metabelian, supersoluble, monomial
Aliases: C15⋊3Dic6, C30.21D6, C32⋊3Dic10, Dic15.2S3, (C3×C15)⋊6Q8, C10.12S32, C3⋊3(C15⋊Q8), C6.27(S3×D5), (C3×C6).12D10, C5⋊2(C32⋊2Q8), C3⋊Dic3.3D5, C2.5(D15⋊S3), (C3×C30).26C22, (C3×Dic15).4C2, (C5×C3⋊Dic3).2C2, SmallGroup(360,88)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊3Dic10
G = < a,b,c,d | a3=b3=c20=1, d2=c10, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 260 in 54 conjugacy classes, 23 normal (11 characteristic)
C1, C2, C3, C3, C4, C5, C6, C6, Q8, C32, C10, Dic3, C12, C15, C15, C3×C6, Dic5, C20, Dic6, C30, C30, C3×Dic3, C3⋊Dic3, Dic10, C3×C15, C5×Dic3, C3×Dic5, Dic15, C32⋊2Q8, C3×C30, C15⋊Q8, C3×Dic15, C5×C3⋊Dic3, C32⋊3Dic10
Quotients: C1, C2, C22, S3, Q8, D5, D6, D10, Dic6, S32, Dic10, S3×D5, C32⋊2Q8, C15⋊Q8, D15⋊S3, C32⋊3Dic10
►On 120 points
Generators in S120
(1 21 50)(2 51 22)(3 23 52)(4 53 24)(5 25 54)(6 55 26)(7 27 56)(8 57 28)(9 29 58)(10 59 30)(11 31 60)(12 41 32)(13 33 42)(14 43 34)(15 35 44)(16 45 36)(17 37 46)(18 47 38)(19 39 48)(20 49 40)(61 92 106)(62 107 93)(63 94 108)(64 109 95)(65 96 110)(66 111 97)(67 98 112)(68 113 99)(69 100 114)(70 115 81)(71 82 116)(72 117 83)(73 84 118)(74 119 85)(75 86 120)(76 101 87)(77 88 102)(78 103 89)(79 90 104)(80 105 91)
(1 50 21)(2 22 51)(3 52 23)(4 24 53)(5 54 25)(6 26 55)(7 56 27)(8 28 57)(9 58 29)(10 30 59)(11 60 31)(12 32 41)(13 42 33)(14 34 43)(15 44 35)(16 36 45)(17 46 37)(18 38 47)(19 48 39)(20 40 49)(61 92 106)(62 107 93)(63 94 108)(64 109 95)(65 96 110)(66 111 97)(67 98 112)(68 113 99)(69 100 114)(70 115 81)(71 82 116)(72 117 83)(73 84 118)(74 119 85)(75 86 120)(76 101 87)(77 88 102)(78 103 89)(79 90 104)(80 105 91)
(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 80 11 70)(2 79 12 69)(3 78 13 68)(4 77 14 67)(5 76 15 66)(6 75 16 65)(7 74 17 64)(8 73 18 63)(9 72 19 62)(10 71 20 61)(21 91 31 81)(22 90 32 100)(23 89 33 99)(24 88 34 98)(25 87 35 97)(26 86 36 96)(27 85 37 95)(28 84 38 94)(29 83 39 93)(30 82 40 92)(41 114 51 104)(42 113 52 103)(43 112 53 102)(44 111 54 101)(45 110 55 120)(46 109 56 119)(47 108 57 118)(48 107 58 117)(49 106 59 116)(50 105 60 115)
G:=sub<Sym(120)| (1,21,50)(2,51,22)(3,23,52)(4,53,24)(5,25,54)(6,55,26)(7,27,56)(8,57,28)(9,29,58)(10,59,30)(11,31,60)(12,41,32)(13,33,42)(14,43,34)(15,35,44)(16,45,36)(17,37,46)(18,47,38)(19,39,48)(20,49,40)(61,92,106)(62,107,93)(63,94,108)(64,109,95)(65,96,110)(66,111,97)(67,98,112)(68,113,99)(69,100,114)(70,115,81)(71,82,116)(72,117,83)(73,84,118)(74,119,85)(75,86,120)(76,101,87)(77,88,102)(78,103,89)(79,90,104)(80,105,91), (1,50,21)(2,22,51)(3,52,23)(4,24,53)(5,54,25)(6,26,55)(7,56,27)(8,28,57)(9,58,29)(10,30,59)(11,60,31)(12,32,41)(13,42,33)(14,34,43)(15,44,35)(16,36,45)(17,46,37)(18,38,47)(19,48,39)(20,40,49)(61,92,106)(62,107,93)(63,94,108)(64,109,95)(65,96,110)(66,111,97)(67,98,112)(68,113,99)(69,100,114)(70,115,81)(71,82,116)(72,117,83)(73,84,118)(74,119,85)(75,86,120)(76,101,87)(77,88,102)(78,103,89)(79,90,104)(80,105,91), (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), (1,80,11,70)(2,79,12,69)(3,78,13,68)(4,77,14,67)(5,76,15,66)(6,75,16,65)(7,74,17,64)(8,73,18,63)(9,72,19,62)(10,71,20,61)(21,91,31,81)(22,90,32,100)(23,89,33,99)(24,88,34,98)(25,87,35,97)(26,86,36,96)(27,85,37,95)(28,84,38,94)(29,83,39,93)(30,82,40,92)(41,114,51,104)(42,113,52,103)(43,112,53,102)(44,111,54,101)(45,110,55,120)(46,109,56,119)(47,108,57,118)(48,107,58,117)(49,106,59,116)(50,105,60,115)>;
