direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C23.1D10, (C6×Dic5)⋊2C4, (C2×C6).24D20, C23.D5⋊1C6, C23.1(C6×D5), C15⋊11(C23⋊C4), (C2×Dic5)⋊1C12, (C2×C30).153D4, (C22×D5)⋊2C12, (C22×C6).1D10, C22.3(D5×C12), C22.2(C3×D20), C30.81(C22⋊C4), C6.34(D10⋊C4), (C22×C30).91C22, (D5×C2×C6)⋊2C4, C5⋊3(C3×C23⋊C4), (C5×C22⋊C4)⋊1C6, C22⋊C4⋊1(C3×D5), (C3×C22⋊C4)⋊1D5, (C2×C6).37(C4×D5), (C2×C5⋊D4).1C6, (C6×C5⋊D4).8C2, (C15×C22⋊C4)⋊1C2, (C2×C10).28(C3×D4), C22.8(C3×C5⋊D4), (C2×C30).118(C2×C4), (C2×C10).21(C2×C12), C2.4(C3×D10⋊C4), (C2×C6).61(C5⋊D4), C10.13(C3×C22⋊C4), (C3×C23.D5)⋊17C2, (C22×C10).10(C2×C6), SmallGroup(480,84)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.1D10
G = < a,b,c,d,e | a3=b2=c2=d20=1, e2=b, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 416 in 104 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C23⋊C4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C3×C22⋊C4, C3×C22⋊C4, C6×D4, C3×Dic5, C60, C6×D5, C2×C30, C2×C30, C23.D5, C5×C22⋊C4, C2×C5⋊D4, C3×C23⋊C4, C6×Dic5, C6×Dic5, C3×C5⋊D4, C2×C60, D5×C2×C6, C22×C30, C23.1D10, C3×C23.D5, C15×C22⋊C4, C6×C5⋊D4, C3×C23.1D10
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, D10, C2×C12, C3×D4, C3×D5, C23⋊C4, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C6×D5, D10⋊C4, C3×C23⋊C4, D5×C12, C3×D20, C3×C5⋊D4, C23.1D10, C3×D10⋊C4, C3×C23.1D10
►On 120 points
Generators in S120
(1 26 49)(2 27 50)(3 28 51)(4 29 52)(5 30 53)(6 31 54)(7 32 55)(8 33 56)(9 34 57)(10 35 58)(11 36 59)(12 37 60)(13 38 41)(14 39 42)(15 40 43)(16 21 44)(17 22 45)(18 23 46)(19 24 47)(20 25 48)(61 113 84)(62 114 85)(63 115 86)(64 116 87)(65 117 88)(66 118 89)(67 119 90)(68 120 91)(69 101 92)(70 102 93)(71 103 94)(72 104 95)(73 105 96)(74 106 97)(75 107 98)(76 108 99)(77 109 100)(78 110 81)(79 111 82)(80 112 83)
(1 81)(3 83)(5 85)(7 87)(9 89)(11 91)(13 93)(15 95)(17 97)(19 99)(22 74)(24 76)(26 78)(28 80)(30 62)(32 64)(34 66)(36 68)(38 70)(40 72)(41 102)(43 104)(45 106)(47 108)(49 110)(51 112)(53 114)(55 116)(57 118)(59 120)
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 97)(18 98)(19 99)(20 100)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 102)(42 103)(43 104)(44 105)(45 106)(46 107)(47 108)(48 109)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 101)
(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 95 81 15)(2 94)(3 13 83 93)(4 12)(5 91 85 11)(6 90)(7 9 87 89)(10 86)(14 82)(16 20)(17 99 97 19)(18 98)(21 25)(22 76 74 24)(23 75)(26 72 78 40)(27 71)(28 38 80 70)(29 37)(30 68 62 36)(31 67)(32 34 64 66)(35 63)(39 79)(41 112 102 51)(42 111)(43 49 104 110)(44 48)(45 108 106 47)(46 107)(50 103)(52 60)(53 120 114 59)(54 119)(55 57 116 118)(58 115)(61 69)(73 77)(84 92)(96 100)(101 113)(105 109)
