C₇×#12157 #13189373

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12155
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12157

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₇
prime radical	C₇×#537	C₂²
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂×C₇×C₄⋊C₄ ◁ G

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

normal subgroups only

This group has 372 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) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(8,9)(10,12)(11,14)(13,15)(16,17)(18,22)(19,29)(20,33)(21,25)(23,32)(24,34)(26,41)(27,40)(28,37)(30,45)(31,44)(35,42)(36,43)(38,46)(39,47);(9,12)(10,15)(13,14)(16,18,21,28)(17,22,25,37)(19,30,31,26)(20,27,34,43)(23,38,39,35)(24,40,33,36)(29,41,44,45)(32,46,47,42);(16,19)(17,23,25,39)(18,26)(20,32)(21,31)(22,35,37,38)(24,29,33,44)(27,42)(28,30)(34,47)(36,41,40,45)(43,46)

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, 5145);