C₇×#7160 #13184376

order	1792
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₄×C₇×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7160

Subgroups

	subgroup	G/H
center	C₂×C₇×C₈	D₈
commutator	C₈	C₄×C₇×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₇
socle	C₂²×C₇	#335
prime radical	C₂³×C₇	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₂²×C₇×C₈ ◁ C₂×C₄×C₇×C₈ ◁ G

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

normal subgroups only

This group has 210 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 — 2 generators on 43 points:

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

Conjugacy classes & character table

616 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 148);