C₇×#7427 #13184643

order	1792
nilpotency class	5
isoclinism	isoclinic class of #7426
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#7427

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1351
commutator	C₂×C₂²⋊C₄	C₂×C₄×C₇
Frattini, cosocle	#381	C₂²×C₇
prime radical	C₂⁴×C₇	C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂³×C₇ ◁ C₂⁴×C₇ ◁ C₇×#381 ◁ G

214 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 39 points:

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 415);