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G = C2×C19⋊C3order 114 = 2·3·19

Direct product of C2 and C19⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C2×C19⋊C3, C38⋊C3, C192C6, SmallGroup(114,2)

Series: Derived Chief Lower central Upper central

C1C19 — C2×C19⋊C3
C1C19C19⋊C3 — C2×C19⋊C3
C19 — C2×C19⋊C3
C1C2

Generators and relations for C2×C19⋊C3
 G = < a,b,c | a2=b19=c3=1, ab=ba, ac=ca, cbc-1=b11 >

19C3
19C6

Character table of C2×C19⋊C3

 class 123A3B6A6B19A19B19C19D19E19F38A38B38C38D38E38F
 size 1119191919333333333333
ρ1111111111111111111    trivial
ρ21-111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ31-1ζ3ζ32ζ6ζ65111111-1-1-1-1-1-1    linear of order 6
ρ411ζ32ζ3ζ3ζ32111111111111    linear of order 3
ρ51-1ζ32ζ3ζ65ζ6111111-1-1-1-1-1-1    linear of order 6
ρ611ζ3ζ32ζ32ζ3111111111111    linear of order 3
ρ7330000ζ199196194ζ19181912198ζ191519131910ζ1914193192ζ19171916195ζ191119719ζ19181912198ζ191519131910ζ19171916195ζ191119719ζ1914193192ζ199196194    complex lifted from C19⋊C3
ρ8330000ζ191119719ζ1914193192ζ19181912198ζ191519131910ζ199196194ζ19171916195ζ1914193192ζ19181912198ζ199196194ζ19171916195ζ191519131910ζ191119719    complex lifted from C19⋊C3
ρ93-30000ζ19171916195ζ191519131910ζ1914193192ζ19181912198ζ191119719ζ19919619419151913191019141931921911197191991961941918191219819171916195    complex faithful
ρ103-30000ζ191519131910ζ191119719ζ199196194ζ19171916195ζ1914193192ζ1918191219819111971919919619419141931921918191219819171916195191519131910    complex faithful
ρ113-30000ζ19181912198ζ19171916195ζ191119719ζ199196194ζ191519131910ζ191419319219171916195191119719191519131910191419319219919619419181912198    complex faithful
ρ12330000ζ19181912198ζ19171916195ζ191119719ζ199196194ζ191519131910ζ1914193192ζ19171916195ζ191119719ζ191519131910ζ1914193192ζ199196194ζ19181912198    complex lifted from C19⋊C3
ρ133-30000ζ191119719ζ1914193192ζ19181912198ζ191519131910ζ199196194ζ1917191619519141931921918191219819919619419171916195191519131910191119719    complex faithful
ρ143-30000ζ199196194ζ19181912198ζ191519131910ζ1914193192ζ19171916195ζ19111971919181912198191519131910191719161951911197191914193192199196194    complex faithful
ρ15330000ζ19171916195ζ191519131910ζ1914193192ζ19181912198ζ191119719ζ199196194ζ191519131910ζ1914193192ζ191119719ζ199196194ζ19181912198ζ19171916195    complex lifted from C19⋊C3
ρ16330000ζ1914193192ζ199196194ζ19171916195ζ191119719ζ19181912198ζ191519131910ζ199196194ζ19171916195ζ19181912198ζ191519131910ζ191119719ζ1914193192    complex lifted from C19⋊C3
ρ17330000ζ191519131910ζ191119719ζ199196194ζ19171916195ζ1914193192ζ19181912198ζ191119719ζ199196194ζ1914193192ζ19181912198ζ19171916195ζ191519131910    complex lifted from C19⋊C3
ρ183-30000ζ1914193192ζ199196194ζ19171916195ζ191119719ζ19181912198ζ19151913191019919619419171916195191819121981915191319101911197191914193192    complex faithful

Smallest permutation representation of C2×C19⋊C3
On 38 points
Generators in S38
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19)(20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38)
(2 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)(21 27 31)(22 34 23)(24 29 26)(25 36 37)(28 38 32)(30 33 35)

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

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

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

C2×C19⋊C3 is a maximal subgroup of   C19⋊C12  C38.A4

Matrix representation of C2×C19⋊C3 in GL3(𝔽7) generated by

600
060
006
,
634
564
335
,
114
020
024
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[6,5,3,3,6,3,4,4,5],[1,0,0,1,2,2,4,0,4] >;

C2×C19⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_{19}\rtimes C_3
% in TeX

G:=Group("C2xC19:C3");
// GroupNames label

G:=SmallGroup(114,2);
// by ID

G=gap.SmallGroup(114,2);
# by ID

G:=PCGroup([3,-2,-3,-19,194]);
// Polycyclic

G:=Group<a,b,c|a^2=b^19=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^11>;
// generators/relations

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Subgroup lattice of C2×C19⋊C3 in TeX
Character table of C2×C19⋊C3 in TeX

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