direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D4.D5, C30.41D4, C15⋊11SD16, Dic10⋊2C6, C12.37D10, C60.37C22, D4.(C3×D5), C5⋊2C8⋊2C6, C4.2(C6×D5), C5⋊2(C3×SD16), C20.2(C2×C6), (C5×D4).1C6, (C3×D4).2D5, C10.8(C3×D4), (D4×C15).2C2, (C3×Dic10)⋊8C2, C6.24(C5⋊D4), (C3×C5⋊2C8)⋊9C2, C2.5(C3×C5⋊D4), SmallGroup(240,45)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4.D5
G = < a,b,c,d,e | a3=b4=c2=d5=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >
Smallest permutation representation of C3×D4.D5
►On 120 points
Generators in S120
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 66)(62 67)(63 68)(64 69)(65 70)(81 86)(82 87)(83 88)(84 89)(85 90)(101 106)(102 107)(103 108)(104 109)(105 110)
(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 68 6 63)(2 67 7 62)(3 66 8 61)(4 70 9 65)(5 69 10 64)(11 78 16 73)(12 77 17 72)(13 76 18 71)(14 80 19 75)(15 79 20 74)(21 88 26 83)(22 87 27 82)(23 86 28 81)(24 90 29 85)(25 89 30 84)(31 98 36 93)(32 97 37 92)(33 96 38 91)(34 100 39 95)(35 99 40 94)(41 108 46 103)(42 107 47 102)(43 106 48 101)(44 110 49 105)(45 109 50 104)(51 118 56 113)(52 117 57 112)(53 116 58 111)(54 120 59 115)(55 119 60 114)
G:=sub<Sym(120)| (1,41,21)(2,42,22)(3,43,23)(4,44,24)(5,45,25)(6,46,26)(7,47,27)(8,48,28)(9,49,29)(10,50,30)(11,51,31)(12,52,32)(13,53,33)(14,54,34)(15,55,35)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(61,101,81)(62,102,82)(63,103,83)(64,104,84)(65,105,85)(66,106,86)(67,107,87)(68,108,88)(69,109,89)(70,110,90)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,66)(62,67)(63,68)(64,69)(65,70)(81,86)(82,87)(83,88)(84,89)(85,90)(101,106)(102,107)(103,108)(104,109)(105,110), (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), (1,68,6,63)(2,67,7,62)(3,66,8,61)(4,70,9,65)(5,69,10,64)(11,78,16,73)(12,77,17,72)(13,76,18,71)(14,80,19,75)(15,79,20,74)(21,88,26,83)(22,87,27,82)(23,86,28,81)(24,90,29,85)(25,89,30,84)(31,98,36,93)(32,97,37,92)(33,96,38,91)(34,100,39,95)(35,99,40,94)(41,108,46,103)(42,107,47,102)(43,106,48,101)(44,110,49,105)(45,109,50,104)(51,118,56,113)(52,117,57,112)(53,116,58,111)(54,120,59,115)(55,119,60,114)>;
G:=Group( (1,41,21)(2,42,22)(3,43,23)(4,44,24)(5,45,25)(6,46,26)(7,47,27)(8,48,28)(9,49,29)(10,50,30)(11,51,31)(12,52,32)(13,53,33)(14,54,34)(15,55,35)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(61,101,81)(62,102,82)(63,103,83)(64,104,84)(65,105,85)(66,106,86)(67,107,87)(68,108,88)(69,109,89)(70,110,90)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,66)(62,67)(63,68)(64,69)(65,70)(81,86)(82,87)(83,88)(84,89)(85,90)(101,106)(102,107)(103,108)(104,109)(105,110), (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), (1,68,6,63)(2,67,7,62)(3,66,8,61)(4,70,9,65)(5,69,10,64)(11,78,16,73)(12,77,17,72)(13,76,18,71)(14,80,19,75)(15,79,20,74)(21,88,26,83)(22,87,27,82)(23,86,28,81)(24,90,29,85)(25,89,30,84)(31,98,36,93)(32,97,37,92)(33,96,38,91)(34,100,39,95)(35,99,40,94)(41,108,46,103)(42,107,47,102)(43,106,48,101)(44,110,49,105)(45,109,50,104)(51,118,56,113)(52,117,57,112)(53,116,58,111)(54,120,59,115)(55,119,60,114) );
G=PermutationGroup([[(1,41,21),(2,42,22),(3,43,23),(4,44,24),(5,45,25),(6,46,26),(7,47,27),(8,48,28),(9,49,29),(10,50,30),(11,51,31),(12,52,32),(13,53,33),(14,54,34),(15,55,35),(16,56,36),(17,57,37),(18,58,38),(19,59,39),(20,60,40),(61,101,81),(62,102,82),(63,103,83),(64,104,84),(65,105,85),(66,106,86),(67,107,87),(68,108,88),(69,109,89),(70,110,90),(71,111,91),(72,112,92),(73,113,93),(74,114,94),(75,115,95),(76,116,96),(77,117,97),(78,118,98),(79,119,99),(80,120,100)], [(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,71,66,76),(62,72,67,77),(63,73,68,78),(64,74,69,79),(65,75,70,80),(81,91,86,96),(82,92,87,97),(83,93,88,98),(84,94,89,99),(85,95,90,100),(101,111,106,116),(102,112,107,117),(103,113,108,118),(104,114,109,119),(105,115,110,120)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,66),(62,67),(63,68),(64,69),(65,70),(81,86),(82,87),(83,88),(84,89),(85,90),(101,106),(102,107),(103,108),(104,109),(105,110)], [(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)], [(1,68,6,63),(2,67,7,62),(3,66,8,61),(4,70,9,65),(5,69,10,64),(11,78,16,73),(12,77,17,72),(13,76,18,71),(14,80,19,75),(15,79,20,74),(21,88,26,83),(22,87,27,82),(23,86,28,81),(24,90,29,85),(25,89,30,84),(31,98,36,93),(32,97,37,92),(33,96,38,91),(34,100,39,95),(35,99,40,94),(41,108,46,103),(42,107,47,102),(43,106,48,101),(44,110,49,105),(45,109,50,104),(51,118,56,113),(52,117,57,112),(53,116,58,111),(54,120,59,115),(55,119,60,114)]])
C3×D4.D5 is a maximal subgroup of
C60.10C23 Dic10⋊D6 D30.9D4 C60.19C23 D12.9D10 D30.11D4 D12⋊5D10 C3×D5×SD16
51 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | 20B | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | ··· | 30L | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 4 | 1 | 1 | 2 | 20 | 2 | 2 | 1 | 1 | 4 | 4 | 10 | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D5 | SD16 | D10 | C3×D4 | C3×D5 | C5⋊D4 | C3×SD16 | C6×D5 | C3×C5⋊D4 | D4.D5 | C3×D4.D5 |
| kernel | C3×D4.D5 | C3×C5⋊2C8 | C3×Dic10 | D4×C15 | D4.D5 | C5⋊2C8 | Dic10 | C5×D4 | C30 | C3×D4 | C15 | C12 | C10 | D4 | C6 | C5 | C4 | C2 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 2 | 4 |
Matrix representation of C3×D4.D5 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 225 |
| 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 85 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 91 | 177 |
| 0 | 0 | 0 | 98 |
| 222 | 19 | 0 | 0 |
| 19 | 19 | 0 | 0 |
| 0 | 0 | 23 | 156 |
| 0 | 0 | 182 | 218 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,225,0,0,0,0,225],[0,240,0,0,1,0,0,0,0,0,240,0,0,0,0,240],[0,1,0,0,1,0,0,0,0,0,240,0,0,0,85,1],[1,0,0,0,0,1,0,0,0,0,91,0,0,0,177,98],[222,19,0,0,19,19,0,0,0,0,23,182,0,0,156,218] >;
C3×D4.D5 in GAP, Magma, Sage, TeX
C_3\times D_4.D_5
% in TeX
G:=Group("C3xD4.D5"); // GroupNames label
G:=SmallGroup(240,45);
// by ID
G=gap.SmallGroup(240,45);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-5,144,169,867,441,69,6917]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^5=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations
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