direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C4○D20, D20⋊5C6, Dic10⋊5C6, C12.60D10, C30.41C23, C60.63C22, (C2×C60)⋊9C2, (C2×C20)⋊4C6, (C4×D5)⋊4C6, (C2×C12)⋊7D5, C5⋊D4⋊3C6, (D5×C12)⋊9C2, C4.16(C6×D5), (C3×D20)⋊11C2, C15⋊12(C4○D4), C20.12(C2×C6), D10.1(C2×C6), (C2×C6).20D10, C22.2(C6×D5), C10.4(C22×C6), Dic5.2(C2×C6), C6.41(C22×D5), (C3×Dic10)⋊11C2, (C2×C30).40C22, (C6×D5).16C22, (C3×Dic5).18C22, C5⋊1(C3×C4○D4), C2.5(D5×C2×C6), (C2×C4)⋊3(C3×D5), (C3×C5⋊D4)⋊7C2, (C2×C10).11(C2×C6), SmallGroup(240,158)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4○D20
G = < a,b,c,d | a3=b4=d2=1, c10=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c9 >
Subgroups: 212 in 80 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C4○D4, Dic5, C20, D10, C2×C10, C2×C12, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C30, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C3×C4○D4, C3×Dic5, C60, C6×D5, C2×C30, C4○D20, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, C3×C4○D20
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, C4○D4, D10, C22×C6, C3×D5, C22×D5, C3×C4○D4, C6×D5, C4○D20, D5×C2×C6, C3×C4○D20
►On 120 points
Generators in S120
(1 82 39)(2 83 40)(3 84 21)(4 85 22)(5 86 23)(6 87 24)(7 88 25)(8 89 26)(9 90 27)(10 91 28)(11 92 29)(12 93 30)(13 94 31)(14 95 32)(15 96 33)(16 97 34)(17 98 35)(18 99 36)(19 100 37)(20 81 38)(41 105 65)(42 106 66)(43 107 67)(44 108 68)(45 109 69)(46 110 70)(47 111 71)(48 112 72)(49 113 73)(50 114 74)(51 115 75)(52 116 76)(53 117 77)(54 118 78)(55 119 79)(56 120 80)(57 101 61)(58 102 62)(59 103 63)(60 104 64)
(1 73 11 63)(2 74 12 64)(3 75 13 65)(4 76 14 66)(5 77 15 67)(6 78 16 68)(7 79 17 69)(8 80 18 70)(9 61 19 71)(10 62 20 72)(21 115 31 105)(22 116 32 106)(23 117 33 107)(24 118 34 108)(25 119 35 109)(26 120 36 110)(27 101 37 111)(28 102 38 112)(29 103 39 113)(30 104 40 114)(41 84 51 94)(42 85 52 95)(43 86 53 96)(44 87 54 97)(45 88 55 98)(46 89 56 99)(47 90 57 100)(48 91 58 81)(49 92 59 82)(50 93 60 83)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 60)(58 59)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 104)(102 103)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)
G:=sub<Sym(120)| (1,82,39)(2,83,40)(3,84,21)(4,85,22)(5,86,23)(6,87,24)(7,88,25)(8,89,26)(9,90,27)(10,91,28)(11,92,29)(12,93,30)(13,94,31)(14,95,32)(15,96,33)(16,97,34)(17,98,35)(18,99,36)(19,100,37)(20,81,38)(41,105,65)(42,106,66)(43,107,67)(44,108,68)(45,109,69)(46,110,70)(47,111,71)(48,112,72)(49,113,73)(50,114,74)(51,115,75)(52,116,76)(53,117,77)(54,118,78)(55,119,79)(56,120,80)(57,101,61)(58,102,62)(59,103,63)(60,104,64), (1,73,11,63)(2,74,12,64)(3,75,13,65)(4,76,14,66)(5,77,15,67)(6,78,16,68)(7,79,17,69)(8,80,18,70)(9,61,19,71)(10,62,20,72)(21,115,31,105)(22,116,32,106)(23,117,33,107)(24,118,34,108)(25,119,35,109)(26,120,36,110)(27,101,37,111)(28,102,38,112)(29,103,39,113)(30,104,40,114)(41,84,51,94)(42,85,52,95)(43,86,53,96)(44,87,54,97)(45,88,55,98)(46,89,56,99)(47,90,57,100)(48,91,58,81)(49,92,59,82)(50,93,60,83), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,60)(58,59)(61,64)(62,63)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(81,82)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)>;
