C₇×#7069 #13184285

order	1792
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#7069

Subgroups

	subgroup	G/H
center	C₂×C₇×C₈	D₈
commutator	C₈	C₂×C₇×C₁₆
Frattini, cosocle	C₈²	C₂²×C₇
socle, prime radical	C₂²×C₇	D₄⋊C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₄×C₇×C₈ ◁ C₇×C₈² ◁ G

146 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 39
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 2 generators on 39 points:

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

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, 57);