direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×C4○D4, D4⋊2C30, Q8⋊3C30, C30.60C23, C60.81C22, (C5×D4)○C60, (C5×Q8)○C60, (C2×C20)⋊7C6, (C2×C4)⋊3C30, C60○(D4×C15), (C5×D4)⋊5C6, C60○(Q8×C15), (C5×Q8)⋊7C6, (C2×C60)⋊15C2, (C2×C12)⋊7C10, (C3×D4)⋊5C10, C4.5(C2×C30), (C3×Q8)⋊5C10, C22.(C2×C30), (D4×C15)⋊11C2, C20.21(C2×C6), (Q8×C15)⋊11C2, C12.21(C2×C10), C2.3(C22×C30), (C2×C30).21C22, C6.13(C22×C10), C10.13(C22×C6), (C2×C10).2(C2×C6), (C2×C6).2(C2×C10), SmallGroup(240,188)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C4○D4
G = < a,b,c,d | a15=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Subgroups: 92 in 80 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, D4, Q8, C10, C10, C12, C12, C2×C6, C15, C4○D4, C20, C20, C2×C10, C2×C12, C3×D4, C3×Q8, C30, C30, C2×C20, C5×D4, C5×Q8, C3×C4○D4, C60, C60, C2×C30, C5×C4○D4, C2×C60, D4×C15, Q8×C15, C15×C4○D4
Quotients: C1, C2, C3, C22, C5, C6, C23, C10, C2×C6, C15, C4○D4, C2×C10, C22×C6, C30, C22×C10, C3×C4○D4, C2×C30, C5×C4○D4, C22×C30, C15×C4○D4
►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 75 33 29)(2 61 34 30)(3 62 35 16)(4 63 36 17)(5 64 37 18)(6 65 38 19)(7 66 39 20)(8 67 40 21)(9 68 41 22)(10 69 42 23)(11 70 43 24)(12 71 44 25)(13 72 45 26)(14 73 31 27)(15 74 32 28)(46 119 105 77)(47 120 91 78)(48 106 92 79)(49 107 93 80)(50 108 94 81)(51 109 95 82)(52 110 96 83)(53 111 97 84)(54 112 98 85)(55 113 99 86)(56 114 100 87)(57 115 101 88)(58 116 102 89)(59 117 103 90)(60 118 104 76)
(1 29 33 75)(2 30 34 61)(3 16 35 62)(4 17 36 63)(5 18 37 64)(6 19 38 65)(7 20 39 66)(8 21 40 67)(9 22 41 68)(10 23 42 69)(11 24 43 70)(12 25 44 71)(13 26 45 72)(14 27 31 73)(15 28 32 74)(46 119 105 77)(47 120 91 78)(48 106 92 79)(49 107 93 80)(50 108 94 81)(51 109 95 82)(52 110 96 83)(53 111 97 84)(54 112 98 85)(55 113 99 86)(56 114 100 87)(57 115 101 88)(58 116 102 89)(59 117 103 90)(60 118 104 76)
(1 59)(2 60)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 76)(31 101)(32 102)(33 103)(34 104)(35 105)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(61 118)(62 119)(63 120)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 113)(72 114)(73 115)(74 116)(75 117)
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), (1,75,33,29)(2,61,34,30)(3,62,35,16)(4,63,36,17)(5,64,37,18)(6,65,38,19)(7,66,39,20)(8,67,40,21)(9,68,41,22)(10,69,42,23)(11,70,43,24)(12,71,44,25)(13,72,45,26)(14,73,31,27)(15,74,32,28)(46,119,105,77)(47,120,91,78)(48,106,92,79)(49,107,93,80)(50,108,94,81)(51,109,95,82)(52,110,96,83)(53,111,97,84)(54,112,98,85)(55,113,99,86)(56,114,100,87)(57,115,101,88)(58,116,102,89)(59,117,103,90)(60,118,104,76), (1,29,33,75)(2,30,34,61)(3,16,35,62)(4,17,36,63)(5,18,37,64)(6,19,38,65)(7,20,39,66)(8,21,40,67)(9,22,41,68)(10,23,42,69)(11,24,43,70)(12,25,44,71)(13,26,45,72)(14,27,31,73)(15,28,32,74)(46,119,105,77)(47,120,91,78)(48,106,92,79)(49,107,93,80)(50,108,94,81)(51,109,95,82)(52,110,96,83)(53,111,97,84)(54,112,98,85)(55,113,99,86)(56,114,100,87)(57,115,101,88)(58,116,102,89)(59,117,103,90)(60,118,104,76), (1,59)(2,60)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,76)(31,101)(32,102)(33,103)(34,104)(35,105)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(61,118)(62,119)(63,120)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,113)(72,114)(73,115)(74,116)(75,117)>;
