direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22⋊C4×D15, D30.46D4, C23.17D30, (C2×C4)⋊5D30, (C2×C20)⋊17D6, C2.1(D4×D15), C6.96(D4×D5), D30⋊27(C2×C4), (C2×C12)⋊17D10, C10.98(S3×D4), C22⋊4(C4×D15), D30⋊3C4⋊9C2, (C2×C60)⋊15C22, C30.304(C2×D4), (C22×D15)⋊7C4, C30.38D4⋊3C2, (C23×D15).1C2, (C22×C6).55D10, (C22×C10).70D6, (C2×C30).277C23, C30.157(C22×C4), (C2×Dic15)⋊22C22, (C22×C30).11C22, C22.13(C22×D15), (C22×D15).125C22, (C2×C6)⋊5(C4×D5), C5⋊4(S3×C22⋊C4), C3⋊3(D5×C22⋊C4), C2.8(C2×C4×D15), C6.62(C2×C4×D5), (C2×C4×D15)⋊14C2, C10.94(S3×C2×C4), (C2×C10)⋊14(C4×S3), (C2×C30)⋊11(C2×C4), (C5×C22⋊C4)⋊8S3, C15⋊14(C2×C22⋊C4), (C3×C22⋊C4)⋊8D5, (C15×C22⋊C4)⋊10C2, (C2×C6).273(C22×D5), (C2×C10).272(C22×S3), SmallGroup(480,845)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×D15
G = < a,b,c,d,e | a2=b2=c4=d15=e2=1, cac-1=ab=ba, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1796 in 264 conjugacy classes, 71 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C22, C22 [×2], C22 [×20], C5, S3 [×6], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], D5 [×6], C10, C10 [×2], C10 [×2], Dic3 [×2], C12 [×2], D6 [×18], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4, C22⋊C4 [×3], C22×C4 [×2], C24, Dic5 [×2], C20 [×2], D10 [×18], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×S3 [×4], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×10], C22×C6, D15 [×4], D15 [×2], C30, C30 [×2], C30 [×2], C2×C22⋊C4, C4×D5 [×4], C2×Dic5 [×2], C2×C20 [×2], C22×D5 [×10], C22×C10, D6⋊C4 [×2], C6.D4, C3×C22⋊C4, S3×C2×C4 [×2], S3×C23, Dic15 [×2], C60 [×2], D30 [×8], D30 [×10], C2×C30, C2×C30 [×2], C2×C30 [×2], D10⋊C4 [×2], C23.D5, C5×C22⋊C4, C2×C4×D5 [×2], C23×D5, S3×C22⋊C4, C4×D15 [×4], C2×Dic15 [×2], C2×C60 [×2], C22×D15 [×2], C22×D15 [×4], C22×D15 [×4], C22×C30, D5×C22⋊C4, D30⋊3C4 [×2], C30.38D4, C15×C22⋊C4, C2×C4×D15 [×2], C23×D15, C22⋊C4×D15
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D5, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C4×S3 [×2], C22×S3, D15, C2×C22⋊C4, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D4 [×2], D30 [×3], C2×C4×D5, D4×D5 [×2], S3×C22⋊C4, C4×D15 [×2], C22×D15, D5×C22⋊C4, C2×C4×D15, D4×D15 [×2], C22⋊C4×D15
►On 120 points
Generators in S120
(1 43)(2 44)(3 45)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 58)(17 59)(18 60)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 102)(77 103)(78 104)(79 105)(80 91)(81 92)(82 93)(83 94)(84 95)(85 96)(86 97)(87 98)(88 99)(89 100)(90 101)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 16)(13 17)(14 18)(15 19)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(41 60)(42 46)(43 47)(44 48)(45 49)(61 87)(62 88)(63 89)(64 90)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)
(1 99 43 62)(2 100 44 63)(3 101 45 64)(4 102 31 65)(5 103 32 66)(6 104 33 67)(7 105 34 68)(8 91 35 69)(9 92 36 70)(10 93 37 71)(11 94 38 72)(12 95 39 73)(13 96 40 74)(14 97 41 75)(15 98 42 61)(16 110 58 84)(17 111 59 85)(18 112 60 86)(19 113 46 87)(20 114 47 88)(21 115 48 89)(22 116 49 90)(23 117 50 76)(24 118 51 77)(25 119 52 78)(26 120 53 79)(27 106 54 80)(28 107 55 81)(29 108 56 82)(30 109 57 83)
(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 19)(2 18)(3 17)(4 16)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 60)(45 59)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)(73 76)(74 90)(75 89)(91 106)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)
G:=sub<Sym(120)| (1,43)(2,44)(3,45)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,58)(17,59)(18,60)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,102)(77,103)(78,104)(79,105)(80,91)(81,92)(82,93)(83,94)(84,95)(85,96)(86,97)(87,98)(88,99)(89,100)(90,101), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,16)(13,17)(14,18)(15,19)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,46)(43,47)(44,48)(45,49)(61,87)(62,88)(63,89)(64,90)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120), (1,99,43,62)(2,100,44,63)(3,101,45,64)(4,102,31,65)(5,103,32,66)(6,104,33,67)(7,105,34,68)(8,91,35,69)(9,92,36,70)(10,93,37,71)(11,94,38,72)(12,95,39,73)(13,96,40,74)(14,97,41,75)(15,98,42,61)(16,110,58,84)(17,111,59,85)(18,112,60,86)(19,113,46,87)(20,114,47,88)(21,115,48,89)(22,116,49,90)(23,117,50,76)(24,118,51,77)(25,119,52,78)(26,120,53,79)(27,106,54,80)(28,107,55,81)(29,108,56,82)(30,109,57,83), (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), (1,19)(2,18)(3,17)(4,16)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,60)(45,59)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,90)(75,89)(91,106)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)>;
G:=Group( (1,43)(2,44)(3,45)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,58)(17,59)(18,60)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,102)(77,103)(78,104)(79,105)(80,91)(81,92)(82,93)(83,94)(84,95)(85,96)(86,97)(87,98)(88,99)(89,100)(90,101), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,16)(13,17)(14,18)(15,19)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,46)(43,47)(44,48)(45,49)(61,87)(62,88)(63,89)(64,90)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120), (1,99,43,62)(2,100,44,63)(3,101,45,64)(4,102,31,65)(5,103,32,66)(6,104,33,67)(7,105,34,68)(8,91,35,69)(9,92,36,70)(10,93,37,71)(11,94,38,72)(12,95,39,73)(13,96,40,74)(14,97,41,75)(15,98,42,61)(16,110,58,84)(17,111,59,85)(18,112,60,86)(19,113,46,87)(20,114,47,88)(21,115,48,89)(22,116,49,90)(23,117,50,76)(24,118,51,77)(25,119,52,78)(26,120,53,79)(27,106,54,80)(28,107,55,81)(29,108,56,82)(30,109,57,83), (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), (1,19)(2,18)(3,17)(4,16)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,60)(45,59)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,90)(75,89)(91,106)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107) );
G=PermutationGroup([(1,43),(2,44),(3,45),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,58),(17,59),(18,60),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(61,113),(62,114),(63,115),(64,116),(65,117),(66,118),(67,119),(68,120),(69,106),(70,107),(71,108),(72,109),(73,110),(74,111),(75,112),(76,102),(77,103),(78,104),(79,105),(80,91),(81,92),(82,93),(83,94),(84,95),(85,96),(86,97),(87,98),(88,99),(89,100),(90,101)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,16),(13,17),(14,18),(15,19),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59),(41,60),(42,46),(43,47),(44,48),(45,49),(61,87),(62,88),(63,89),(64,90),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83),(73,84),(74,85),(75,86),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119),(105,120)], [(1,99,43,62),(2,100,44,63),(3,101,45,64),(4,102,31,65),(5,103,32,66),(6,104,33,67),(7,105,34,68),(8,91,35,69),(9,92,36,70),(10,93,37,71),(11,94,38,72),(12,95,39,73),(13,96,40,74),(14,97,41,75),(15,98,42,61),(16,110,58,84),(17,111,59,85),(18,112,60,86),(19,113,46,87),(20,114,47,88),(21,115,48,89),(22,116,49,90),(23,117,50,76),(24,118,51,77),(25,119,52,78),(26,120,53,79),(27,106,54,80),(28,107,55,81),(29,108,56,82),(30,109,57,83)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,21),(15,20),(31,58),(32,57),(33,56),(34,55),(35,54),(36,53),(37,52),(38,51),(39,50),(40,49),(41,48),(42,47),(43,46),(44,60),(45,59),(61,88),(62,87),(63,86),(64,85),(65,84),(66,83),(67,82),(68,81),(69,80),(70,79),(71,78),(72,77),(73,76),(74,90),(75,89),(91,106),(92,120),(93,119),(94,118),(95,117),(96,116),(97,115),(98,114),(99,113),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107)])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 15 | 15 | 15 | 15 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D6 | D10 | D10 | C4×S3 | D15 | C4×D5 | D30 | D30 | C4×D15 | S3×D4 | D4×D5 | D4×D15 |
| kernel | C22⋊C4×D15 | D30⋊3C4 | C30.38D4 | C15×C22⋊C4 | C2×C4×D15 | C23×D15 | C22×D15 | C5×C22⋊C4 | D30 | C3×C22⋊C4 | C2×C20 | C22×C10 | C2×C12 | C22×C6 | C2×C10 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C22 | C10 | C6 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 4 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 16 | 2 | 4 | 8 |
Matrix representation of C22⋊C4×D15 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 30 | 23 | 0 | 0 | 0 | 0 |
| 2 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 37 | 8 | 0 | 0 | 0 | 0 |
| 12 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[30,2,0,0,0,0,23,28,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[37,12,0,0,0,0,8,24,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
C22⋊C4×D15 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times D_{15} % in TeX
G:=Group("C2^2:C4xD15"); // GroupNames label
G:=SmallGroup(480,845);
// by ID
G=gap.SmallGroup(480,845);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,58,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^15=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations