direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C15×C4≀C2, D4⋊2C60, Q8⋊3C60, C42⋊6C30, C60.251D4, M4(2)⋊4C30, (C4×C20)⋊16C6, (C4×C60)⋊24C2, (C3×D4)⋊5C20, (C5×D4)⋊8C12, C4.3(C2×C60), (C3×Q8)⋊5C20, (C4×C12)⋊10C10, (D4×C15)⋊17C4, (C5×Q8)⋊11C12, (Q8×C15)⋊17C4, C4○D4.3C30, C12.66(C5×D4), C4.17(D4×C15), C20.66(C3×D4), C20.52(C2×C12), C60.227(C2×C4), C12.31(C2×C20), (C2×C30).129D4, C22.3(D4×C15), (C5×M4(2))⋊10C6, (C3×M4(2))⋊10C10, (C15×M4(2))⋊22C2, (C2×C60).571C22, C30.130(C22⋊C4), (C5×C4○D4).8C6, (C2×C6).22(C5×D4), (C2×C4).18(C2×C30), (C3×C4○D4).4C10, (C2×C10).23(C3×D4), C2.8(C15×C22⋊C4), C6.26(C5×C22⋊C4), (C2×C20).119(C2×C6), (C15×C4○D4).10C2, C10.37(C3×C22⋊C4), (C2×C12).122(C2×C10), SmallGroup(480,207)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C4≀C2
G = < a,b,c,d | a15=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >
Subgroups: 136 in 88 conjugacy classes, 48 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C10, C10, C12, C12, C2×C6, C2×C6, C15, C42, M4(2), C4○D4, C20, C20, C2×C10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C30, C30, C4≀C2, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C4×C12, C3×M4(2), C3×C4○D4, C60, C60, C2×C30, C2×C30, C4×C20, C5×M4(2), C5×C4○D4, C3×C4≀C2, C120, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, C5×C4≀C2, C4×C60, C15×M4(2), C15×C4○D4, C15×C4≀C2
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, D4, C10, C12, C2×C6, C15, C22⋊C4, C20, C2×C10, C2×C12, C3×D4, C30, C4≀C2, C2×C20, C5×D4, C3×C22⋊C4, C60, C2×C30, C5×C22⋊C4, C3×C4≀C2, C2×C60, D4×C15, C5×C4≀C2, C15×C22⋊C4, C15×C4≀C2
►On 120 points
Generators in S120
(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 67 55 21)(2 68 56 22)(3 69 57 23)(4 70 58 24)(5 71 59 25)(6 72 60 26)(7 73 46 27)(8 74 47 28)(9 75 48 29)(10 61 49 30)(11 62 50 16)(12 63 51 17)(13 64 52 18)(14 65 53 19)(15 66 54 20)(31 96 120 87)(32 97 106 88)(33 98 107 89)(34 99 108 90)(35 100 109 76)(36 101 110 77)(37 102 111 78)(38 103 112 79)(39 104 113 80)(40 105 114 81)(41 91 115 82)(42 92 116 83)(43 93 117 84)(44 94 118 85)(45 95 119 86)
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 106)(13 107)(14 108)(15 109)(16 87)(17 88)(18 89)(19 90)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(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 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 105)(72 91)(73 92)(74 93)(75 94)
(1 21 55 67)(2 22 56 68)(3 23 57 69)(4 24 58 70)(5 25 59 71)(6 26 60 72)(7 27 46 73)(8 28 47 74)(9 29 48 75)(10 30 49 61)(11 16 50 62)(12 17 51 63)(13 18 52 64)(14 19 53 65)(15 20 54 66)
G:=sub<Sym(120)| (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), (1,67,55,21)(2,68,56,22)(3,69,57,23)(4,70,58,24)(5,71,59,25)(6,72,60,26)(7,73,46,27)(8,74,47,28)(9,75,48,29)(10,61,49,30)(11,62,50,16)(12,63,51,17)(13,64,52,18)(14,65,53,19)(15,66,54,20)(31,96,120,87)(32,97,106,88)(33,98,107,89)(34,99,108,90)(35,100,109,76)(36,101,110,77)(37,102,111,78)(38,103,112,79)(39,104,113,80)(40,105,114,81)(41,91,115,82)(42,92,116,83)(43,93,117,84)(44,94,118,85)(45,95,119,86), (1,110)(2,111)(3,112)(4,113)(5,114)(6,115)(7,116)(8,117)(9,118)(10,119)(11,120)(12,106)(13,107)(14,108)(15,109)(16,87)(17,88)(18,89)(19,90)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(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)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,105)(72,91)(73,92)(74,93)(75,94), (1,21,55,67)(2,22,56,68)(3,23,57,69)(4,24,58,70)(5,25,59,71)(6,26,60,72)(7,27,46,73)(8,28,47,74)(9,29,48,75)(10,30,49,61)(11,16,50,62)(12,17,51,63)(13,18,52,64)(14,19,53,65)(15,20,54,66)>;
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)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(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), (1,67,55,21)(2,68,56,22)(3,69,57,23)(4,70,58,24)(5,71,59,25)(6,72,60,26)(7,73,46,27)(8,74,47,28)(9,75,48,29)(10,61,49,30)(11,62,50,16)(12,63,51,17)(13,64,52,18)(14,65,53,19)(15,66,54,20)(31,96,120,87)(32,97,106,88)(33,98,107,89)(34,99,108,90)(35,100,109,76)(36,101,110,77)(37,102,111,78)(38,103,112,79)(39,104,113,80)(40,105,114,81)(41,91,115,82)(42,92,116,83)(43,93,117,84)(44,94,118,85)(45,95,119,86), (1,110)(2,111)(3,112)(4,113)(5,114)(6,115)(7,116)(8,117)(9,118)(10,119)(11,120)(12,106)(13,107)(14,108)(15,109)(16,87)(17,88)(18,89)(19,90)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(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)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,105)(72,91)(73,92)(74,93)(75,94), (1,21,55,67)(2,22,56,68)(3,23,57,69)(4,24,58,70)(5,25,59,71)(6,26,60,72)(7,27,46,73)(8,28,47,74)(9,29,48,75)(10,30,49,61)(11,16,50,62)(12,17,51,63)(13,18,52,64)(14,19,53,65)(15,20,54,66) );
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),(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90),(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)], [(1,67,55,21),(2,68,56,22),(3,69,57,23),(4,70,58,24),(5,71,59,25),(6,72,60,26),(7,73,46,27),(8,74,47,28),(9,75,48,29),(10,61,49,30),(11,62,50,16),(12,63,51,17),(13,64,52,18),(14,65,53,19),(15,66,54,20),(31,96,120,87),(32,97,106,88),(33,98,107,89),(34,99,108,90),(35,100,109,76),(36,101,110,77),(37,102,111,78),(38,103,112,79),(39,104,113,80),(40,105,114,81),(41,91,115,82),(42,92,116,83),(43,93,117,84),(44,94,118,85),(45,95,119,86)], [(1,110),(2,111),(3,112),(4,113),(5,114),(6,115),(7,116),(8,117),(9,118),(10,119),(11,120),(12,106),(13,107),(14,108),(15,109),(16,87),(17,88),(18,89),(19,90),(20,76),(21,77),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(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),(61,95),(62,96),(63,97),(64,98),(65,99),(66,100),(67,101),(68,102),(69,103),(70,104),(71,105),(72,91),(73,92),(74,93),(75,94)], [(1,21,55,67),(2,22,56,68),(3,23,57,69),(4,24,58,70),(5,25,59,71),(6,26,60,72),(7,27,46,73),(8,28,47,74),(9,29,48,75),(10,30,49,61),(11,16,50,62),(12,17,51,63),(13,18,52,64),(14,19,53,65),(15,20,54,66)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | ··· | 4G | 4H | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 12A | 12B | 12C | 12D | 12E | ··· | 12N | 12O | 12P | 15A | ··· | 15H | 20A | ··· | 20H | 20I | ··· | 20AB | 20AC | 20AD | 20AE | 20AF | 24A | 24B | 24C | 24D | 30A | ··· | 30H | 30I | ··· | 30P | 30Q | ··· | 30X | 40A | ··· | 40H | 60A | ··· | 60P | 60Q | ··· | 60BD | 60BE | ··· | 60BL | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 |
| type | + | + | + | + | + | + | ||||||||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C5 | C6 | C6 | C6 | C10 | C10 | C10 | C12 | C12 | C15 | C20 | C20 | C30 | C30 | C30 | C60 | C60 | D4 | D4 | C3×D4 | C3×D4 | C4≀C2 | C5×D4 | C5×D4 | C3×C4≀C2 | D4×C15 | D4×C15 | C5×C4≀C2 | C15×C4≀C2 |
| kernel | C15×C4≀C2 | C4×C60 | C15×M4(2) | C15×C4○D4 | C5×C4≀C2 | D4×C15 | Q8×C15 | C3×C4≀C2 | C4×C20 | C5×M4(2) | C5×C4○D4 | C4×C12 | C3×M4(2) | C3×C4○D4 | C5×D4 | C5×Q8 | C4≀C2 | C3×D4 | C3×Q8 | C42 | M4(2) | C4○D4 | D4 | Q8 | C60 | C2×C30 | C20 | C2×C10 | C15 | C12 | C2×C6 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | 16 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C15×C4≀C2 ►in GL2(𝔽241) generated by
| 94 | 0 |
| 0 | 94 |
| 64 | 0 |
| 152 | 177 |
| 89 | 128 |
| 149 | 152 |
| 177 | 0 |
| 209 | 1 |
G:=sub<GL(2,GF(241))| [94,0,0,94],[64,152,0,177],[89,149,128,152],[177,209,0,1] >;
C15×C4≀C2 in GAP, Magma, Sage, TeX
C_{15}\times C_4\wr C_2 % in TeX
G:=Group("C15xC4wrC2"); // GroupNames label
G:=SmallGroup(480,207);
// by ID
G=gap.SmallGroup(480,207);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,10504,5261,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
// generators/relations