C₇×#11892 #13189108

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#11892

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂⁴×C₇	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

396 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 3 generators on 47 points:

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

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, 4880);