C₇×#7066 #13184282

order	1792
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7066

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#99
commutator	C₂²×C₄	C₂×C₇×C₈
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₇
socle	C₂²×C₇	#324
prime radical	C₂³×C₇	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×C₄⋊C₈ ◁ G

176 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 2 generators on 47 points:

(1,2,3,4,5,6,7)(8,9)(10,13,15,12)(11,14)(16,17,19,24,20,25,30,38,21,23,29,35,31,36,42,45)(18,22,27,32,28,34,41,43,26,33,39,37,40,44,47,46);(9,12,14,13)(16,18,21,26)(17,22)(19,27,29,39)(20,28,31,40)(23,33)(24,37)(25,34)(30,41,42,47)(32,35)(36,44)(38,46)(43,45)

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, 54);