C₇×#11920 #13189136

order	1792
nilpotency class	4
isoclinism	isoclinic class of #1357
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#11920

Subgroups

	subgroup	G/H
center	C₄×C₇	#331
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂×C₇	C₂×#331
prime radical	C₇×#3147	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

400 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 138 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)(8,9,12,16,11,15,24,31,13,18,27,34,25,17,35,33)(10,19,21,36,22,30,37,39,23,28,38,32,20,14,26,29);(8,10)(9,14)(11,20)(12,26)(13,23)(15,28)(16,32)(17,19)(18,30)(21,35)(22,25)(24,38)(27,37)(29,33)(31,39)(34,36);(8,11,13,25)(9,15,18,17)(10,20,23,22)(12,24,27,35)(14,28,30,19)(16,31,34,33)(21,26,38,37)(29,32,39,36)

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4908);