direct product, metabelian, nilpotent (class 3), monomial
Aliases: C32×D4⋊C4, C62.138D4, (C2×C24)⋊4C6, (C6×C24)⋊6C2, (C3×D4)⋊4C12, D4⋊1(C3×C12), C4.1(C6×C12), C6.19(C3×D8), (C3×C6).42D8, (C6×D4).20C6, C12.77(C3×D4), C2.1(C32×D8), C12.36(C2×C12), (C3×C12).178D4, (D4×C32)⋊10C4, (C2×C4).14C62, (C3×C6).31SD16, C6.13(C3×SD16), C4.11(D4×C32), C2.1(C32×SD16), C22.8(D4×C32), (C6×C12).361C22, C4⋊C4⋊1(C3×C6), (C2×C8)⋊2(C3×C6), (C3×C4⋊C4)⋊10C6, (D4×C3×C6).15C2, (C2×D4).3(C3×C6), (C2×C6).65(C3×D4), (C32×C4⋊C4)⋊19C2, C6.33(C3×C22⋊C4), (C2×C12).148(C2×C6), (C3×C12).118(C2×C4), C2.6(C32×C22⋊C4), (C3×C6).82(C22⋊C4), SmallGroup(288,320)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×D4⋊C4
G = < a,b,c,d,e | a3=b3=c4=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=ece-1=c-1, ede-1=cd >
Subgroups: 276 in 150 conjugacy classes, 84 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, C32, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×D4, C3×C6, C3×C6, C24, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, D4⋊C4, C3×C12, C3×C12, C62, C62, C3×C4⋊C4, C2×C24, C6×D4, C3×C24, C6×C12, C6×C12, D4×C32, D4×C32, C2×C62, C3×D4⋊C4, C32×C4⋊C4, C6×C24, D4×C3×C6, C32×D4⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C32, C12, C2×C6, C22⋊C4, D8, SD16, C3×C6, C2×C12, C3×D4, D4⋊C4, C3×C12, C62, C3×C22⋊C4, C3×D8, C3×SD16, C6×C12, D4×C32, C3×D4⋊C4, C32×C22⋊C4, C32×D8, C32×SD16, C32×D4⋊C4
►On 144 points
Generators in S144
(1 72 40)(2 69 37)(3 70 38)(4 71 39)(5 114 82)(6 115 83)(7 116 84)(8 113 81)(9 76 41)(10 73 42)(11 74 43)(12 75 44)(13 53 45)(14 54 46)(15 55 47)(16 56 48)(17 57 49)(18 58 50)(19 59 51)(20 60 52)(21 61 29)(22 62 30)(23 63 31)(24 64 32)(25 65 33)(26 66 34)(27 67 35)(28 68 36)(77 141 109)(78 142 110)(79 143 111)(80 144 112)(85 125 117)(86 126 118)(87 127 119)(88 128 120)(89 129 121)(90 130 122)(91 131 123)(92 132 124)(93 133 101)(94 134 102)(95 135 103)(96 136 104)(97 137 105)(98 138 106)(99 139 107)(100 140 108)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 138 130)(6 139 131)(7 140 132)(8 137 129)(9 25 17)(10 26 18)(11 27 19)(12 28 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 49 41)(34 50 42)(35 51 43)(36 52 44)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(57 76 65)(58 73 66)(59 74 67)(60 75 68)(77 93 85)(78 94 86)(79 95 87)(80 96 88)(81 97 89)(82 98 90)(83 99 91)(84 100 92)(101 117 109)(102 118 110)(103 119 111)(104 120 112)(105 121 113)(106 122 114)(107 123 115)(108 124 116)(125 141 133)(126 142 134)(127 143 135)(128 144 136)
(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 4)(2 3)(5 7)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)(25 28)(26 27)(29 30)(31 32)(33 36)(34 35)(37 38)(39 40)(41 44)(42 43)(45 46)(47 48)(49 52)(50 51)(53 54)(55 56)(57 60)(58 59)(61 62)(63 64)(65 68)(66 67)(69 70)(71 72)(73 74)(75 76)(78 80)(82 84)(86 88)(90 92)(94 96)(98 100)(102 104)(106 108)(110 112)(114 116)(118 120)(122 124)(126 128)(130 132)(134 136)(138 140)(142 144)
(1 84 11 80)(2 83 12 79)(3 82 9 78)(4 81 10 77)(5 76 142 70)(6 75 143 69)(7 74 144 72)(8 73 141 71)(13 91 20 87)(14 90 17 86)(15 89 18 85)(16 92 19 88)(21 99 28 95)(22 98 25 94)(23 97 26 93)(24 100 27 96)(29 107 36 103)(30 106 33 102)(31 105 34 101)(32 108 35 104)(37 115 44 111)(38 114 41 110)(39 113 42 109)(40 116 43 112)(45 123 52 119)(46 122 49 118)(47 121 50 117)(48 124 51 120)(53 131 60 127)(54 130 57 126)(55 129 58 125)(56 132 59 128)(61 139 68 135)(62 138 65 134)(63 137 66 133)(64 140 67 136)
