direct product, metabelian, nilpotent (class 2), monomial
Aliases: C32×C4⋊1D4, C122⋊20C2, C23.5C62, C62.296C23, C12⋊6(C3×D4), (C4×C12)⋊19C6, (C6×D4)⋊12C6, (C3×C12)⋊21D4, C42⋊9(C3×C6), C6.89(C6×D4), C4⋊1(D4×C32), (C2×C4).22C62, (C2×C62).5C22, (C6×C12).372C22, C22.17(C2×C62), C2.9(D4×C3×C6), (D4×C3×C6)⋊21C2, (C2×D4)⋊3(C3×C6), (C3×C6).306(C2×D4), (C2×C12).158(C2×C6), (C22×C6).15(C2×C6), (C2×C6).102(C22×C6), SmallGroup(288,824)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C4⋊1D4
G = < a,b,c,d,e | a3=b3=c4=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 540 in 324 conjugacy classes, 156 normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, D4, C23, C32, C12, C2×C6, C2×C6, C42, C2×D4, C3×C6, C3×C6, C2×C12, C3×D4, C22×C6, C4⋊1D4, C3×C12, C62, C62, C4×C12, C6×D4, C6×C12, D4×C32, C2×C62, C3×C4⋊1D4, C122, D4×C3×C6, C32×C4⋊1D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2×C6, C2×D4, C3×C6, C3×D4, C22×C6, C4⋊1D4, C62, C6×D4, D4×C32, C2×C62, C3×C4⋊1D4, D4×C3×C6, C32×C4⋊1D4
►On 144 points
Generators in S144
(1 86 78)(2 87 79)(3 88 80)(4 85 77)(5 122 13)(6 123 14)(7 124 15)(8 121 16)(9 114 17)(10 115 18)(11 116 19)(12 113 20)(21 109 29)(22 110 30)(23 111 31)(24 112 32)(25 142 134)(26 143 135)(27 144 136)(28 141 133)(33 138 130)(34 139 131)(35 140 132)(36 137 129)(37 81 45)(38 82 46)(39 83 47)(40 84 48)(41 73 49)(42 74 50)(43 75 51)(44 76 52)(53 98 90)(54 99 91)(55 100 92)(56 97 89)(57 101 65)(58 102 66)(59 103 67)(60 104 68)(61 93 69)(62 94 70)(63 95 71)(64 96 72)(105 125 117)(106 126 118)(107 127 119)(108 128 120)
(1 74 38)(2 75 39)(3 76 40)(4 73 37)(5 106 114)(6 107 115)(7 108 116)(8 105 113)(9 13 118)(10 14 119)(11 15 120)(12 16 117)(17 122 126)(18 123 127)(19 124 128)(20 121 125)(21 25 130)(22 26 131)(23 27 132)(24 28 129)(29 134 138)(30 135 139)(31 136 140)(32 133 137)(33 109 142)(34 110 143)(35 111 144)(36 112 141)(41 45 77)(42 46 78)(43 47 79)(44 48 80)(49 81 85)(50 82 86)(51 83 87)(52 84 88)(53 70 102)(54 71 103)(55 72 104)(56 69 101)(57 89 93)(58 90 94)(59 91 95)(60 92 96)(61 65 97)(62 66 98)(63 67 99)(64 68 100)
(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)(121 122 123 124)(125 126 127 128)(129 130 131 132)(133 134 135 136)(137 138 139 140)(141 142 143 144)
(1 91 22 107)(2 92 23 108)(3 89 24 105)(4 90 21 106)(5 37 58 130)(6 38 59 131)(7 39 60 132)(8 40 57 129)(9 41 62 134)(10 42 63 135)(11 43 64 136)(12 44 61 133)(13 45 66 138)(14 46 67 139)(15 47 68 140)(16 48 65 137)(17 49 70 142)(18 50 71 143)(19 51 72 144)(20 52 69 141)(25 114 73 94)(26 115 74 95)(27 116 75 96)(28 113 76 93)(29 118 77 98)(30 119 78 99)(31 120 79 100)(32 117 80 97)(33 122 81 102)(34 123 82 103)(35 124 83 104)(36 121 84 101)(53 109 126 85)(54 110 127 86)(55 111 128 87)(56 112 125 88)
