C₇₉×#11486 #34123175

order	20224
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#11486

Subgroups

	subgroup	G/H
center	C₄×C₇₉	C₂×#99
commutator	C₂×C₄	C₂²×C₈×C₇₉
Frattini, cosocle	C₂×M₁₆	C₂³×C₇₉
socle	C₂×C₇₉	#1459
prime radical	#532×C₇₉	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₉ ◁ C₂×C₈×C₇₉ ◁ C₂×#131×C₇₉ ◁ G

364 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

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, 4474);