C₇×#13049 #13190265

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#13049

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×D₄	C₂³×C₇
prime radical	C₂×C₇×#453	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂²×C₇×D₄ ◁ G

796 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 27
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3,4,5,6,7)(9,11)(12,13)(14,17)(15,19,24,18)(16,20)(21,26)(22,25,23,27);(8,9,10,11)(12,14)(13,17)(15,23)(16,21)(18,27)(19,25)(20,26)(22,24);(13,18)(14,21)(15,24)(17,25)(19,20)(26,27)

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, 6037);