C₇×#12455 #13189671

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12433
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#12455

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂×C₇×D₄	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×C₈×D₄ ◁ G

206 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 896

A minimal faithful permutation representation — 3 generators on 47 points:

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

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5443);