G:=sub<Sym(120)| (1,26,49)(2,27,50)(3,28,51)(4,29,52)(5,30,53)(6,31,54)(7,32,55)(8,33,56)(9,34,57)(10,35,58)(11,36,59)(12,37,60)(13,38,41)(14,39,42)(15,40,43)(16,21,44)(17,22,45)(18,23,46)(19,24,47)(20,25,48)(61,113,84)(62,114,85)(63,115,86)(64,116,87)(65,117,88)(66,118,89)(67,119,90)(68,120,91)(69,101,92)(70,102,93)(71,103,94)(72,104,95)(73,105,96)(74,106,97)(75,107,98)(76,108,99)(77,109,100)(78,110,81)(79,111,82)(80,112,83), (1,81)(3,83)(5,85)(7,87)(9,89)(11,91)(13,93)(15,95)(17,97)(19,99)(22,74)(24,76)(26,78)(28,80)(30,62)(32,64)(34,66)(36,68)(38,70)(40,72)(41,102)(43,104)(45,106)(47,108)(49,110)(51,112)(53,114)(55,116)(57,118)(59,120), (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,98)(19,99)(20,100)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,102)(42,103)(43,104)(44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,101), (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), (1,95,81,15)(2,94)(3,13,83,93)(4,12)(5,91,85,11)(6,90)(7,9,87,89)(10,86)(14,82)(16,20)(17,99,97,19)(18,98)(21,25)(22,76,74,24)(23,75)(26,72,78,40)(27,71)(28,38,80,70)(29,37)(30,68,62,36)(31,67)(32,34,64,66)(35,63)(39,79)(41,112,102,51)(42,111)(43,49,104,110)(44,48)(45,108,106,47)(46,107)(50,103)(52,60)(53,120,114,59)(54,119)(55,57,116,118)(58,115)(61,69)(73,77)(84,92)(96,100)(101,113)(105,109)>;
G:=Group( (1,26,49)(2,27,50)(3,28,51)(4,29,52)(5,30,53)(6,31,54)(7,32,55)(8,33,56)(9,34,57)(10,35,58)(11,36,59)(12,37,60)(13,38,41)(14,39,42)(15,40,43)(16,21,44)(17,22,45)(18,23,46)(19,24,47)(20,25,48)(61,113,84)(62,114,85)(63,115,86)(64,116,87)(65,117,88)(66,118,89)(67,119,90)(68,120,91)(69,101,92)(70,102,93)(71,103,94)(72,104,95)(73,105,96)(74,106,97)(75,107,98)(76,108,99)(77,109,100)(78,110,81)(79,111,82)(80,112,83), (1,81)(3,83)(5,85)(7,87)(9,89)(11,91)(13,93)(15,95)(17,97)(19,99)(22,74)(24,76)(26,78)(28,80)(30,62)(32,64)(34,66)(36,68)(38,70)(40,72)(41,102)(43,104)(45,106)(47,108)(49,110)(51,112)(53,114)(55,116)(57,118)(59,120), (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,98)(19,99)(20,100)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,102)(42,103)(43,104)(44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,101), (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), (1,95,81,15)(2,94)(3,13,83,93)(4,12)(5,91,85,11)(6,90)(7,9,87,89)(10,86)(14,82)(16,20)(17,99,97,19)(18,98)(21,25)(22,76,74,24)(23,75)(26,72,78,40)(27,71)(28,38,80,70)(29,37)(30,68,62,36)(31,67)(32,34,64,66)(35,63)(39,79)(41,112,102,51)(42,111)(43,49,104,110)(44,48)(45,108,106,47)(46,107)(50,103)(52,60)(53,120,114,59)(54,119)(55,57,116,118)(58,115)(61,69)(73,77)(84,92)(96,100)(101,113)(105,109) );