G:=sub<Sym(144)| (1,72,40)(2,69,37)(3,70,38)(4,71,39)(5,114,82)(6,115,83)(7,116,84)(8,113,81)(9,76,41)(10,73,42)(11,74,43)(12,75,44)(13,53,45)(14,54,46)(15,55,47)(16,56,48)(17,57,49)(18,58,50)(19,59,51)(20,60,52)(21,61,29)(22,62,30)(23,63,31)(24,64,32)(25,65,33)(26,66,34)(27,67,35)(28,68,36)(77,141,109)(78,142,110)(79,143,111)(80,144,112)(85,125,117)(86,126,118)(87,127,119)(88,128,120)(89,129,121)(90,130,122)(91,131,123)(92,132,124)(93,133,101)(94,134,102)(95,135,103)(96,136,104)(97,137,105)(98,138,106)(99,139,107)(100,140,108), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,138,130)(6,139,131)(7,140,132)(8,137,129)(9,25,17)(10,26,18)(11,27,19)(12,28,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,49,41)(34,50,42)(35,51,43)(36,52,44)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,76,65)(58,73,66)(59,74,67)(60,75,68)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(81,97,89)(82,98,90)(83,99,91)(84,100,92)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(105,121,113)(106,122,114)(107,123,115)(108,124,116)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (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), (1,4)(2,3)(5,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,28)(26,27)(29,30)(31,32)(33,36)(34,35)(37,38)(39,40)(41,44)(42,43)(45,46)(47,48)(49,52)(50,51)(53,54)(55,56)(57,60)(58,59)(61,62)(63,64)(65,68)(66,67)(69,70)(71,72)(73,74)(75,76)(78,80)(82,84)(86,88)(90,92)(94,96)(98,100)(102,104)(106,108)(110,112)(114,116)(118,120)(122,124)(126,128)(130,132)(134,136)(138,140)(142,144), (1,84,11,80)(2,83,12,79)(3,82,9,78)(4,81,10,77)(5,76,142,70)(6,75,143,69)(7,74,144,72)(8,73,141,71)(13,91,20,87)(14,90,17,86)(15,89,18,85)(16,92,19,88)(21,99,28,95)(22,98,25,94)(23,97,26,93)(24,100,27,96)(29,107,36,103)(30,106,33,102)(31,105,34,101)(32,108,35,104)(37,115,44,111)(38,114,41,110)(39,113,42,109)(40,116,43,112)(45,123,52,119)(46,122,49,118)(47,121,50,117)(48,124,51,120)(53,131,60,127)(54,130,57,126)(55,129,58,125)(56,132,59,128)(61,139,68,135)(62,138,65,134)(63,137,66,133)(64,140,67,136)>;
G:=Group( (1,72,40)(2,69,37)(3,70,38)(4,71,39)(5,114,82)(6,115,83)(7,116,84)(8,113,81)(9,76,41)(10,73,42)(11,74,43)(12,75,44)(13,53,45)(14,54,46)(15,55,47)(16,56,48)(17,57,49)(18,58,50)(19,59,51)(20,60,52)(21,61,29)(22,62,30)(23,63,31)(24,64,32)(25,65,33)(26,66,34)(27,67,35)(28,68,36)(77,141,109)(78,142,110)(79,143,111)(80,144,112)(85,125,117)(86,126,118)(87,127,119)(88,128,120)(89,129,121)(90,130,122)(91,131,123)(92,132,124)(93,133,101)(94,134,102)(95,135,103)(96,136,104)(97,137,105)(98,138,106)(99,139,107)(100,140,108), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,138,130)(6,139,131)(7,140,132)(8,137,129)(9,25,17)(10,26,18)(11,27,19)(12,28,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,49,41)(34,50,42)(35,51,43)(36,52,44)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,76,65)(58,73,66)(59,74,67)(60,75,68)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(81,97,89)(82,98,90)(83,99,91)(84,100,92)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(105,121,113)(106,122,114)(107,123,115)(108,124,116)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (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), (1,4)(2,3)(5,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,28)(26,27)(29,30)(31,32)(33,36)(34,35)(37,38)(39,40)(41,44)(42,43)(45,46)(47,48)(49,52)(50,51)(53,54)(55,56)(57,60)(58,59)(61,62)(63,64)(65,68)(66,67)(69,70)(71,72)(73,74)(75,76)(78,80)(82,84)(86,88)(90,92)(94,96)(98,100)(102,104)(106,108)(110,112)(114,116)(118,120)(122,124)(126,128)(130,132)(134,136)(138,140)(142,144), (1,84,11,80)(2,83,12,79)(3,82,9,78)(4,81,10,77)(5,76,142,70)(6,75,143,69)(7,74,144,72)(8,73,141,71)(13,91,20,87)(14,90,17,86)(15,89,18,85)(16,92,19,88)(21,99,28,95)(22,98,25,94)(23,97,26,93)(24,100,27,96)(29,107,36,103)(30,106,33,102)(31,105,34,101)(32,108,35,104)(37,115,44,111)(38,114,41,110)(39,113,42,109)(40,116,43,112)(45,123,52,119)(46,122,49,118)(47,121,50,117)(48,124,51,120)(53,131,60,127)(54,130,57,126)(55,129,58,125)(56,132,59,128)(61,139,68,135)(62,138,65,134)(63,137,66,133)(64,140,67,136) );
