C₇₉×#13565 #34125254

order	20224
nilpotency class	5
isoclinism	isoclinic class of #13561
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#13565

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₇₉
Frattini, cosocle	#124	C₂³×C₇₉
prime radical	C₂×H(5,𝔽₂)×C₇₉	C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂³×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×#124×C₇₉ ◁ G

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

normal subgroups only

This group has 496 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:

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

Conjugacy classes & character table

2212 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6553);