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G = C19⋊C12order 228 = 22·3·19

The semidirect product of C19 and C12 acting via C12/C2=C6

metacyclic, supersoluble, monomial, Z-group

Aliases: C19⋊C12, C38.C6, Dic19⋊C3, C19⋊C3⋊C4, C2.(C19⋊C6), (C2×C19⋊C3).C2, SmallGroup(228,1)

Series: Derived Chief Lower central Upper central

C1C19 — C19⋊C12
C1C19C38C2×C19⋊C3 — C19⋊C12
C19 — C19⋊C12
C1C2

Generators and relations for C19⋊C12
 G = < a,b | a19=b12=1, bab-1=a8 >

19C3
19C4
19C6
19C12

Character table of C19⋊C12

 class 123A3B4A4B6A6B12A12B12C12D19A19B19C38A38B38C
 size 1119191919191919191919666666
ρ1111111111111111111    trivial
ρ21111-1-111-1-1-1-1111111    linear of order 2
ρ311ζ3ζ3211ζ3ζ32ζ32ζ3ζ3ζ32111111    linear of order 3
ρ411ζ32ζ311ζ32ζ3ζ3ζ32ζ32ζ3111111    linear of order 3
ρ511ζ32ζ3-1-1ζ32ζ3ζ65ζ6ζ6ζ65111111    linear of order 6
ρ611ζ3ζ32-1-1ζ3ζ32ζ6ζ65ζ65ζ6111111    linear of order 6
ρ71-111-ii-1-1-ii-ii111-1-1-1    linear of order 4
ρ81-111i-i-1-1i-ii-i111-1-1-1    linear of order 4
ρ91-1ζ32ζ3-iiζ6ζ65ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3111-1-1-1    linear of order 12
ρ101-1ζ3ζ32-iiζ65ζ6ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32111-1-1-1    linear of order 12
ρ111-1ζ3ζ32i-iζ65ζ6ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32111-1-1-1    linear of order 12
ρ121-1ζ32ζ3i-iζ6ζ65ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3111-1-1-1    linear of order 12
ρ13660000000000ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194    orthogonal lifted from C19⋊C6
ρ14660000000000ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192    orthogonal lifted from C19⋊C6
ρ15660000000000ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719    orthogonal lifted from C19⋊C6
ρ166-60000000000ζ191719161914195193192ζ191519131910199196194ζ1918191219111981971919171916191419519319219151913191019919619419181912191119819719    symplectic faithful, Schur index 2
ρ176-60000000000ζ191519131910199196194ζ19181912191119819719ζ19171916191419519319219151913191019919619419181912191119819719191719161914195193192    symplectic faithful, Schur index 2
ρ186-60000000000ζ19181912191119819719ζ191719161914195193192ζ19151913191019919619419181912191119819719191719161914195193192191519131910199196194    symplectic faithful, Schur index 2

Smallest permutation representation of C19⋊C12
On 76 points
Generators in S76
(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)(39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57)(58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76)
(1 58 20 39)(2 70 31 57 8 66 21 51 12 76 27 47)(3 63 23 56 15 74 22 44 4 75 34 55)(5 68 26 54 10 71 24 49 7 73 29 52)(6 61 37 53 17 60 25 42 18 72 36 41)(9 59 32 50 19 65 28 40 13 69 38 46)(11 64 35 48 14 62 30 45 16 67 33 43)

G:=sub<Sym(76)| (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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76), (1,58,20,39)(2,70,31,57,8,66,21,51,12,76,27,47)(3,63,23,56,15,74,22,44,4,75,34,55)(5,68,26,54,10,71,24,49,7,73,29,52)(6,61,37,53,17,60,25,42,18,72,36,41)(9,59,32,50,19,65,28,40,13,69,38,46)(11,64,35,48,14,62,30,45,16,67,33,43)>;

G:=Group( (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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76), (1,58,20,39)(2,70,31,57,8,66,21,51,12,76,27,47)(3,63,23,56,15,74,22,44,4,75,34,55)(5,68,26,54,10,71,24,49,7,73,29,52)(6,61,37,53,17,60,25,42,18,72,36,41)(9,59,32,50,19,65,28,40,13,69,38,46)(11,64,35,48,14,62,30,45,16,67,33,43) );

G=PermutationGroup([(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),(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57),(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76)], [(1,58,20,39),(2,70,31,57,8,66,21,51,12,76,27,47),(3,63,23,56,15,74,22,44,4,75,34,55),(5,68,26,54,10,71,24,49,7,73,29,52),(6,61,37,53,17,60,25,42,18,72,36,41),(9,59,32,50,19,65,28,40,13,69,38,46),(11,64,35,48,14,62,30,45,16,67,33,43)])

C19⋊C12 is a maximal subgroup of   Dic38⋊C3  C4×C19⋊C6  D38⋊C6
C19⋊C12 is a maximal quotient of   C19⋊C24

Matrix representation of C19⋊C12 in GL6(𝔽229)

010000
001000
000100
000010
000001
22810817817108
,
10115817018111479
14109200101113
207219135191176197
150115485971137
117831859222773
56561241377956

G:=sub<GL(6,GF(229))| [0,0,0,0,0,228,1,0,0,0,0,108,0,1,0,0,0,17,0,0,1,0,0,8,0,0,0,1,0,17,0,0,0,0,1,108],[101,14,207,150,117,56,158,109,219,115,83,56,170,200,135,48,185,124,181,101,191,59,92,137,114,1,176,71,227,79,79,13,197,137,73,56] >;

C19⋊C12 in GAP, Magma, Sage, TeX

C_{19}\rtimes C_{12}
% in TeX

G:=Group("C19:C12");
// GroupNames label

G:=SmallGroup(228,1);
// by ID

G=gap.SmallGroup(228,1);
# by ID

G:=PCGroup([4,-2,-3,-2,-19,24,3459,679]);
// Polycyclic

G:=Group<a,b|a^19=b^12=1,b*a*b^-1=a^8>;
// generators/relations

Export

Subgroup lattice of C19⋊C12 in TeX
Character table of C19⋊C12 in TeX

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