C₇₉×#7319 #34119008

order	20224
nilpotency class	4
isoclinism	isoclinic class of #7318
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#7319

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#1304
commutator	C₂²×C₄	C₄²×C₇₉
Frattini, cosocle	C₂²×#41	C₂²×C₇₉
prime radical	C₂²×#41×C₇₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂³×C₇₉ ◁ C₂²×#41×C₇₉ ◁ G

456 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 2 generators on 111 points:

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

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, 307);