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G = C3×C5⋊C8order 120 = 23·3·5

Direct product of C3 and C5⋊C8

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3×C5⋊C8, C5⋊C24, C152C8, C10.C12, C6.2F5, C30.2C4, Dic5.2C6, C2.(C3×F5), (C3×Dic5).4C2, SmallGroup(120,6)

Series: Derived Chief Lower central Upper central

C1C5 — C3×C5⋊C8
C1C5C10Dic5C3×Dic5 — C3×C5⋊C8
C5 — C3×C5⋊C8
C1C6

Generators and relations for C3×C5⋊C8
 G = < a,b,c | a3=b5=c8=1, ab=ba, ac=ca, cbc-1=b3 >

5C4
5C8
5C12
5C24

Character table of C3×C5⋊C8

 class 123A3B4A4B56A6B8A8B8C8D1012A12B12C12D15A15B24A24B24C24D24E24F24G24H30A30B
 size 111155411555545555445555555544
ρ1111111111111111111111111111111    trivial
ρ2111111111-1-1-1-11111111-1-1-1-1-1-1-1-111    linear of order 2
ρ311ζ32ζ3111ζ3ζ32-1-1-1-11ζ3ζ32ζ32ζ3ζ3ζ32ζ6ζ65ζ65ζ65ζ6ζ6ζ6ζ65ζ3ζ32    linear of order 6
ρ411ζ3ζ32111ζ32ζ311111ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ511ζ32ζ3111ζ3ζ3211111ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ611ζ3ζ32111ζ32ζ3-1-1-1-11ζ32ζ3ζ3ζ32ζ32ζ3ζ65ζ6ζ6ζ6ζ65ζ65ζ65ζ6ζ32ζ3    linear of order 6
ρ71111-1-1111i-ii-i1-1-1-1-111i-iii-i-ii-i11    linear of order 4
ρ81111-1-1111-ii-ii1-1-1-1-111-ii-i-iii-ii11    linear of order 4
ρ91-111i-i1-1-1ζ85ζ87ζ8ζ83-1-ii-ii11ζ85ζ87ζ8ζ85ζ83ζ87ζ8ζ83-1-1    linear of order 8
ρ101-111-ii1-1-1ζ83ζ8ζ87ζ85-1i-ii-i11ζ83ζ8ζ87ζ83ζ85ζ8ζ87ζ85-1-1    linear of order 8
ρ111-111i-i1-1-1ζ8ζ83ζ85ζ87-1-ii-ii11ζ8ζ83ζ85ζ8ζ87ζ83ζ85ζ87-1-1    linear of order 8
ρ121-111-ii1-1-1ζ87ζ85ζ83ζ8-1i-ii-i11ζ87ζ85ζ83ζ87ζ8ζ85ζ83ζ8-1-1    linear of order 8
ρ1311ζ32ζ3-1-11ζ3ζ32i-ii-i1ζ65ζ6ζ6ζ65ζ3ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ3ζ32    linear of order 12
ρ1411ζ3ζ32-1-11ζ32ζ3i-ii-i1ζ6ζ65ζ65ζ6ζ32ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ32ζ3    linear of order 12
ρ1511ζ32ζ3-1-11ζ3ζ32-ii-ii1ζ65ζ6ζ6ζ65ζ3ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ3ζ32    linear of order 12
ρ1611ζ3ζ32-1-11ζ32ζ3-ii-ii1ζ6ζ65ζ65ζ6ζ32ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ32ζ3    linear of order 12
ρ171-1ζ3ζ32-ii1ζ6ζ65ζ83ζ8ζ87ζ85-1ζ82ζ32ζ86ζ3ζ82ζ3ζ86ζ32ζ32ζ3ζ83ζ3ζ8ζ32ζ87ζ32ζ83ζ32ζ85ζ3ζ8ζ3ζ87ζ3ζ85ζ32ζ6ζ65    linear of order 24
ρ181-1ζ32ζ3-ii1ζ65ζ6ζ87ζ85ζ83ζ8-1ζ82ζ3ζ86ζ32ζ82ζ32ζ86ζ3ζ3ζ32ζ87ζ32ζ85ζ3ζ83ζ3ζ87ζ3ζ8ζ32ζ85ζ32ζ83ζ32ζ8ζ3ζ65ζ6    linear of order 24
ρ191-1ζ32ζ3i-i1ζ65ζ6ζ8ζ83ζ85ζ87-1ζ86ζ3ζ82ζ32ζ86ζ32ζ82ζ3ζ3ζ32ζ8ζ32ζ83ζ3ζ85ζ3ζ8ζ3ζ87ζ32ζ83ζ32ζ85ζ32ζ87ζ3ζ65ζ6    linear of order 24
ρ201-1ζ32ζ3-ii1ζ65ζ6ζ83ζ8ζ87ζ85-1ζ82ζ3ζ86ζ32ζ82ζ32ζ86ζ3ζ3ζ32ζ83ζ32ζ8ζ3ζ87ζ3ζ83ζ3ζ85ζ32ζ8ζ32ζ87ζ32ζ85ζ3ζ65ζ6    linear of order 24
ρ211-1ζ32ζ3i-i1ζ65ζ6ζ85ζ87ζ8ζ83-1ζ86ζ3ζ82ζ32ζ86ζ32ζ82ζ3ζ3ζ32ζ85ζ32ζ87ζ3ζ8ζ3ζ85ζ3ζ83ζ32ζ87ζ32ζ8ζ32ζ83ζ3ζ65ζ6    linear of order 24
ρ221-1ζ3ζ32-ii1ζ6ζ65ζ87ζ85ζ83ζ8-1ζ82ζ32ζ86ζ3ζ82ζ3ζ86ζ32ζ32ζ3ζ87ζ3ζ85ζ32ζ83ζ32ζ87ζ32ζ8ζ3ζ85ζ3ζ83ζ3ζ8ζ32ζ6ζ65    linear of order 24
ρ231-1ζ3ζ32i-i1ζ6ζ65ζ85ζ87ζ8ζ83-1ζ86ζ32ζ82ζ3ζ86ζ3ζ82ζ32ζ32ζ3ζ85ζ3ζ87ζ32ζ8ζ32ζ85ζ32ζ83ζ3ζ87ζ3ζ8ζ3ζ83ζ32ζ6ζ65    linear of order 24
ρ241-1ζ3ζ32i-i1ζ6ζ65ζ8ζ83ζ85ζ87-1ζ86ζ32ζ82ζ3ζ86ζ3ζ82ζ32ζ32ζ3ζ8ζ3ζ83ζ32ζ85ζ32ζ8ζ32ζ87ζ3ζ83ζ3ζ85ζ3ζ87ζ32ζ6ζ65    linear of order 24
ρ25444400-1440000-10000-1-100000000-1-1    orthogonal lifted from F5
ρ264-44400-1-4-4000010000-1-10000000011    symplectic lifted from C5⋊C8, Schur index 2
ρ2744-2-2-3-2+2-300-1-2+2-3-2-2-30000-10000ζ65ζ600000000ζ65ζ6    complex lifted from C3×F5
ρ284-4-2-2-3-2+2-300-12-2-32+2-3000010000ζ65ζ600000000ζ3ζ32    complex faithful
ρ294-4-2+2-3-2-2-300-12+2-32-2-3000010000ζ6ζ6500000000ζ32ζ3    complex faithful
ρ3044-2+2-3-2-2-300-1-2-2-3-2+2-30000-10000ζ6ζ6500000000ζ6ζ65    complex lifted from C3×F5