G:=Group( (1,21,50)(2,51,22)(3,23,52)(4,53,24)(5,25,54)(6,55,26)(7,27,56)(8,57,28)(9,29,58)(10,59,30)(11,31,60)(12,41,32)(13,33,42)(14,43,34)(15,35,44)(16,45,36)(17,37,46)(18,47,38)(19,39,48)(20,49,40)(61,92,106)(62,107,93)(63,94,108)(64,109,95)(65,96,110)(66,111,97)(67,98,112)(68,113,99)(69,100,114)(70,115,81)(71,82,116)(72,117,83)(73,84,118)(74,119,85)(75,86,120)(76,101,87)(77,88,102)(78,103,89)(79,90,104)(80,105,91), (1,50,21)(2,22,51)(3,52,23)(4,24,53)(5,54,25)(6,26,55)(7,56,27)(8,28,57)(9,58,29)(10,30,59)(11,60,31)(12,32,41)(13,42,33)(14,34,43)(15,44,35)(16,36,45)(17,46,37)(18,38,47)(19,48,39)(20,40,49)(61,92,106)(62,107,93)(63,94,108)(64,109,95)(65,96,110)(66,111,97)(67,98,112)(68,113,99)(69,100,114)(70,115,81)(71,82,116)(72,117,83)(73,84,118)(74,119,85)(75,86,120)(76,101,87)(77,88,102)(78,103,89)(79,90,104)(80,105,91), (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), (1,80,11,70)(2,79,12,69)(3,78,13,68)(4,77,14,67)(5,76,15,66)(6,75,16,65)(7,74,17,64)(8,73,18,63)(9,72,19,62)(10,71,20,61)(21,91,31,81)(22,90,32,100)(23,89,33,99)(24,88,34,98)(25,87,35,97)(26,86,36,96)(27,85,37,95)(28,84,38,94)(29,83,39,93)(30,82,40,92)(41,114,51,104)(42,113,52,103)(43,112,53,102)(44,111,54,101)(45,110,55,120)(46,109,56,119)(47,108,57,118)(48,107,58,117)(49,106,59,116)(50,105,60,115) );
G=PermutationGroup([[(1,21,50),(2,51,22),(3,23,52),(4,53,24),(5,25,54),(6,55,26),(7,27,56),(8,57,28),(9,29,58),(10,59,30),(11,31,60),(12,41,32),(13,33,42),(14,43,34),(15,35,44),(16,45,36),(17,37,46),(18,47,38),(19,39,48),(20,49,40),(61,92,106),(62,107,93),(63,94,108),(64,109,95),(65,96,110),(66,111,97),(67,98,112),(68,113,99),(69,100,114),(70,115,81),(71,82,116),(72,117,83),(73,84,118),(74,119,85),(75,86,120),(76,101,87),(77,88,102),(78,103,89),(79,90,104),(80,105,91)], [(1,50,21),(2,22,51),(3,52,23),(4,24,53),(5,54,25),(6,26,55),(7,56,27),(8,28,57),(9,58,29),(10,30,59),(11,60,31),(12,32,41),(13,42,33),(14,34,43),(15,44,35),(16,36,45),(17,46,37),(18,38,47),(19,48,39),(20,40,49),(61,92,106),(62,107,93),(63,94,108),(64,109,95),(65,96,110),(66,111,97),(67,98,112),(68,113,99),(69,100,114),(70,115,81),(71,82,116),(72,117,83),(73,84,118),(74,119,85),(75,86,120),(76,101,87),(77,88,102),(78,103,89),(79,90,104),(80,105,91)], [(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)], [(1,80,11,70),(2,79,12,69),(3,78,13,68),(4,77,14,67),(5,76,15,66),(6,75,16,65),(7,74,17,64),(8,73,18,63),(9,72,19,62),(10,71,20,61),(21,91,31,81),(22,90,32,100),(23,89,33,99),(24,88,34,98),(25,87,35,97),(26,86,36,96),(27,85,37,95),(28,84,38,94),(29,83,39,93),(30,82,40,92),(41,114,51,104),(42,113,52,103),(43,112,53,102),(44,111,54,101),(45,110,55,120),(46,109,56,119),(47,108,57,118),(48,107,58,117),(49,106,59,116),(50,105,60,115)]])
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 12A | 12B | 12C | 12D | 15A | ··· | 15H | 20A | 20B | 20C | 20D | 30A | ··· | 30H |
| order | 1 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 2 | 4 | 18 | 30 | 30 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 30 | 30 | 30 | 30 | 4 | ··· | 4 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | - | - | + | + | - | - | ||
| image | C1 | C2 | C2 | S3 | Q8 | D5 | D6 | D10 | Dic6 | Dic10 | S32 | S3×D5 | C32⋊2Q8 | C15⋊Q8 | D15⋊S3 | C32⋊3Dic10 |
| kernel | C32⋊3Dic10 | C3×Dic15 | C5×C3⋊Dic3 | Dic15 | C3×C15 | C3⋊Dic3 | C30 | C3×C6 | C15 | C32 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 4 | 1 | 4 | 4 | 4 |
Matrix representation of C32⋊3Dic10 ►in GL8(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 27 | 25 | 0 | 0 | 0 | 0 | 0 | 0 |
| 27 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,60,18,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,59,1,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[27,27,0,0,0,0,0,0,25,34,0,0,0,0,0,0,0,0,9,21,0,0,0,0,0,0,60,52,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C32⋊3Dic10 in GAP, Magma, Sage, TeX
C_3^2\rtimes_3{\rm Dic}_{10} % in TeX
G:=Group("C3^2:3Dic10"); // GroupNames label
G:=SmallGroup(360,88);
// by ID
G=gap.SmallGroup(360,88);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,73,31,387,201,730,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^20=1,d^2=c^10,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations