direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C40⋊C4, C24⋊6F5, C120⋊6C4, C40⋊2C12, C8⋊2(C3×F5), C5⋊2C8⋊5C12, C4⋊F5.3C6, C4.8(C6×F5), (C8×D5).5C6, C15⋊3(C4.Q8), C20.8(C2×C12), C60.61(C2×C4), D10.8(C3×D4), (C6×D5).54D4, C6.15(C4⋊F5), C30.15(C4⋊C4), C12.61(C2×F5), (D5×C24).16C2, D5.1(C3×SD16), (C3×D5).7SD16, Dic5.2(C3×Q8), (C3×Dic5).13Q8, (D5×C12).127C22, C5⋊(C3×C4.Q8), C2.4(C3×C4⋊F5), C10.1(C3×C4⋊C4), (C3×C5⋊2C8)⋊15C4, (C3×C4⋊F5).7C2, (C4×D5).25(C2×C6), SmallGroup(480,273)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C40⋊C4
G = < a,b,c | a3=b40=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Subgroups: 280 in 72 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, F5, D10, C24, C24, C2×C12, C3×D5, C30, C4.Q8, C5⋊2C8, C40, C4×D5, C2×F5, C3×C4⋊C4, C2×C24, C3×Dic5, C60, C3×F5, C6×D5, C8×D5, C4⋊F5, C3×C4.Q8, C3×C5⋊2C8, C120, D5×C12, C6×F5, C40⋊C4, D5×C24, C3×C4⋊F5, C3×C40⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C4⋊C4, SD16, F5, C2×C12, C3×D4, C3×Q8, C4.Q8, C2×F5, C3×C4⋊C4, C3×SD16, C3×F5, C4⋊F5, C3×C4.Q8, C6×F5, C40⋊C4, C3×C4⋊F5, C3×C40⋊C4
►On 120 points
Generators in S120
(1 109 52)(2 110 53)(3 111 54)(4 112 55)(5 113 56)(6 114 57)(7 115 58)(8 116 59)(9 117 60)(10 118 61)(11 119 62)(12 120 63)(13 81 64)(14 82 65)(15 83 66)(16 84 67)(17 85 68)(18 86 69)(19 87 70)(20 88 71)(21 89 72)(22 90 73)(23 91 74)(24 92 75)(25 93 76)(26 94 77)(27 95 78)(28 96 79)(29 97 80)(30 98 41)(31 99 42)(32 100 43)(33 101 44)(34 102 45)(35 103 46)(36 104 47)(37 105 48)(38 106 49)(39 107 50)(40 108 51)
(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)
(1 36 21 16)(2 23 30 19)(3 10 39 22)(4 37 8 25)(5 24 17 28)(6 11 26 31)(7 38 35 34)(9 12 13 40)(14 27 18 15)(20 29 32 33)(41 70 53 74)(42 57 62 77)(43 44 71 80)(45 58 49 46)(47 72 67 52)(48 59 76 55)(50 73 54 61)(51 60 63 64)(56 75 68 79)(65 78 69 66)(81 108 117 120)(82 95 86 83)(84 109 104 89)(85 96 113 92)(87 110 91 98)(88 97 100 101)(90 111 118 107)(93 112 105 116)(94 99 114 119)(102 115 106 103)
G:=sub<Sym(120)| (1,109,52)(2,110,53)(3,111,54)(4,112,55)(5,113,56)(6,114,57)(7,115,58)(8,116,59)(9,117,60)(10,118,61)(11,119,62)(12,120,63)(13,81,64)(14,82,65)(15,83,66)(16,84,67)(17,85,68)(18,86,69)(19,87,70)(20,88,71)(21,89,72)(22,90,73)(23,91,74)(24,92,75)(25,93,76)(26,94,77)(27,95,78)(28,96,79)(29,97,80)(30,98,41)(31,99,42)(32,100,43)(33,101,44)(34,102,45)(35,103,46)(36,104,47)(37,105,48)(38,106,49)(39,107,50)(40,108,51), (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), (1,36,21,16)(2,23,30,19)(3,10,39,22)(4,37,8,25)(5,24,17,28)(6,11,26,31)(7,38,35,34)(9,12,13,40)(14,27,18,15)(20,29,32,33)(41,70,53,74)(42,57,62,77)(43,44,71,80)(45,58,49,46)(47,72,67,52)(48,59,76,55)(50,73,54,61)(51,60,63,64)(56,75,68,79)(65,78,69,66)(81,108,117,120)(82,95,86,83)(84,109,104,89)(85,96,113,92)(87,110,91,98)(88,97,100,101)(90,111,118,107)(93,112,105,116)(94,99,114,119)(102,115,106,103)>;
G:=Group( (1,109,52)(2,110,53)(3,111,54)(4,112,55)(5,113,56)(6,114,57)(7,115,58)(8,116,59)(9,117,60)(10,118,61)(11,119,62)(12,120,63)(13,81,64)(14,82,65)(15,83,66)(16,84,67)(17,85,68)(18,86,69)(19,87,70)(20,88,71)(21,89,72)(22,90,73)(23,91,74)(24,92,75)(25,93,76)(26,94,77)(27,95,78)(28,96,79)(29,97,80)(30,98,41)(31,99,42)(32,100,43)(33,101,44)(34,102,45)(35,103,46)(36,104,47)(37,105,48)(38,106,49)(39,107,50)(40,108,51), (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), (1,36,21,16)(2,23,30,19)(3,10,39,22)(4,37,8,25)(5,24,17,28)(6,11,26,31)(7,38,35,34)(9,12,13,40)(14,27,18,15)(20,29,32,33)(41,70,53,74)(42,57,62,77)(43,44,71,80)(45,58,49,46)(47,72,67,52)(48,59,76,55)(50,73,54,61)(51,60,63,64)(56,75,68,79)(65,78,69,66)(81,108,117,120)(82,95,86,83)(84,109,104,89)(85,96,113,92)(87,110,91,98)(88,97,100,101)(90,111,118,107)(93,112,105,116)(94,99,114,119)(102,115,106,103) );
G=PermutationGroup([[(1,109,52),(2,110,53),(3,111,54),(4,112,55),(5,113,56),(6,114,57),(7,115,58),(8,116,59),(9,117,60),(10,118,61),(11,119,62),(12,120,63),(13,81,64),(14,82,65),(15,83,66),(16,84,67),(17,85,68),(18,86,69),(19,87,70),(20,88,71),(21,89,72),(22,90,73),(23,91,74),(24,92,75),(25,93,76),(26,94,77),(27,95,78),(28,96,79),(29,97,80),(30,98,41),(31,99,42),(32,100,43),(33,101,44),(34,102,45),(35,103,46),(36,104,47),(37,105,48),(38,106,49),(39,107,50),(40,108,51)], [(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)], [(1,36,21,16),(2,23,30,19),(3,10,39,22),(4,37,8,25),(5,24,17,28),(6,11,26,31),(7,38,35,34),(9,12,13,40),(14,27,18,15),(20,29,32,33),(41,70,53,74),(42,57,62,77),(43,44,71,80),(45,58,49,46),(47,72,67,52),(48,59,76,55),(50,73,54,61),(51,60,63,64),(56,75,68,79),(65,78,69,66),(81,108,117,120),(82,95,86,83),(84,109,104,89),(85,96,113,92),(87,110,91,98),(88,97,100,101),(90,111,118,107),(93,112,105,116),(94,99,114,119),(102,115,106,103)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 10 | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | 20B | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 2 | 10 | 20 | 20 | 20 | 20 | 4 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 10 | 10 | 4 | 2 | 2 | 10 | 10 | 20 | ··· | 20 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | Q8 | D4 | SD16 | C3×Q8 | C3×D4 | C3×SD16 | F5 | C2×F5 | C3×F5 | C4⋊F5 | C6×F5 | C40⋊C4 | C3×C4⋊F5 | C3×C40⋊C4 |
| kernel | C3×C40⋊C4 | D5×C24 | C3×C4⋊F5 | C40⋊C4 | C3×C5⋊2C8 | C120 | C8×D5 | C4⋊F5 | C5⋊2C8 | C40 | C3×Dic5 | C6×D5 | C3×D5 | Dic5 | D10 | D5 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 4 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C3×C40⋊C4 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 222 | 19 | 0 | 0 | 0 | 0 |
| 222 | 222 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 207 | 17 | 224 |
| 0 | 0 | 34 | 207 | 224 | 0 |
| 0 | 0 | 17 | 0 | 224 | 207 |
| 0 | 0 | 34 | 224 | 17 | 207 |
| 11 | 230 | 0 | 0 | 0 | 0 |
| 230 | 230 | 0 | 0 | 0 | 0 |
| 0 | 0 | 207 | 34 | 17 | 0 |
| 0 | 0 | 224 | 34 | 0 | 207 |
| 0 | 0 | 207 | 0 | 34 | 224 |
| 0 | 0 | 0 | 17 | 34 | 207 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[222,222,0,0,0,0,19,222,0,0,0,0,0,0,0,34,17,34,0,0,207,207,0,224,0,0,17,224,224,17,0,0,224,0,207,207],[11,230,0,0,0,0,230,230,0,0,0,0,0,0,207,224,207,0,0,0,34,34,0,17,0,0,17,0,34,34,0,0,0,207,224,207] >;
C3×C40⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_{40}\rtimes C_4 % in TeX
G:=Group("C3xC40:C4"); // GroupNames label
G:=SmallGroup(480,273);
// by ID
G=gap.SmallGroup(480,273);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,176,2524,102,9414,1595]);
// Polycyclic
G:=Group<a,b,c|a^3=b^40=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations