C₂×C₇₉×#1407 #34125265

order	20224
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×C₇₉×#1407

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#353
commutator	C₄²	C₂²×C₄×C₇₉
Frattini, cosocle	#124	C₂³×C₇₉
prime radical	C₂²×D₄×C₇₉	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂³×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×#124×C₇₉ ◁ G

416 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 113
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 113 points:

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

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, 6564);