C₇×#12216 #13189432

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12090
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12216

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇
prime radical	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂×C₇×C₂²⋊C₄ ◁ G

388 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 74 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)(8,9,14,20)(10,23,27,39)(11,28,30,37)(12,32,25,18)(13,19,33,35)(15,22,24,21)(16,31,36,26)(17,34,38,29);(8,10,24,33)(9,16,21,28)(11,25,26,34)(12,31,29,30)(13,14,27,15)(17,35,32,23)(18,39,38,19)(20,36,22,37);(8,11,15,31)(9,17,22,18)(10,25,13,29)(12,33,34,27)(14,30,24,26)(16,35,37,39)(19,28,23,36)(20,38,21,32)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5204);