C₇×#13642 #13190858

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#13642

Subgroups

	subgroup	G/H
center	C₇×C₈	#117
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂×C₇	#2163
prime radical	C₂³×C₇	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ C₇×#346 ◁ G

170 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

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

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6630);