C₇×#11770 #13188986

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#11770

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1505
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂×M₁₆	C₂³×C₇
prime radical	C₇×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#405 ◁ G

316 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 94 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)(10,15)(11,24,26,14)(12,17)(13,19)(16,25,28,35)(18,21)(20,30)(22,36,37,27)(23,32)(29,33,38,39)(31,34);(9,14)(10,20)(11,25)(12,21)(15,29)(16,18)(17,28)(19,24)(22,37)(23,34)(26,35)(27,30)(31,36)(32,38);(8,10,21,34,13,23,12,20)(9,15,18,31,19,32,17,30)(11,22,35,39,26,37,25,33)(14,27,28,38,24,36,16,29)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4758);