G=PermutationGroup([[(1,72,40),(2,69,37),(3,70,38),(4,71,39),(5,114,82),(6,115,83),(7,116,84),(8,113,81),(9,76,41),(10,73,42),(11,74,43),(12,75,44),(13,53,45),(14,54,46),(15,55,47),(16,56,48),(17,57,49),(18,58,50),(19,59,51),(20,60,52),(21,61,29),(22,62,30),(23,63,31),(24,64,32),(25,65,33),(26,66,34),(27,67,35),(28,68,36),(77,141,109),(78,142,110),(79,143,111),(80,144,112),(85,125,117),(86,126,118),(87,127,119),(88,128,120),(89,129,121),(90,130,122),(91,131,123),(92,132,124),(93,133,101),(94,134,102),(95,135,103),(96,136,104),(97,137,105),(98,138,106),(99,139,107),(100,140,108)], [(1,24,16),(2,21,13),(3,22,14),(4,23,15),(5,138,130),(6,139,131),(7,140,132),(8,137,129),(9,25,17),(10,26,18),(11,27,19),(12,28,20),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(33,49,41),(34,50,42),(35,51,43),(36,52,44),(53,69,61),(54,70,62),(55,71,63),(56,72,64),(57,76,65),(58,73,66),(59,74,67),(60,75,68),(77,93,85),(78,94,86),(79,95,87),(80,96,88),(81,97,89),(82,98,90),(83,99,91),(84,100,92),(101,117,109),(102,118,110),(103,119,111),(104,120,112),(105,121,113),(106,122,114),(107,123,115),(108,124,116),(125,141,133),(126,142,134),(127,143,135),(128,144,136)], [(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)], [(1,4),(2,3),(5,7),(9,12),(10,11),(13,14),(15,16),(17,20),(18,19),(21,22),(23,24),(25,28),(26,27),(29,30),(31,32),(33,36),(34,35),(37,38),(39,40),(41,44),(42,43),(45,46),(47,48),(49,52),(50,51),(53,54),(55,56),(57,60),(58,59),(61,62),(63,64),(65,68),(66,67),(69,70),(71,72),(73,74),(75,76),(78,80),(82,84),(86,88),(90,92),(94,96),(98,100),(102,104),(106,108),(110,112),(114,116),(118,120),(122,124),(126,128),(130,132),(134,136),(138,140),(142,144)], [(1,84,11,80),(2,83,12,79),(3,82,9,78),(4,81,10,77),(5,76,142,70),(6,75,143,69),(7,74,144,72),(8,73,141,71),(13,91,20,87),(14,90,17,86),(15,89,18,85),(16,92,19,88),(21,99,28,95),(22,98,25,94),(23,97,26,93),(24,100,27,96),(29,107,36,103),(30,106,33,102),(31,105,34,101),(32,108,35,104),(37,115,44,111),(38,114,41,110),(39,113,42,109),(40,116,43,112),(45,123,52,119),(46,122,49,118),(47,121,50,117),(48,124,51,120),(53,131,60,127),(54,130,57,126),(55,129,58,125),(56,132,59,128),(61,139,68,135),(62,138,65,134),(63,137,66,133),(64,140,67,136)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 6A | ··· | 6X | 6Y | ··· | 6AN | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 12Q | ··· | 12AF | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D4 | D8 | SD16 | C3×D4 | C3×D4 | C3×D8 | C3×SD16 |
| kernel | C32×D4⋊C4 | C32×C4⋊C4 | C6×C24 | D4×C3×C6 | C3×D4⋊C4 | D4×C32 | C3×C4⋊C4 | C2×C24 | C6×D4 | C3×D4 | C3×C12 | C62 | C3×C6 | C3×C6 | C12 | C2×C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 8 | 4 | 8 | 8 | 8 | 32 | 1 | 1 | 2 | 2 | 8 | 8 | 16 | 16 |
Matrix representation of C32×D4⋊C4 ►in GL3(𝔽73) generated by
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 8 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 72 | 0 |
| 72 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 27 | 0 | 0 |
| 0 | 57 | 16 |
| 0 | 16 | 16 |
G:=sub<GL(3,GF(73))| [8,0,0,0,8,0,0,0,8],[8,0,0,0,64,0,0,0,64],[1,0,0,0,0,72,0,1,0],[72,0,0,0,0,1,0,1,0],[27,0,0,0,57,16,0,16,16] >;
C32×D4⋊C4 in GAP, Magma, Sage, TeX
C_3^2\times D_4\rtimes C_4
% in TeX
G:=Group("C3^2xD4:C4"); // GroupNames label
G:=SmallGroup(288,320);
// by ID
G=gap.SmallGroup(288,320);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,6304,3161,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=d^2=e^4=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,d*c*d=e*c*e^-1=c^-1,e*d*e^-1=c*d>;
// generators/relations