G:=Group( (1,82,39)(2,83,40)(3,84,21)(4,85,22)(5,86,23)(6,87,24)(7,88,25)(8,89,26)(9,90,27)(10,91,28)(11,92,29)(12,93,30)(13,94,31)(14,95,32)(15,96,33)(16,97,34)(17,98,35)(18,99,36)(19,100,37)(20,81,38)(41,105,65)(42,106,66)(43,107,67)(44,108,68)(45,109,69)(46,110,70)(47,111,71)(48,112,72)(49,113,73)(50,114,74)(51,115,75)(52,116,76)(53,117,77)(54,118,78)(55,119,79)(56,120,80)(57,101,61)(58,102,62)(59,103,63)(60,104,64), (1,73,11,63)(2,74,12,64)(3,75,13,65)(4,76,14,66)(5,77,15,67)(6,78,16,68)(7,79,17,69)(8,80,18,70)(9,61,19,71)(10,62,20,72)(21,115,31,105)(22,116,32,106)(23,117,33,107)(24,118,34,108)(25,119,35,109)(26,120,36,110)(27,101,37,111)(28,102,38,112)(29,103,39,113)(30,104,40,114)(41,84,51,94)(42,85,52,95)(43,86,53,96)(44,87,54,97)(45,88,55,98)(46,89,56,99)(47,90,57,100)(48,91,58,81)(49,92,59,82)(50,93,60,83), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,60)(58,59)(61,64)(62,63)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(81,82)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113) );
G=PermutationGroup([[(1,82,39),(2,83,40),(3,84,21),(4,85,22),(5,86,23),(6,87,24),(7,88,25),(8,89,26),(9,90,27),(10,91,28),(11,92,29),(12,93,30),(13,94,31),(14,95,32),(15,96,33),(16,97,34),(17,98,35),(18,99,36),(19,100,37),(20,81,38),(41,105,65),(42,106,66),(43,107,67),(44,108,68),(45,109,69),(46,110,70),(47,111,71),(48,112,72),(49,113,73),(50,114,74),(51,115,75),(52,116,76),(53,117,77),(54,118,78),(55,119,79),(56,120,80),(57,101,61),(58,102,62),(59,103,63),(60,104,64)], [(1,73,11,63),(2,74,12,64),(3,75,13,65),(4,76,14,66),(5,77,15,67),(6,78,16,68),(7,79,17,69),(8,80,18,70),(9,61,19,71),(10,62,20,72),(21,115,31,105),(22,116,32,106),(23,117,33,107),(24,118,34,108),(25,119,35,109),(26,120,36,110),(27,101,37,111),(28,102,38,112),(29,103,39,113),(30,104,40,114),(41,84,51,94),(42,85,52,95),(43,86,53,96),(44,87,54,97),(45,88,55,98),(46,89,56,99),(47,90,57,100),(48,91,58,81),(49,92,59,82),(50,93,60,83)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(37,40),(38,39),(41,56),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(57,60),(58,59),(61,64),(62,63),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73),(81,82),(83,100),(84,99),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(101,104),(102,103),(105,120),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113)]])
C3×C4○D20 is a maximal subgroup of
C60.96D4 C60.97D4 D20.3Dic3 D20.2Dic3 D20.34D6 D20⋊21D6 D20⋊19D6 D20.37D6 D20.31D6 C60.63D4 D20.38D6 D20.39D6 D20⋊24D6 D20⋊26D6 D20⋊29D6 C3×D5×C4○D4
C3×C4○D20 is a maximal quotient of
C12×Dic10 C12×D20 C12×C5⋊D4
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 10 | 10 | 2 | 2 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D5 | C4○D4 | D10 | D10 | C3×D5 | C3×C4○D4 | C6×D5 | C6×D5 | C4○D20 | C3×C4○D20 |
| kernel | C3×C4○D20 | C3×Dic10 | D5×C12 | C3×D20 | C3×C5⋊D4 | C2×C60 | C4○D20 | Dic10 | C4×D5 | D20 | C5⋊D4 | C2×C20 | C2×C12 | C15 | C12 | C2×C6 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 8 | 16 |
Matrix representation of C3×C4○D20 ►in GL2(𝔽61) generated by
| 47 | 0 |
| 0 | 47 |
| 11 | 0 |
| 0 | 11 |
| 7 | 32 |
| 29 | 2 |
| 7 | 32 |
| 29 | 54 |
G:=sub<GL(2,GF(61))| [47,0,0,47],[11,0,0,11],[7,29,32,2],[7,29,32,54] >;
C3×C4○D20 in GAP, Magma, Sage, TeX
C_3\times C_4\circ D_{20} % in TeX
G:=Group("C3xC4oD20"); // GroupNames label
G:=SmallGroup(240,158);
// by ID
G=gap.SmallGroup(240,158);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-5,151,506,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=d^2=1,c^10=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^9>;
// generators/relations