C₇₉×#7466 #34119155

order	20224
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#7466

Subgroups

	subgroup	G/H
center	C₂×C₈×C₇₉	C₄²
commutator	C₄	C₄×C₁₆×C₇₉
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇₉
socle	C₂²×C₇₉	#346
prime radical	C₂³×C₇₉	M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₇₉ ◁ G

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

normal subgroups only

This group has 182 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:

(1,2)(3,4)(5,6,8,13,9,15,20,28,7,12,16,25,17,26,30,35,10,14,19,27,21,29,32,36,18,11,22,23,31,24,33,34)(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);(2,4)(6,11,14,12)(8,19)(13,25,27,23)(15,24,29,26)(16,22)(20,32)(28,35,36,34)(30,33)

Conjugacy classes & character table

6952 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 454);