #34124054

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12364
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#12365

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#1629
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₉
prime radical	C₇₉×#3291	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₄×C₈×C₇₉ ◁ G

346 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,11)(4,16,19,32)(5,10)(6,21)(7,23)(8,14)(9,18,28,13)(12,20)(15,26)(17,29,24,25)(22,30,31,27)(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);(1,3,5,14)(2,8,10,11)(4,13,19,18)(6,20,15,7)(9,25,28,29)(12,21,23,26)(16,30,32,27)(17,31,24,22);(1,4)(2,9)(3,13)(5,19)(6,22)(7,24)(8,25)(10,28)(11,29)(12,30)(14,18)(15,31)(16,26)(17,20)(21,32)(23,27)

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, 5353);