C₇×#12932 #13190148

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12932

Subgroups

	subgroup	G/H
center	C₄×C₇	#481
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1853
prime radical	C₂⁴×C₇	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×M₁₆ ◁ G

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

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 146 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)(8,9,12,16,13,18,28,17)(10,19,22,37,21,36,27,33)(11,23,26,15,25,35,38,34)(14,29,31,20,30,24,32,39);(8,10,13,21)(9,14)(11,24,25,29)(12,27,28,22)(15,33)(16,32)(17,31)(18,30)(19,23)(20,38,39,26)(34,37)(35,36);(11,25)(15,34)(19,36)(20,39)(23,35)(24,29)(26,38)(33,37)

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5920);