C₇×#13665 #13190881

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13665

Subgroups

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

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

normal subgroups only

This group has 284 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) = 75
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 75 points:

(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,18,23)(14,24,28,55)(15,29,34,53)(16,35,38,69)(17,39,37,22)(19,41,44,25)(20,45,48,32)(21,49,51,31)(26,57,58,74)(27,59,40,54)(30,50,64,75)(33,67,66,61)(36,46,70,73)(42,68,71,63)(43,56,52,62)(47,65,60,72);(12,14,18,28)(13,19,23,44)(15,30,34,64)(16,36,38,70)(17,40,37,27)(20,46,48,73)(21,50,51,75)(22,52,39,43)(24,53,55,29)(25,45,41,32)(26,49,58,31)(33,68,66,63)(35,42,69,71)(47,74,60,57)(54,61,59,67)(56,65,62,72);(9,11)

Conjugacy classes & character table

343 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6653);