C₇₉×#7467 #34119156

order	20224
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#7467

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₇₉	D₄
commutator	C₄	C₄×C₁₆×C₇₉
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇₉
socle	C₂²×C₇₉	C₄⋊C₁₆
prime radical	C₂³×C₇₉	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₇₉ ◁ C₂²×C₁₆×C₇₉ ◁ G

190 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 115
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 2 generators on 115 points:

(5,6,8,13,9,15,19,28,7,11,16,22,17,23,30,33,10,14,20,27,21,29,32,36,18,12,24,25,31,26,34,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,2,3,4)(6,11,14,12)(13,22,27,25)(15,23,29,26)(28,33,36,35)

Conjugacy classes & character table

8848 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 455);