C₇×#13051 #13190267

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13051

Subgroups

	subgroup	G/H
center	C₄×C₇	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×#41	C₂³×C₇
socle	C₂×C₇	#1889
prime radical	C₇×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#41 ◁ G

632 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 114 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)(8,9,13,18,11,16,20,29)(10,19,23,35,21,14,28,32)(12,17,26,34,24,15,27,33)(22,30,36,38,25,31,37,39);(8,10,20,28,11,21,13,23)(9,14,29,35,16,19,18,32)(12,25,26,37,24,22,27,36)(15,30,33,38,17,31,34,39);(9,15)(12,24)(14,30)(16,17)(18,34)(19,31)(22,25)(26,27)(29,33)(32,39)(35,38)(36,37)

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, 6039);