direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic15, C6⋊Dic5, C30⋊3C4, C2.2D30, C22.D15, C10⋊2Dic3, C10.11D6, C6.11D10, C30.11C22, (C2×C6).D5, (C2×C10).S3, C15⋊10(C2×C4), C5⋊3(C2×Dic3), C3⋊2(C2×Dic5), (C2×C30).1C2, SmallGroup(120,29)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C2×Dic15 |
Generators and relations for C2×Dic15
G = < a,b,c | a2=b30=1, c2=b15, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic15
►Regular action on 120 points
Generators in S120
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(61 116)(62 117)(63 118)(64 119)(65 120)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 97)(73 98)(74 99)(75 100)(76 101)(77 102)(78 103)(79 104)(80 105)(81 106)(82 107)(83 108)(84 109)(85 110)(86 111)(87 112)(88 113)(89 114)(90 115)
(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 91 16 106)(2 120 17 105)(3 119 18 104)(4 118 19 103)(5 117 20 102)(6 116 21 101)(7 115 22 100)(8 114 23 99)(9 113 24 98)(10 112 25 97)(11 111 26 96)(12 110 27 95)(13 109 28 94)(14 108 29 93)(15 107 30 92)(31 81 46 66)(32 80 47 65)(33 79 48 64)(34 78 49 63)(35 77 50 62)(36 76 51 61)(37 75 52 90)(38 74 53 89)(39 73 54 88)(40 72 55 87)(41 71 56 86)(42 70 57 85)(43 69 58 84)(44 68 59 83)(45 67 60 82)
G:=sub<Sym(120)| (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(61,116)(62,117)(63,118)(64,119)(65,120)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(87,112)(88,113)(89,114)(90,115), (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), (1,91,16,106)(2,120,17,105)(3,119,18,104)(4,118,19,103)(5,117,20,102)(6,116,21,101)(7,115,22,100)(8,114,23,99)(9,113,24,98)(10,112,25,97)(11,111,26,96)(12,110,27,95)(13,109,28,94)(14,108,29,93)(15,107,30,92)(31,81,46,66)(32,80,47,65)(33,79,48,64)(34,78,49,63)(35,77,50,62)(36,76,51,61)(37,75,52,90)(38,74,53,89)(39,73,54,88)(40,72,55,87)(41,71,56,86)(42,70,57,85)(43,69,58,84)(44,68,59,83)(45,67,60,82)>;
G:=Group( (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(61,116)(62,117)(63,118)(64,119)(65,120)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(87,112)(88,113)(89,114)(90,115), (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), (1,91,16,106)(2,120,17,105)(3,119,18,104)(4,118,19,103)(5,117,20,102)(6,116,21,101)(7,115,22,100)(8,114,23,99)(9,113,24,98)(10,112,25,97)(11,111,26,96)(12,110,27,95)(13,109,28,94)(14,108,29,93)(15,107,30,92)(31,81,46,66)(32,80,47,65)(33,79,48,64)(34,78,49,63)(35,77,50,62)(36,76,51,61)(37,75,52,90)(38,74,53,89)(39,73,54,88)(40,72,55,87)(41,71,56,86)(42,70,57,85)(43,69,58,84)(44,68,59,83)(45,67,60,82) );
G=PermutationGroup([[(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,31),(17,32),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,43),(29,44),(30,45),(61,116),(62,117),(63,118),(64,119),(65,120),(66,91),(67,92),(68,93),(69,94),(70,95),(71,96),(72,97),(73,98),(74,99),(75,100),(76,101),(77,102),(78,103),(79,104),(80,105),(81,106),(82,107),(83,108),(84,109),(85,110),(86,111),(87,112),(88,113),(89,114),(90,115)], [(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)], [(1,91,16,106),(2,120,17,105),(3,119,18,104),(4,118,19,103),(5,117,20,102),(6,116,21,101),(7,115,22,100),(8,114,23,99),(9,113,24,98),(10,112,25,97),(11,111,26,96),(12,110,27,95),(13,109,28,94),(14,108,29,93),(15,107,30,92),(31,81,46,66),(32,80,47,65),(33,79,48,64),(34,78,49,63),(35,77,50,62),(36,76,51,61),(37,75,52,90),(38,74,53,89),(39,73,54,88),(40,72,55,87),(41,71,56,86),(42,70,57,85),(43,69,58,84),(44,68,59,83),(45,67,60,82)]])
C2×Dic15 is a maximal subgroup of
Dic3×Dic5 D10⋊Dic3 D6⋊Dic5 C30.Q8 Dic15⋊5C4 C6.Dic10 C30.4Q8 C60⋊5C4 D30⋊3C4 C30.38D4 C2×D5×Dic3 C30.C23 C2×S3×Dic5 C2×C4×D15 D4⋊2D15 Q8⋊Dic15
C2×Dic15 is a maximal quotient of
C60.7C4 C60⋊5C4 C30.38D4
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 15A | 15B | 15C | 15D | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | D5 | Dic3 | D6 | Dic5 | D10 | D15 | Dic15 | D30 |
| kernel | C2×Dic15 | Dic15 | C2×C30 | C30 | C2×C10 | C2×C6 | C10 | C10 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 2 | 1 | 4 | 2 | 4 | 8 | 4 |
Matrix representation of C2×Dic15 ►in GL3(𝔽61) generated by
| 60 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 8 | 25 |
| 0 | 13 | 56 |
| 1 | 0 | 0 |
| 0 | 38 | 3 |
| 0 | 47 | 23 |
G:=sub<GL(3,GF(61))| [60,0,0,0,1,0,0,0,1],[1,0,0,0,8,13,0,25,56],[1,0,0,0,38,47,0,3,23] >;
C2×Dic15 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{15} % in TeX
G:=Group("C2xDic15"); // GroupNames label
G:=SmallGroup(120,29);
// by ID
G=gap.SmallGroup(120,29);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-5,20,323,2404]);
// Polycyclic
G:=Group<a,b,c|a^2=b^30=1,c^2=b^15,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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