(1 21)(2 24)(3 23)(4 22)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18)(19 20)(25 74)(26 73)(27 76)(28 75)(29 78)(30 77)(31 80)(32 79)(33 82)(34 81)(35 84)(36 83)(37 131)(38 130)(39 129)(40 132)(41 135)(42 134)(43 133)(44 136)(45 139)(46 138)(47 137)(48 140)(49 143)(50 142)(51 141)(52 144)(53 54)(55 56)(57 60)(58 59)(61 64)(62 63)(65 68)(66 67)(69 72)(70 71)(85 110)(86 109)(87 112)(88 111)(89 92)(90 91)(93 96)(94 95)(97 100)(98 99)(101 104)(102 103)(105 108)(106 107)(113 116)(114 115)(117 120)(118 119)(121 124)(122 123)(125 128)(126 127)
G:=sub<Sym(144)| (1,86,78)(2,87,79)(3,88,80)(4,85,77)(5,122,13)(6,123,14)(7,124,15)(8,121,16)(9,114,17)(10,115,18)(11,116,19)(12,113,20)(21,109,29)(22,110,30)(23,111,31)(24,112,32)(25,142,134)(26,143,135)(27,144,136)(28,141,133)(33,138,130)(34,139,131)(35,140,132)(36,137,129)(37,81,45)(38,82,46)(39,83,47)(40,84,48)(41,73,49)(42,74,50)(43,75,51)(44,76,52)(53,98,90)(54,99,91)(55,100,92)(56,97,89)(57,101,65)(58,102,66)(59,103,67)(60,104,68)(61,93,69)(62,94,70)(63,95,71)(64,96,72)(105,125,117)(106,126,118)(107,127,119)(108,128,120), (1,74,38)(2,75,39)(3,76,40)(4,73,37)(5,106,114)(6,107,115)(7,108,116)(8,105,113)(9,13,118)(10,14,119)(11,15,120)(12,16,117)(17,122,126)(18,123,127)(19,124,128)(20,121,125)(21,25,130)(22,26,131)(23,27,132)(24,28,129)(29,134,138)(30,135,139)(31,136,140)(32,133,137)(33,109,142)(34,110,143)(35,111,144)(36,112,141)(41,45,77)(42,46,78)(43,47,79)(44,48,80)(49,81,85)(50,82,86)(51,83,87)(52,84,88)(53,70,102)(54,71,103)(55,72,104)(56,69,101)(57,89,93)(58,90,94)(59,91,95)(60,92,96)(61,65,97)(62,66,98)(63,67,99)(64,68,100), (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)(121,122,123,124)(125,126,127,128)(129,130,131,132)(133,134,135,136)(137,138,139,140)(141,142,143,144), (1,91,22,107)(2,92,23,108)(3,89,24,105)(4,90,21,106)(5,37,58,130)(6,38,59,131)(7,39,60,132)(8,40,57,129)(9,41,62,134)(10,42,63,135)(11,43,64,136)(12,44,61,133)(13,45,66,138)(14,46,67,139)(15,47,68,140)(16,48,65,137)(17,49,70,142)(18,50,71,143)(19,51,72,144)(20,52,69,141)(25,114,73,94)(26,115,74,95)(27,116,75,96)(28,113,76,93)(29,118,77,98)(30,119,78,99)(31,120,79,100)(32,117,80,97)(33,122,81,102)(34,123,82,103)(35,124,83,104)(36,121,84,101)(53,109,126,85)(54,110,127,86)(55,111,128,87)(56,112,125,88), (1,21)(2,24)(3,23)(4,22)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(25,74)(26,73)(27,76)(28,75)(29,78)(30,77)(31,80)(32,79)(33,82)(34,81)(35,84)(36,83)(37,131)(38,130)(39,129)(40,132)(41,135)(42,134)(43,133)(44,136)(45,139)(46,138)(47,137)(48,140)(49,143)(50,142)(51,141)(52,144)(53,54)(55,56)(57,60)(58,59)(61,64)(62,63)(65,68)(66,67)(69,72)(70,71)(85,110)(86,109)(87,112)(88,111)(89,92)(90,91)(93,96)(94,95)(97,100)(98,99)(101,104)(102,103)(105,108)(106,107)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)>;
G:=Group( (1,86,78)(2,87,79)(3,88,80)(4,85,77)(5,122,13)(6,123,14)(7,124,15)(8,121,16)(9,114,17)(10,115,18)(11,116,19)(12,113,20)(21,109,29)(22,110,30)(23,111,31)(24,112,32)(25,142,134)(26,143,135)(27,144,136)(28,141,133)(33,138,130)(34,139,131)(35,140,132)(36,137,129)(37,81,45)(38,82,46)(39,83,47)(40,84,48)(41,73,49)(42,74,50)(43,75,51)(44,76,52)(53,98,90)(54,99,91)(55,100,92)(56,97,89)(57,101,65)(58,102,66)(59,103,67)(60,104,68)(61,93,69)(62,94,70)(63,95,71)(64,96,72)(105,125,117)(106,126,118)(107,127,119)(108,128,120), (1,74,38)(2,75,39)(3,76,40)(4,73,37)(5,106,114)(6,107,115)(7,108,116)(8,105,113)(9,13,118)(10,14,119)(11,15,120)(12,16,117)(17,122,126)(18,123,127)(19,124,128)(20,121,125)(21,25,130)(22,26,131)(23,27,132)(24,28,129)(29,134,138)(30,135,139)(31,136,140)(32,133,137)(33,109,142)(34,110,143)(35,111,144)(36,112,141)(41,45,77)(42,46,78)(43,47,79)(44,48,80)(49,81,85)(50,82,86)(51,83,87)(52,84,88)(53,70,102)(54,71,103)(55,72,104)(56,69,101)(57,89,93)(58,90,94)(59,91,95)(60,92,96)(61,65,97)(62,66,98)(63,67,99)(64,68,100), (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)(121,122,123,124)(125,126,127,128)(129,130,131,132)(133,134,135,136)(137,138,139,140)(141,142,143,144), (1,91,22,107)(2,92,23,108)(3,89,24,105)(4,90,21,106)(5,37,58,130)(6,38,59,131)(7,39,60,132)(8,40,57,129)(9,41,62,134)(10,42,63,135)(11,43,64,136)(12,44,61,133)(13,45,66,138)(14,46,67,139)(15,47,68,140)(16,48,65,137)(17,49,70,142)(18,50,71,143)(19,51,72,144)(20,52,69,141)(25,114,73,94)(26,115,74,95)(27,116,75,96)(28,113,76,93)(29,118,77,98)(30,119,78,99)(31,120,79,100)(32,117,80,97)(33,122,81,102)(34,123,82,103)(35,124,83,104)(36,121,84,101)(53,109,126,85)(54,110,127,86)(55,111,128,87)(56,112,125,88), (1,21)(2,24)(3,23)(4,22)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(25,74)(26,73)(27,76)(28,75)(29,78)(30,77)(31,80)(32,79)(33,82)(34,81)(35,84)(36,83)(37,131)(38,130)(39,129)(40,132)(41,135)(42,134)(43,133)(44,136)(45,139)(46,138)(47,137)(48,140)(49,143)(50,142)(51,141)(52,144)(53,54)(55,56)(57,60)(58,59)(61,64)(62,63)(65,68)(66,67)(69,72)(70,71)(85,110)(86,109)(87,112)(88,111)(89,92)(90,91)(93,96)(94,95)(97,100)(98,99)(101,104)(102,103)(105,108)(106,107)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127) );
G=PermutationGroup([[(1,86,78),(2,87,79),(3,88,80),(4,85,77),(5,122,13),(6,123,14),(7,124,15),(8,121,16),(9,114,17),(10,115,18),(11,116,19),(12,113,20),(21,109,29),(22,110,30),(23,111,31),(24,112,32),(25,142,134),(26,143,135),(27,144,136),(28,141,133),(33,138,130),(34,139,131),(35,140,132),(36,137,129),(37,81,45),(38,82,46),(39,83,47),(40,84,48),(41,73,49),(42,74,50),(43,75,51),(44,76,52),(53,98,90),(54,99,91),(55,100,92),(56,97,89),(57,101,65),(58,102,66),(59,103,67),(60,104,68),(61,93,69),(62,94,70),(63,95,71),(64,96,72),(105,125,117),(106,126,118),(107,127,119),(108,128,120)], [(1,74,38),(2,75,39),(3,76,40),(4,73,37),(5,106,114),(6,107,115),(7,108,116),(8,105,113),(9,13,118),(10,14,119),(11,15,120),(12,16,117),(17,122,126),(18,123,127),(19,124,128),(20,121,125),(21,25,130),(22,26,131),(23,27,132),(24,28,129),(29,134,138),(30,135,139),(31,136,140),(32,133,137),(33,109,142),(34,110,143),(35,111,144),(36,112,141),(41,45,77),(42,46,78),(43,47,79),(44,48,80),(49,81,85),(50,82,86),(51,83,87),(52,84,88),(53,70,102),(54,71,103),(55,72,104),(56,69,101),(57,89,93),(58,90,94),(59,91,95),(60,92,96),(61,65,97),(62,66,98),(63,67,99),(64,68,100)], [(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),(121,122,123,124),(125,126,127,128),(129,130,131,132),(133,134,135,136),(137,138,139,140),(141,142,143,144)], [(1,91,22,107),(2,92,23,108),(3,89,24,105),(4,90,21,106),(5,37,58,130),(6,38,59,131),(7,39,60,132),(8,40,57,129),(9,41,62,134),(10,42,63,135),(11,43,64,136),(12,44,61,133),(13,45,66,138),(14,46,67,139),(15,47,68,140),(16,48,65,137),(17,49,70,142),(18,50,71,143),(19,51,72,144),(20,52,69,141),(25,114,73,94),(26,115,74,95),(27,116,75,96),(28,113,76,93),(29,118,77,98),(30,119,78,99),(31,120,79,100),(32,117,80,97),(33,122,81,102),(34,123,82,103),(35,124,83,104),(36,121,84,101),(53,109,126,85),(54,110,127,86),(55,111,128,87),(56,112,125,88)], [(1,21),(2,24),(3,23),(4,22),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(25,74),(26,73),(27,76),(28,75),(29,78),(30,77),(31,80),(32,79),(33,82),(34,81),(35,84),(36,83),(37,131),(38,130),(39,129),(40,132),(41,135),(42,134),(43,133),(44,136),(45,139),(46,138),(47,137),(48,140),(49,143),(50,142),(51,141),(52,144),(53,54),(55,56),(57,60),(58,59),(61,64),(62,63),(65,68),(66,67),(69,72),(70,71),(85,110),(86,109),(87,112),(88,111),(89,92),(90,91),(93,96),(94,95),(97,100),(98,99),(101,104),(102,103),(105,108),(106,107),(113,116),(114,115),(117,120),(118,119),(121,124),(122,123),(125,128),(126,127)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | ··· | 3H | 4A | ··· | 4F | 6A | ··· | 6X | 6Y | ··· | 6BD | 12A | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | C3×D4 |
| kernel | C32×C4⋊1D4 | C122 | D4×C3×C6 | C3×C4⋊1D4 | C4×C12 | C6×D4 | C3×C12 | C12 |
| # reps | 1 | 1 | 6 | 8 | 8 | 48 | 6 | 48 |
Matrix representation of C32×C4⋊1D4 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 4 |
| 0 | 0 | 7 | 7 |
| 4 | 11 | 0 | 0 |
| 2 | 9 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 9 | 1 | 0 | 0 |
| 0 | 0 | 6 | 4 |
| 0 | 0 | 1 | 7 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,6,7,0,0,4,7],[4,2,0,0,11,9,0,0,0,0,12,0,0,0,0,12],[12,9,0,0,0,1,0,0,0,0,6,1,0,0,4,7] >;
C32×C4⋊1D4 in GAP, Magma, Sage, TeX
C_3^2\times C_4\rtimes_1D_4
% in TeX
G:=Group("C3^2xC4:1D4"); // GroupNames label
G:=SmallGroup(288,824);
// by ID
G=gap.SmallGroup(288,824);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,512,3110,772]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations