direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×C22⋊C4, C22⋊2C60, C30.52D4, C23.2C30, (C2×C6)⋊1C20, (C2×C60)⋊4C2, (C2×C4)⋊1C30, (C2×C20)⋊2C6, (C2×C30)⋊5C4, (C2×C12)⋊2C10, (C2×C10)⋊6C12, C2.1(C2×C60), C6.12(C5×D4), C2.1(D4×C15), C30.62(C2×C4), C6.10(C2×C20), C10.12(C3×D4), C10.17(C2×C12), (C22×C10).3C6, (C22×C30).1C2, (C22×C6).1C10, C22.2(C2×C30), (C2×C30).52C22, (C2×C6).13(C2×C10), (C2×C10).13(C2×C6), SmallGroup(240,82)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C22⋊C4
G = < a,b,c,d | a15=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 92 in 68 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C23, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C20, C2×C10, C2×C10, C2×C10, C2×C12, C22×C6, C30, C30, C30, C2×C20, C22×C10, C3×C22⋊C4, C60, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C2×C60, C22×C30, C15×C22⋊C4
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, C2×C20, C5×D4, C3×C22⋊C4, C60, C2×C30, C5×C22⋊C4, C2×C60, D4×C15, C15×C22⋊C4
►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)
(16 81)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 76)(27 77)(28 78)(29 79)(30 80)(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(73 91)(74 92)(75 93)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 31)(13 32)(14 33)(15 34)(16 81)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 76)(27 77)(28 78)(29 79)(30 80)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 106)(55 107)(56 108)(57 109)(58 110)(59 111)(60 112)(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(73 91)(74 92)(75 93)
(1 72 108 81)(2 73 109 82)(3 74 110 83)(4 75 111 84)(5 61 112 85)(6 62 113 86)(7 63 114 87)(8 64 115 88)(9 65 116 89)(10 66 117 90)(11 67 118 76)(12 68 119 77)(13 69 120 78)(14 70 106 79)(15 71 107 80)(16 35 105 56)(17 36 91 57)(18 37 92 58)(19 38 93 59)(20 39 94 60)(21 40 95 46)(22 41 96 47)(23 42 97 48)(24 43 98 49)(25 44 99 50)(26 45 100 51)(27 31 101 52)(28 32 102 53)(29 33 103 54)(30 34 104 55)
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), (16,81)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,76)(27,77)(28,78)(29,79)(30,80)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,91)(74,92)(75,93), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,31)(13,32)(14,33)(15,34)(16,81)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,76)(27,77)(28,78)(29,79)(30,80)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,91)(74,92)(75,93), (1,72,108,81)(2,73,109,82)(3,74,110,83)(4,75,111,84)(5,61,112,85)(6,62,113,86)(7,63,114,87)(8,64,115,88)(9,65,116,89)(10,66,117,90)(11,67,118,76)(12,68,119,77)(13,69,120,78)(14,70,106,79)(15,71,107,80)(16,35,105,56)(17,36,91,57)(18,37,92,58)(19,38,93,59)(20,39,94,60)(21,40,95,46)(22,41,96,47)(23,42,97,48)(24,43,98,49)(25,44,99,50)(26,45,100,51)(27,31,101,52)(28,32,102,53)(29,33,103,54)(30,34,104,55)>;
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), (16,81)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,76)(27,77)(28,78)(29,79)(30,80)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,91)(74,92)(75,93), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,31)(13,32)(14,33)(15,34)(16,81)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,76)(27,77)(28,78)(29,79)(30,80)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,91)(74,92)(75,93), (1,72,108,81)(2,73,109,82)(3,74,110,83)(4,75,111,84)(5,61,112,85)(6,62,113,86)(7,63,114,87)(8,64,115,88)(9,65,116,89)(10,66,117,90)(11,67,118,76)(12,68,119,77)(13,69,120,78)(14,70,106,79)(15,71,107,80)(16,35,105,56)(17,36,91,57)(18,37,92,58)(19,38,93,59)(20,39,94,60)(21,40,95,46)(22,41,96,47)(23,42,97,48)(24,43,98,49)(25,44,99,50)(26,45,100,51)(27,31,101,52)(28,32,102,53)(29,33,103,54)(30,34,104,55) );
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)], [(16,81),(17,82),(18,83),(19,84),(20,85),(21,86),(22,87),(23,88),(24,89),(25,90),(26,76),(27,77),(28,78),(29,79),(30,80),(61,94),(62,95),(63,96),(64,97),(65,98),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,105),(73,91),(74,92),(75,93)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,31),(13,32),(14,33),(15,34),(16,81),(17,82),(18,83),(19,84),(20,85),(21,86),(22,87),(23,88),(24,89),(25,90),(26,76),(27,77),(28,78),(29,79),(30,80),(46,113),(47,114),(48,115),(49,116),(50,117),(51,118),(52,119),(53,120),(54,106),(55,107),(56,108),(57,109),(58,110),(59,111),(60,112),(61,94),(62,95),(63,96),(64,97),(65,98),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,105),(73,91),(74,92),(75,93)], [(1,72,108,81),(2,73,109,82),(3,74,110,83),(4,75,111,84),(5,61,112,85),(6,62,113,86),(7,63,114,87),(8,64,115,88),(9,65,116,89),(10,66,117,90),(11,67,118,76),(12,68,119,77),(13,69,120,78),(14,70,106,79),(15,71,107,80),(16,35,105,56),(17,36,91,57),(18,37,92,58),(19,38,93,59),(20,39,94,60),(21,40,95,46),(22,41,96,47),(23,42,97,48),(24,43,98,49),(25,44,99,50),(26,45,100,51),(27,31,101,52),(28,32,102,53),(29,33,103,54),(30,34,104,55)]])
C15×C22⋊C4 is a maximal subgroup of
C23.6D30 C23.15D30 C22⋊2Dic30 C23.8D30 Dic15⋊19D4 D30⋊16D4 D30.28D4 D30⋊9D4 C23.11D30 C22.D60 D4×C60
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 10A | ··· | 10L | 10M | ··· | 10T | 12A | ··· | 12H | 15A | ··· | 15H | 20A | ··· | 20P | 30A | ··· | 30X | 30Y | ··· | 30AN | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C60 | D4 | C3×D4 | C5×D4 | D4×C15 |
| kernel | C15×C22⋊C4 | C2×C60 | C22×C30 | C5×C22⋊C4 | C2×C30 | C3×C22⋊C4 | C2×C20 | C22×C10 | C2×C12 | C22×C6 | C2×C10 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C22 | C30 | C10 | C6 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 4 | 8 | 8 | 16 | 16 | 8 | 32 | 2 | 4 | 8 | 16 |
Matrix representation of C15×C22⋊C4 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 34 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 29 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 29 | 54 |
| 0 | 0 | 59 | 32 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,29,60],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,50,0,0,0,0,29,59,0,0,54,32] >;
C15×C22⋊C4 in GAP, Magma, Sage, TeX
C_{15}\times C_2^2\rtimes C_4 % in TeX
G:=Group("C15xC2^2:C4"); // GroupNames label
G:=SmallGroup(240,82);
// by ID
G=gap.SmallGroup(240,82);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,-2,720,745]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations