C₇×#9022 #13186238

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#9022

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	#120
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇ ◁ G

572 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 43
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 43 points:

(1,2,3,4,5,6,7)(9,11)(12,13,16,20,17,21,15,19)(14,22,23,18,25,26,24,27)(28,29)(30,33)(31,38,39,34)(32,35)(36,40,42,43)(37,41);(8,9,10,11)(12,14)(13,18)(15,23)(16,24)(17,25)(19,26)(20,22)(21,27)(28,30,32,37)(29,33,35,41)(31,36,39,42)(34,40,38,43);(12,15,17,16)(13,19,21,20)(14,23,25,24)(18,26,27,22)(28,31)(29,34)(30,36)(32,39)(33,40)(35,38)(37,42)(41,43)

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 2010);