G=PermutationGroup([[(1,26,49),(2,27,50),(3,28,51),(4,29,52),(5,30,53),(6,31,54),(7,32,55),(8,33,56),(9,34,57),(10,35,58),(11,36,59),(12,37,60),(13,38,41),(14,39,42),(15,40,43),(16,21,44),(17,22,45),(18,23,46),(19,24,47),(20,25,48),(61,113,84),(62,114,85),(63,115,86),(64,116,87),(65,117,88),(66,118,89),(67,119,90),(68,120,91),(69,101,92),(70,102,93),(71,103,94),(72,104,95),(73,105,96),(74,106,97),(75,107,98),(76,108,99),(77,109,100),(78,110,81),(79,111,82),(80,112,83)], [(1,81),(3,83),(5,85),(7,87),(9,89),(11,91),(13,93),(15,95),(17,97),(19,99),(22,74),(24,76),(26,78),(28,80),(30,62),(32,64),(34,66),(36,68),(38,70),(40,72),(41,102),(43,104),(45,106),(47,108),(49,110),(51,112),(53,114),(55,116),(57,118),(59,120)], [(1,81),(2,82),(3,83),(4,84),(5,85),(6,86),(7,87),(8,88),(9,89),(10,90),(11,91),(12,92),(13,93),(14,94),(15,95),(16,96),(17,97),(18,98),(19,99),(20,100),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,61),(30,62),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72),(41,102),(42,103),(43,104),(44,105),(45,106),(46,107),(47,108),(48,109),(49,110),(50,111),(51,112),(52,113),(53,114),(54,115),(55,116),(56,117),(57,118),(58,119),(59,120),(60,101)], [(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)], [(1,95,81,15),(2,94),(3,13,83,93),(4,12),(5,91,85,11),(6,90),(7,9,87,89),(10,86),(14,82),(16,20),(17,99,97,19),(18,98),(21,25),(22,76,74,24),(23,75),(26,72,78,40),(27,71),(28,38,80,70),(29,37),(30,68,62,36),(31,67),(32,34,64,66),(35,63),(39,79),(41,112,102,51),(42,111),(43,49,104,110),(44,48),(45,108,106,47),(46,107),(50,103),(52,60),(53,120,114,59),(54,119),(55,57,116,118),(58,115),(61,69),(73,77),(84,92),(96,100),(101,113),(105,109)]])
93 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 20 | 1 | 1 | 4 | 4 | 20 | 20 | 20 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
93 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 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D4 | D5 | D10 | C3×D4 | C3×D5 | C4×D5 | D20 | C5⋊D4 | C6×D5 | D5×C12 | C3×D20 | C3×C5⋊D4 | C23⋊C4 | C3×C23⋊C4 | C23.1D10 | C3×C23.1D10 |
| kernel | C3×C23.1D10 | C3×C23.D5 | C15×C22⋊C4 | C6×C5⋊D4 | C23.1D10 | C6×Dic5 | D5×C2×C6 | C23.D5 | C5×C22⋊C4 | C2×C5⋊D4 | C2×Dic5 | C22×D5 | C2×C30 | C3×C22⋊C4 | C22×C6 | C2×C10 | C22⋊C4 | C2×C6 | C2×C6 | C2×C6 | C23 | C22 | C22 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C3×C23.1D10 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 60 | 0 | 1 | 0 |
| 0 | 60 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 43 | 60 | 36 | 2 |
| 1 | 0 | 59 | 0 |
| 56 | 45 | 18 | 1 |
| 16 | 39 | 60 | 0 |
| 53 | 31 | 0 | 0 |
| 53 | 8 | 0 | 0 |
| 5 | 16 | 43 | 60 |
| 48 | 56 | 18 | 18 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[60,0,60,0,0,60,0,60,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[43,1,56,16,60,0,45,39,36,59,18,60,2,0,1,0],[53,53,5,48,31,8,16,56,0,0,43,18,0,0,60,18] >;
C3×C23.1D10 in GAP, Magma, Sage, TeX
C_3\times C_2^3._1D_{10} % in TeX
G:=Group("C3xC2^3.1D10"); // GroupNames label
G:=SmallGroup(480,84);
// by ID
G=gap.SmallGroup(480,84);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,365,92,1683,1271,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^20=1,e^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations