direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C22.2D20, (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), (C6×C5⋊D4).8C2, (C2×C5⋊D4).1C6, (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×C22.2D20
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 [×4], C3, C4 [×3], C22 [×3], C22 [×3], C5, C6, C6 [×4], C2×C4 [×3], D4 [×2], C23, C23, D5, C10, C10 [×3], C12 [×3], C2×C6 [×3], C2×C6 [×3], C15, C22⋊C4, C22⋊C4, C2×D4, Dic5 [×2], C20, D10 [×2], C2×C10 [×3], C2×C10, C2×C12 [×3], C3×D4 [×2], C22×C6, C22×C6, C3×D5, C30, C30 [×3], C23⋊C4, C2×Dic5, C2×Dic5, C5⋊D4 [×2], C2×C20, C22×D5, C22×C10, C3×C22⋊C4, C3×C22⋊C4, C6×D4, C3×Dic5 [×2], C60, C6×D5 [×2], C2×C30 [×3], C2×C30, C23.D5, C5×C22⋊C4, C2×C5⋊D4, C3×C23⋊C4, C6×Dic5, C6×Dic5, C3×C5⋊D4 [×2], C2×C60, D5×C2×C6, C22×C30, C22.2D20, C3×C23.D5, C15×C22⋊C4, C6×C5⋊D4, C3×C22.2D20
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, D10, C2×C12, C3×D4 [×2], 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, C22.2D20, C3×D10⋊C4, C3×C22.2D20
►On 120 points
Generators in S120
(1 80 108)(2 61 109)(3 62 110)(4 63 111)(5 64 112)(6 65 113)(7 66 114)(8 67 115)(9 68 116)(10 69 117)(11 70 118)(12 71 119)(13 72 120)(14 73 101)(15 74 102)(16 75 103)(17 76 104)(18 77 105)(19 78 106)(20 79 107)(21 82 48)(22 83 49)(23 84 50)(24 85 51)(25 86 52)(26 87 53)(27 88 54)(28 89 55)(29 90 56)(30 91 57)(31 92 58)(32 93 59)(33 94 60)(34 95 41)(35 96 42)(36 97 43)(37 98 44)(38 99 45)(39 100 46)(40 81 47)
(1 95)(3 97)(5 99)(7 81)(9 83)(11 85)(13 87)(15 89)(17 91)(19 93)(22 116)(24 118)(26 120)(28 102)(30 104)(32 106)(34 108)(36 110)(38 112)(40 114)(41 80)(43 62)(45 64)(47 66)(49 68)(51 70)(53 72)(55 74)(57 76)(59 78)
(1 95)(2 96)(3 97)(4 98)(5 99)(6 100)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 115)(22 116)(23 117)(24 118)(25 119)(26 120)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 80)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)
(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 89 95 15)(2 88)(3 13 97 87)(4 12)(5 85 99 11)(6 84)(7 9 81 83)(10 100)(14 96)(16 20)(17 93 91 19)(18 92)(22 114 116 40)(23 113)(24 38 118 112)(25 37)(26 110 120 36)(27 109)(28 34 102 108)(29 33)(30 106 104 32)(31 105)(35 101)(39 117)(41 74 80 55)(42 73)(43 53 62 72)(44 52)(45 70 64 51)(46 69)(47 49 66 68)(50 65)(54 61)(56 60)(57 78 76 59)(58 77)(63 71)(75 79)(86 98)(90 94)(103 107)(111 119)
G:=sub<Sym(120)| (1,80,108)(2,61,109)(3,62,110)(4,63,111)(5,64,112)(6,65,113)(7,66,114)(8,67,115)(9,68,116)(10,69,117)(11,70,118)(12,71,119)(13,72,120)(14,73,101)(15,74,102)(16,75,103)(17,76,104)(18,77,105)(19,78,106)(20,79,107)(21,82,48)(22,83,49)(23,84,50)(24,85,51)(25,86,52)(26,87,53)(27,88,54)(28,89,55)(29,90,56)(30,91,57)(31,92,58)(32,93,59)(33,94,60)(34,95,41)(35,96,42)(36,97,43)(37,98,44)(38,99,45)(39,100,46)(40,81,47), (1,95)(3,97)(5,99)(7,81)(9,83)(11,85)(13,87)(15,89)(17,91)(19,93)(22,116)(24,118)(26,120)(28,102)(30,104)(32,106)(34,108)(36,110)(38,112)(40,114)(41,80)(43,62)(45,64)(47,66)(49,68)(51,70)(53,72)(55,74)(57,76)(59,78), (1,95)(2,96)(3,97)(4,98)(5,99)(6,100)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,80)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79), (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), (1,89,95,15)(2,88)(3,13,97,87)(4,12)(5,85,99,11)(6,84)(7,9,81,83)(10,100)(14,96)(16,20)(17,93,91,19)(18,92)(22,114,116,40)(23,113)(24,38,118,112)(25,37)(26,110,120,36)(27,109)(28,34,102,108)(29,33)(30,106,104,32)(31,105)(35,101)(39,117)(41,74,80,55)(42,73)(43,53,62,72)(44,52)(45,70,64,51)(46,69)(47,49,66,68)(50,65)(54,61)(56,60)(57,78,76,59)(58,77)(63,71)(75,79)(86,98)(90,94)(103,107)(111,119)>;
G:=Group( (1,80,108)(2,61,109)(3,62,110)(4,63,111)(5,64,112)(6,65,113)(7,66,114)(8,67,115)(9,68,116)(10,69,117)(11,70,118)(12,71,119)(13,72,120)(14,73,101)(15,74,102)(16,75,103)(17,76,104)(18,77,105)(19,78,106)(20,79,107)(21,82,48)(22,83,49)(23,84,50)(24,85,51)(25,86,52)(26,87,53)(27,88,54)(28,89,55)(29,90,56)(30,91,57)(31,92,58)(32,93,59)(33,94,60)(34,95,41)(35,96,42)(36,97,43)(37,98,44)(38,99,45)(39,100,46)(40,81,47), (1,95)(3,97)(5,99)(7,81)(9,83)(11,85)(13,87)(15,89)(17,91)(19,93)(22,116)(24,118)(26,120)(28,102)(30,104)(32,106)(34,108)(36,110)(38,112)(40,114)(41,80)(43,62)(45,64)(47,66)(49,68)(51,70)(53,72)(55,74)(57,76)(59,78), (1,95)(2,96)(3,97)(4,98)(5,99)(6,100)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,80)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79), (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), (1,89,95,15)(2,88)(3,13,97,87)(4,12)(5,85,99,11)(6,84)(7,9,81,83)(10,100)(14,96)(16,20)(17,93,91,19)(18,92)(22,114,116,40)(23,113)(24,38,118,112)(25,37)(26,110,120,36)(27,109)(28,34,102,108)(29,33)(30,106,104,32)(31,105)(35,101)(39,117)(41,74,80,55)(42,73)(43,53,62,72)(44,52)(45,70,64,51)(46,69)(47,49,66,68)(50,65)(54,61)(56,60)(57,78,76,59)(58,77)(63,71)(75,79)(86,98)(90,94)(103,107)(111,119) );
G=PermutationGroup([(1,80,108),(2,61,109),(3,62,110),(4,63,111),(5,64,112),(6,65,113),(7,66,114),(8,67,115),(9,68,116),(10,69,117),(11,70,118),(12,71,119),(13,72,120),(14,73,101),(15,74,102),(16,75,103),(17,76,104),(18,77,105),(19,78,106),(20,79,107),(21,82,48),(22,83,49),(23,84,50),(24,85,51),(25,86,52),(26,87,53),(27,88,54),(28,89,55),(29,90,56),(30,91,57),(31,92,58),(32,93,59),(33,94,60),(34,95,41),(35,96,42),(36,97,43),(37,98,44),(38,99,45),(39,100,46),(40,81,47)], [(1,95),(3,97),(5,99),(7,81),(9,83),(11,85),(13,87),(15,89),(17,91),(19,93),(22,116),(24,118),(26,120),(28,102),(30,104),(32,106),(34,108),(36,110),(38,112),(40,114),(41,80),(43,62),(45,64),(47,66),(49,68),(51,70),(53,72),(55,74),(57,76),(59,78)], [(1,95),(2,96),(3,97),(4,98),(5,99),(6,100),(7,81),(8,82),(9,83),(10,84),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,91),(18,92),(19,93),(20,94),(21,115),(22,116),(23,117),(24,118),(25,119),(26,120),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,113),(40,114),(41,80),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,68),(50,69),(51,70),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79)], [(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)], [(1,89,95,15),(2,88),(3,13,97,87),(4,12),(5,85,99,11),(6,84),(7,9,81,83),(10,100),(14,96),(16,20),(17,93,91,19),(18,92),(22,114,116,40),(23,113),(24,38,118,112),(25,37),(26,110,120,36),(27,109),(28,34,102,108),(29,33),(30,106,104,32),(31,105),(35,101),(39,117),(41,74,80,55),(42,73),(43,53,62,72),(44,52),(45,70,64,51),(46,69),(47,49,66,68),(50,65),(54,61),(56,60),(57,78,76,59),(58,77),(63,71),(75,79),(86,98),(90,94),(103,107),(111,119)])
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 | C22.2D20 | C3×C22.2D20 |
| kernel | C3×C22.2D20 | C3×C23.D5 | C15×C22⋊C4 | C6×C5⋊D4 | C22.2D20 | 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×C22.2D20 ►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×C22.2D20 in GAP, Magma, Sage, TeX
C_3\times C_2^2._2D_{20} % in TeX
G:=Group("C3xC2^2.2D20"); // 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