Smallest permutation representation of C3×C5⋊C8
Regular action on 120 points
Generators in S120
(1 100 60)(2 101 61)(3 102 62)(4 103 63)(5 104 64)(6 97 57)(7 98 58)(8 99 59)(9 86 46)(10 87 47)(11 88 48)(12 81 41)(13 82 42)(14 83 43)(15 84 44)(16 85 45)(17 90 115)(18 91 116)(19 92 117)(20 93 118)(21 94 119)(22 95 120)(23 96 113)(24 89 114)(25 105 66)(26 106 67)(27 107 68)(28 108 69)(29 109 70)(30 110 71)(31 111 72)(32 112 65)(33 53 74)(34 54 75)(35 55 76)(36 56 77)(37 49 78)(38 50 79)(39 51 80)(40 52 73)
(1 105 19 46 54)(2 47 106 55 20)(3 56 48 21 107)(4 22 49 108 41)(5 109 23 42 50)(6 43 110 51 24)(7 52 44 17 111)(8 18 53 112 45)(9 75 100 66 92)(10 67 76 93 101)(11 94 68 102 77)(12 103 95 78 69)(13 79 104 70 96)(14 71 80 89 97)(15 90 72 98 73)(16 99 91 74 65)(25 117 86 34 60)(26 35 118 61 87)(27 62 36 88 119)(28 81 63 120 37)(29 113 82 38 64)(30 39 114 57 83)(31 58 40 84 115)(32 85 59 116 33)
(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 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)

G:=sub<Sym(120)| (1,100,60)(2,101,61)(3,102,62)(4,103,63)(5,104,64)(6,97,57)(7,98,58)(8,99,59)(9,86,46)(10,87,47)(11,88,48)(12,81,41)(13,82,42)(14,83,43)(15,84,44)(16,85,45)(17,90,115)(18,91,116)(19,92,117)(20,93,118)(21,94,119)(22,95,120)(23,96,113)(24,89,114)(25,105,66)(26,106,67)(27,107,68)(28,108,69)(29,109,70)(30,110,71)(31,111,72)(32,112,65)(33,53,74)(34,54,75)(35,55,76)(36,56,77)(37,49,78)(38,50,79)(39,51,80)(40,52,73), (1,105,19,46,54)(2,47,106,55,20)(3,56,48,21,107)(4,22,49,108,41)(5,109,23,42,50)(6,43,110,51,24)(7,52,44,17,111)(8,18,53,112,45)(9,75,100,66,92)(10,67,76,93,101)(11,94,68,102,77)(12,103,95,78,69)(13,79,104,70,96)(14,71,80,89,97)(15,90,72,98,73)(16,99,91,74,65)(25,117,86,34,60)(26,35,118,61,87)(27,62,36,88,119)(28,81,63,120,37)(29,113,82,38,64)(30,39,114,57,83)(31,58,40,84,115)(32,85,59,116,33), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)>;

G:=Group( (1,100,60)(2,101,61)(3,102,62)(4,103,63)(5,104,64)(6,97,57)(7,98,58)(8,99,59)(9,86,46)(10,87,47)(11,88,48)(12,81,41)(13,82,42)(14,83,43)(15,84,44)(16,85,45)(17,90,115)(18,91,116)(19,92,117)(20,93,118)(21,94,119)(22,95,120)(23,96,113)(24,89,114)(25,105,66)(26,106,67)(27,107,68)(28,108,69)(29,109,70)(30,110,71)(31,111,72)(32,112,65)(33,53,74)(34,54,75)(35,55,76)(36,56,77)(37,49,78)(38,50,79)(39,51,80)(40,52,73), (1,105,19,46,54)(2,47,106,55,20)(3,56,48,21,107)(4,22,49,108,41)(5,109,23,42,50)(6,43,110,51,24)(7,52,44,17,111)(8,18,53,112,45)(9,75,100,66,92)(10,67,76,93,101)(11,94,68,102,77)(12,103,95,78,69)(13,79,104,70,96)(14,71,80,89,97)(15,90,72,98,73)(16,99,91,74,65)(25,117,86,34,60)(26,35,118,61,87)(27,62,36,88,119)(28,81,63,120,37)(29,113,82,38,64)(30,39,114,57,83)(31,58,40,84,115)(32,85,59,116,33), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120) );

G=PermutationGroup([(1,100,60),(2,101,61),(3,102,62),(4,103,63),(5,104,64),(6,97,57),(7,98,58),(8,99,59),(9,86,46),(10,87,47),(11,88,48),(12,81,41),(13,82,42),(14,83,43),(15,84,44),(16,85,45),(17,90,115),(18,91,116),(19,92,117),(20,93,118),(21,94,119),(22,95,120),(23,96,113),(24,89,114),(25,105,66),(26,106,67),(27,107,68),(28,108,69),(29,109,70),(30,110,71),(31,111,72),(32,112,65),(33,53,74),(34,54,75),(35,55,76),(36,56,77),(37,49,78),(38,50,79),(39,51,80),(40,52,73)], [(1,105,19,46,54),(2,47,106,55,20),(3,56,48,21,107),(4,22,49,108,41),(5,109,23,42,50),(6,43,110,51,24),(7,52,44,17,111),(8,18,53,112,45),(9,75,100,66,92),(10,67,76,93,101),(11,94,68,102,77),(12,103,95,78,69),(13,79,104,70,96),(14,71,80,89,97),(15,90,72,98,73),(16,99,91,74,65),(25,117,86,34,60),(26,35,118,61,87),(27,62,36,88,119),(28,81,63,120,37),(29,113,82,38,64),(30,39,114,57,83),(31,58,40,84,115),(32,85,59,116,33)], [(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),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120)])

C3×C5⋊C8 is a maximal subgroup of   D15⋊C8  D6.F5  Dic3.F5  SL2(𝔽3).F5

Matrix representation of C3×C5⋊C8 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
2333
6366
1332
6055
,
6404
5643
0013
6531
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,6,1,6,3,3,3,0,3,6,3,5,3,6,2,5],[6,5,0,6,4,6,0,5,0,4,1,3,4,3,3,1] >;

C3×C5⋊C8 in GAP, Magma, Sage, TeX

C_3\times C_5\rtimes C_8
% in TeX

G:=Group("C3xC5:C8");
// GroupNames label

G:=SmallGroup(120,6);
// by ID

G=gap.SmallGroup(120,6);
# by ID

G:=PCGroup([5,-2,-3,-2,-2,-5,30,42,1204,414]);
// Polycyclic

G:=Group<a,b,c|a^3=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C3×C5⋊C8 in TeX
Character table of C3×C5⋊C8 in TeX

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