C₇₉×#12408 #34124097

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#12408

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₉
prime radical	C₂×D₈×C₇₉	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₄²×C₇₉ ◁ C₈×Q₈×C₇₉ ◁ G

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

normal subgroups only

This group has 310 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) = 10112

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

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

Conjugacy classes & character table

3160 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5396);