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), (1,75,33,29)(2,61,34,30)(3,62,35,16)(4,63,36,17)(5,64,37,18)(6,65,38,19)(7,66,39,20)(8,67,40,21)(9,68,41,22)(10,69,42,23)(11,70,43,24)(12,71,44,25)(13,72,45,26)(14,73,31,27)(15,74,32,28)(46,119,105,77)(47,120,91,78)(48,106,92,79)(49,107,93,80)(50,108,94,81)(51,109,95,82)(52,110,96,83)(53,111,97,84)(54,112,98,85)(55,113,99,86)(56,114,100,87)(57,115,101,88)(58,116,102,89)(59,117,103,90)(60,118,104,76), (1,29,33,75)(2,30,34,61)(3,16,35,62)(4,17,36,63)(5,18,37,64)(6,19,38,65)(7,20,39,66)(8,21,40,67)(9,22,41,68)(10,23,42,69)(11,24,43,70)(12,25,44,71)(13,26,45,72)(14,27,31,73)(15,28,32,74)(46,119,105,77)(47,120,91,78)(48,106,92,79)(49,107,93,80)(50,108,94,81)(51,109,95,82)(52,110,96,83)(53,111,97,84)(54,112,98,85)(55,113,99,86)(56,114,100,87)(57,115,101,88)(58,116,102,89)(59,117,103,90)(60,118,104,76), (1,59)(2,60)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,76)(31,101)(32,102)(33,103)(34,104)(35,105)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(61,118)(62,119)(63,120)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,113)(72,114)(73,115)(74,116)(75,117) );
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)], [(1,75,33,29),(2,61,34,30),(3,62,35,16),(4,63,36,17),(5,64,37,18),(6,65,38,19),(7,66,39,20),(8,67,40,21),(9,68,41,22),(10,69,42,23),(11,70,43,24),(12,71,44,25),(13,72,45,26),(14,73,31,27),(15,74,32,28),(46,119,105,77),(47,120,91,78),(48,106,92,79),(49,107,93,80),(50,108,94,81),(51,109,95,82),(52,110,96,83),(53,111,97,84),(54,112,98,85),(55,113,99,86),(56,114,100,87),(57,115,101,88),(58,116,102,89),(59,117,103,90),(60,118,104,76)], [(1,29,33,75),(2,30,34,61),(3,16,35,62),(4,17,36,63),(5,18,37,64),(6,19,38,65),(7,20,39,66),(8,21,40,67),(9,22,41,68),(10,23,42,69),(11,24,43,70),(12,25,44,71),(13,26,45,72),(14,27,31,73),(15,28,32,74),(46,119,105,77),(47,120,91,78),(48,106,92,79),(49,107,93,80),(50,108,94,81),(51,109,95,82),(52,110,96,83),(53,111,97,84),(54,112,98,85),(55,113,99,86),(56,114,100,87),(57,115,101,88),(58,116,102,89),(59,117,103,90),(60,118,104,76)], [(1,59),(2,60),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,57),(15,58),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,85),(25,86),(26,87),(27,88),(28,89),(29,90),(30,76),(31,101),(32,102),(33,103),(34,104),(35,105),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,98),(44,99),(45,100),(61,118),(62,119),(63,120),(64,106),(65,107),(66,108),(67,109),(68,110),(69,111),(70,112),(71,113),(72,114),(73,115),(74,116),(75,117)]])
C15×C4○D4 is a maximal subgroup of
Q8⋊3Dic15 D4.Dic15 D4⋊D30 D4.8D30 D4.9D30 D4⋊8D30 D4.10D30
C15×C4○D4 is a maximal quotient of D4×C60 Q8×C60
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 6A | 6B | 6C | ··· | 6H | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 15A | ··· | 15H | 20A | ··· | 20H | 20I | ··· | 20T | 30A | ··· | 30H | 30I | ··· | 30AF | 60A | ··· | 60P | 60Q | ··· | 60AN |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | ··· | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 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 | C2 | C3 | C5 | C6 | C6 | C6 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | C4○D4 | C3×C4○D4 | C5×C4○D4 | C15×C4○D4 |
| kernel | C15×C4○D4 | C2×C60 | D4×C15 | Q8×C15 | C5×C4○D4 | C3×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C15 | C5 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 4 | 6 | 6 | 2 | 12 | 12 | 4 | 8 | 24 | 24 | 8 | 2 | 4 | 8 | 16 |
Matrix representation of C15×C4○D4 ►in GL3(𝔽61) generated by
| 13 | 0 | 0 |
| 0 | 58 | 0 |
| 0 | 0 | 58 |
| 1 | 0 | 0 |
| 0 | 50 | 0 |
| 0 | 0 | 50 |
| 60 | 0 | 0 |
| 0 | 11 | 11 |
| 0 | 0 | 50 |
| 60 | 0 | 0 |
| 0 | 11 | 11 |
| 0 | 39 | 50 |
G:=sub<GL(3,GF(61))| [13,0,0,0,58,0,0,0,58],[1,0,0,0,50,0,0,0,50],[60,0,0,0,11,0,0,11,50],[60,0,0,0,11,39,0,11,50] >;
C15×C4○D4 in GAP, Magma, Sage, TeX
C_{15}\times C_4\circ D_4 % in TeX
G:=Group("C15xC4oD4"); // GroupNames label
G:=SmallGroup(240,188);
// by ID
G=gap.SmallGroup(240,188);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-5,-2,1465,554]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations