direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D40⋊C2, D40⋊6C6, C24⋊18D10, C120⋊18C22, C60.194C23, C8⋊3(C6×D5), C40⋊3(C2×C6), D4⋊D5⋊3C6, (D4×D5)⋊3C6, Q8⋊D5⋊2C6, Q8⋊3(C6×D5), D20⋊2(C2×C6), C8⋊D5⋊1C6, D4.3(C6×D5), (C3×D40)⋊14C2, (C3×Q8)⋊15D10, Q8⋊2D5⋊4C6, SD16⋊1(C3×D5), (C3×SD16)⋊5D5, (C5×SD16)⋊1C6, (C6×D5).72D4, C10.31(C6×D4), C6.185(D4×D5), C15⋊32(C8⋊C22), (C15×SD16)⋊5C2, (C3×D4).27D10, D10.14(C3×D4), C30.344(C2×D4), C20.5(C22×C6), (C3×D20)⋊19C22, Dic5.17(C3×D4), (C3×Dic5).78D4, (Q8×C15)⋊15C22, (D5×C12).77C22, (D4×C15).27C22, C12.194(C22×D5), C4.5(D5×C2×C6), (C3×D4×D5)⋊10C2, C5⋊3(C3×C8⋊C22), C2.19(C3×D4×D5), C5⋊2C8⋊2(C2×C6), (C5×Q8)⋊3(C2×C6), (C3×D4⋊D5)⋊11C2, (C3×C8⋊D5)⋊5C2, (C3×Q8⋊D5)⋊10C2, (C5×D4).3(C2×C6), (C4×D5).2(C2×C6), (C3×Q8⋊2D5)⋊8C2, (C3×C5⋊2C8)⋊24C22, SmallGroup(480,707)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D40⋊C2
G = < a,b,c,d | a3=b40=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b11, cd=dc >
Subgroups: 576 in 136 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C12, C12, C2×C6, C15, M4(2), D8, SD16, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C24, C24, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C3×D5, C30, C30, C8⋊C22, C5⋊2C8, C40, C4×D5, C4×D5, D20, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C3×M4(2), C3×D8, C3×SD16, C3×SD16, C6×D4, C3×C4○D4, C3×Dic5, C60, C60, C6×D5, C6×D5, C2×C30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q8⋊2D5, C3×C8⋊C22, C3×C5⋊2C8, C120, D5×C12, D5×C12, C3×D20, C3×D20, C3×C5⋊D4, D4×C15, Q8×C15, D5×C2×C6, D40⋊C2, C3×C8⋊D5, C3×D40, C3×D4⋊D5, C3×Q8⋊D5, C15×SD16, C3×D4×D5, C3×Q8⋊2D5, C3×D40⋊C2
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C8⋊C22, C22×D5, C6×D4, C6×D5, D4×D5, C3×C8⋊C22, D5×C2×C6, D40⋊C2, C3×D4×D5, C3×D40⋊C2
►On 120 points
(1 90 66)(2 91 67)(3 92 68)(4 93 69)(5 94 70)(6 95 71)(7 96 72)(8 97 73)(9 98 74)(10 99 75)(11 100 76)(12 101 77)(13 102 78)(14 103 79)(15 104 80)(16 105 41)(17 106 42)(18 107 43)(19 108 44)(20 109 45)(21 110 46)(22 111 47)(23 112 48)(24 113 49)(25 114 50)(26 115 51)(27 116 52)(28 117 53)(29 118 54)(30 119 55)(31 120 56)(32 81 57)(33 82 58)(34 83 59)(35 84 60)(36 85 61)(37 86 62)(38 87 63)(39 88 64)(40 89 65)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 65)(42 64)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(81 113)(82 112)(83 111)(84 110)(85 109)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 100)(95 99)(96 98)(114 120)(115 119)(116 118)
(1 31)(3 13)(4 24)(5 35)(7 17)(8 28)(9 39)(11 21)(12 32)(15 25)(16 36)(19 29)(20 40)(23 33)(27 37)(41 61)(42 72)(44 54)(45 65)(46 76)(48 58)(49 69)(50 80)(52 62)(53 73)(56 66)(57 77)(60 70)(64 74)(68 78)(81 101)(82 112)(84 94)(85 105)(86 116)(88 98)(89 109)(90 120)(92 102)(93 113)(96 106)(97 117)(100 110)(104 114)(108 118)
G:=sub<Sym(120)| (1,90,66)(2,91,67)(3,92,68)(4,93,69)(5,94,70)(6,95,71)(7,96,72)(8,97,73)(9,98,74)(10,99,75)(11,100,76)(12,101,77)(13,102,78)(14,103,79)(15,104,80)(16,105,41)(17,106,42)(18,107,43)(19,108,44)(20,109,45)(21,110,46)(22,111,47)(23,112,48)(24,113,49)(25,114,50)(26,115,51)(27,116,52)(28,117,53)(29,118,54)(30,119,55)(31,120,56)(32,81,57)(33,82,58)(34,83,59)(35,84,60)(36,85,61)(37,86,62)(38,87,63)(39,88,64)(40,89,65), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,65)(42,64)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(114,120)(115,119)(116,118), (1,31)(3,13)(4,24)(5,35)(7,17)(8,28)(9,39)(11,21)(12,32)(15,25)(16,36)(19,29)(20,40)(23,33)(27,37)(41,61)(42,72)(44,54)(45,65)(46,76)(48,58)(49,69)(50,80)(52,62)(53,73)(56,66)(57,77)(60,70)(64,74)(68,78)(81,101)(82,112)(84,94)(85,105)(86,116)(88,98)(89,109)(90,120)(92,102)(93,113)(96,106)(97,117)(100,110)(104,114)(108,118)>;
G:=Group( (1,90,66)(2,91,67)(3,92,68)(4,93,69)(5,94,70)(6,95,71)(7,96,72)(8,97,73)(9,98,74)(10,99,75)(11,100,76)(12,101,77)(13,102,78)(14,103,79)(15,104,80)(16,105,41)(17,106,42)(18,107,43)(19,108,44)(20,109,45)(21,110,46)(22,111,47)(23,112,48)(24,113,49)(25,114,50)(26,115,51)(27,116,52)(28,117,53)(29,118,54)(30,119,55)(31,120,56)(32,81,57)(33,82,58)(34,83,59)(35,84,60)(36,85,61)(37,86,62)(38,87,63)(39,88,64)(40,89,65), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,65)(42,64)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(114,120)(115,119)(116,118), (1,31)(3,13)(4,24)(5,35)(7,17)(8,28)(9,39)(11,21)(12,32)(15,25)(16,36)(19,29)(20,40)(23,33)(27,37)(41,61)(42,72)(44,54)(45,65)(46,76)(48,58)(49,69)(50,80)(52,62)(53,73)(56,66)(57,77)(60,70)(64,74)(68,78)(81,101)(82,112)(84,94)(85,105)(86,116)(88,98)(89,109)(90,120)(92,102)(93,113)(96,106)(97,117)(100,110)(104,114)(108,118) );
G=PermutationGroup([[(1,90,66),(2,91,67),(3,92,68),(4,93,69),(5,94,70),(6,95,71),(7,96,72),(8,97,73),(9,98,74),(10,99,75),(11,100,76),(12,101,77),(13,102,78),(14,103,79),(15,104,80),(16,105,41),(17,106,42),(18,107,43),(19,108,44),(20,109,45),(21,110,46),(22,111,47),(23,112,48),(24,113,49),(25,114,50),(26,115,51),(27,116,52),(28,117,53),(29,118,54),(30,119,55),(31,120,56),(32,81,57),(33,82,58),(34,83,59),(35,84,60),(36,85,61),(37,86,62),(38,87,63),(39,88,64),(40,89,65)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(41,65),(42,64),(43,63),(44,62),(45,61),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(81,113),(82,112),(83,111),(84,110),(85,109),(86,108),(87,107),(88,106),(89,105),(90,104),(91,103),(92,102),(93,101),(94,100),(95,99),(96,98),(114,120),(115,119),(116,118)], [(1,31),(3,13),(4,24),(5,35),(7,17),(8,28),(9,39),(11,21),(12,32),(15,25),(16,36),(19,29),(20,40),(23,33),(27,37),(41,61),(42,72),(44,54),(45,65),(46,76),(48,58),(49,69),(50,80),(52,62),(53,73),(56,66),(57,77),(60,70),(64,74),(68,78),(81,101),(82,112),(84,94),(85,105),(86,116),(88,98),(89,109),(90,120),(92,102),(93,113),(96,106),(97,117),(100,110),(104,114),(108,118)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 60E | 60F | 60G | 60H | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 4 | 10 | 20 | 20 | 1 | 1 | 2 | 4 | 10 | 2 | 2 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 4 | 20 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | D10 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | C6×D5 | C6×D5 | C6×D5 | C8⋊C22 | D4×D5 | C3×C8⋊C22 | D40⋊C2 | C3×D4×D5 | C3×D40⋊C2 |
| kernel | C3×D40⋊C2 | C3×C8⋊D5 | C3×D40 | C3×D4⋊D5 | C3×Q8⋊D5 | C15×SD16 | C3×D4×D5 | C3×Q8⋊2D5 | D40⋊C2 | C8⋊D5 | D40 | D4⋊D5 | Q8⋊D5 | C5×SD16 | D4×D5 | Q8⋊2D5 | C3×Dic5 | C6×D5 | C3×SD16 | C24 | C3×D4 | C3×Q8 | Dic5 | D10 | SD16 | C8 | D4 | Q8 | C15 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C3×D40⋊C2 ►in GL4(𝔽241) generated by
| 225 | 0 | 0 | 0 |
| 0 | 225 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 225 |
| 210 | 104 | 31 | 137 |
| 137 | 104 | 104 | 137 |
| 210 | 104 | 210 | 104 |
| 137 | 104 | 137 | 104 |
| 0 | 0 | 240 | 189 |
| 0 | 0 | 0 | 1 |
| 240 | 189 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,225,0,0,0,0,225],[210,137,210,137,104,104,104,104,31,104,210,137,137,137,104,104],[0,0,240,0,0,0,189,1,240,0,0,0,189,1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;
C3×D40⋊C2 in GAP, Magma, Sage, TeX
C_3\times D_{40}\rtimes C_2 % in TeX
G:=Group("C3xD40:C2"); // GroupNames label
G:=SmallGroup(480,707);
// by ID
G=gap.SmallGroup(480,707);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,1094,303,268,1271,648,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^40=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^11,c*d=d*c>;
// generators/relations