C₇×#13617 #13190833

order	1792
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#13617

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
prime radical	C₂²×C₇×D₁₆	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

330 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 126 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)(8,9,10,11)(12,13)(14,20)(15,19)(16,30)(17,32)(18,23)(21,26)(22,27)(24,33)(25,39)(28,36)(29,35)(31,43)(34,40)(37,41)(38,42);(13,19)(14,17)(15,26)(16,27)(20,36)(21,22)(23,35)(24,40)(25,31)(28,41)(29,42)(30,34)(32,33)(37,43)(38,39);(12,14,18,17)(13,20,23,32)(15,24,29,28)(16,25,31,27)(19,33,35,36)(21,37,38,34)(22,30,39,43)(26,41,42,40)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6605);