direct product, metabelian, nilpotent (class 3), monomial
Aliases: C32×C4○D8, C8.6C62, D4.2C62, C62.76D4, Q8.5C62, (C2×C24)⋊9C6, (C3×D8)⋊7C6, D8⋊3(C3×C6), (C6×C24)⋊14C2, (C3×Q16)⋊7C6, Q16⋊3(C3×C6), C6.94(C6×D4), C24.31(C2×C6), SD16⋊3(C3×C6), (C3×SD16)⋊7C6, C12.92(C3×D4), C4.4(C2×C62), (C3×C12).187D4, (C32×D8)⋊11C2, (C2×C4).28C62, C4.20(D4×C32), C12.58(C22×C6), (C3×C24).74C22, (C32×Q16)⋊11C2, C22.1(D4×C32), (C6×C12).377C22, (C3×C12).188C23, (C32×SD16)⋊11C2, (D4×C32).33C22, (Q8×C32).36C22, (C2×C8)⋊4(C3×C6), C2.14(D4×C3×C6), C4○D4⋊3(C3×C6), (C3×C4○D4)⋊10C6, (C2×C6).15(C3×D4), (C3×D4).17(C2×C6), (C3×C6).311(C2×D4), (C3×Q8).30(C2×C6), (C2×C12).164(C2×C6), (C32×C4○D4)⋊11C2, SmallGroup(288,832)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C4○D8
G = < a,b,c,d,e | a3=b3=c4=e2=1, d4=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d3 >
Subgroups: 276 in 186 conjugacy classes, 120 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C32, C12, C12, C2×C6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×C6, C3×C6, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4○D8, C3×C12, C3×C12, C62, C62, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, C3×C24, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, C3×C4○D8, C6×C24, C32×D8, C32×SD16, C32×Q16, C32×C4○D4, C32×C4○D8
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2×C6, C2×D4, C3×C6, C3×D4, C22×C6, C4○D8, C62, C6×D4, D4×C32, C2×C62, C3×C4○D8, D4×C3×C6, C32×C4○D8
►On 144 points
Generators in S144
(1 26 18)(2 27 19)(3 28 20)(4 29 21)(5 30 22)(6 31 23)(7 32 24)(8 25 17)(9 33 134)(10 34 135)(11 35 136)(12 36 129)(13 37 130)(14 38 131)(15 39 132)(16 40 133)(41 94 49)(42 95 50)(43 96 51)(44 89 52)(45 90 53)(46 91 54)(47 92 55)(48 93 56)(57 78 65)(58 79 66)(59 80 67)(60 73 68)(61 74 69)(62 75 70)(63 76 71)(64 77 72)(81 125 117)(82 126 118)(83 127 119)(84 128 120)(85 121 113)(86 122 114)(87 123 115)(88 124 116)(97 141 105)(98 142 106)(99 143 107)(100 144 108)(101 137 109)(102 138 110)(103 139 111)(104 140 112)
(1 83 46)(2 84 47)(3 85 48)(4 86 41)(5 87 42)(6 88 43)(7 81 44)(8 82 45)(9 142 74)(10 143 75)(11 144 76)(12 137 77)(13 138 78)(14 139 79)(15 140 80)(16 141 73)(17 118 53)(18 119 54)(19 120 55)(20 113 56)(21 114 49)(22 115 50)(23 116 51)(24 117 52)(25 126 90)(26 127 91)(27 128 92)(28 121 93)(29 122 94)(30 123 95)(31 124 96)(32 125 89)(33 106 69)(34 107 70)(35 108 71)(36 109 72)(37 110 65)(38 111 66)(39 112 67)(40 105 68)(57 130 102)(58 131 103)(59 132 104)(60 133 97)(61 134 98)(62 135 99)(63 136 100)(64 129 101)
(1 58 5 62)(2 59 6 63)(3 60 7 64)(4 61 8 57)(9 126 13 122)(10 127 14 123)(11 128 15 124)(12 121 16 125)(17 65 21 69)(18 66 22 70)(19 67 23 71)(20 68 24 72)(25 78 29 74)(26 79 30 75)(27 80 31 76)(28 73 32 77)(33 118 37 114)(34 119 38 115)(35 120 39 116)(36 113 40 117)(41 98 45 102)(42 99 46 103)(43 100 47 104)(44 101 48 97)(49 106 53 110)(50 107 54 111)(51 108 55 112)(52 109 56 105)(81 129 85 133)(82 130 86 134)(83 131 87 135)(84 132 88 136)(89 137 93 141)(90 138 94 142)(91 139 95 143)(92 140 96 144)
(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 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 18)(19 24)(20 23)(21 22)(25 26)(27 32)(28 31)(29 30)(33 34)(35 40)(36 39)(37 38)(41 42)(43 48)(44 47)(45 46)(49 50)(51 56)(52 55)(53 54)(57 58)(59 64)(60 63)(61 62)(65 66)(67 72)(68 71)(69 70)(73 76)(74 75)(77 80)(78 79)(81 84)(82 83)(85 88)(86 87)(89 92)(90 91)(93 96)(94 95)(97 100)(98 99)(101 104)(102 103)(105 108)(106 107)(109 112)(110 111)(113 116)(114 115)(117 120)(118 119)(121 124)(122 123)(125 128)(126 127)(129 132)(130 131)(133 136)(134 135)(137 140)(138 139)(141 144)(142 143)
G:=sub<Sym(144)| (1,26,18)(2,27,19)(3,28,20)(4,29,21)(5,30,22)(6,31,23)(7,32,24)(8,25,17)(9,33,134)(10,34,135)(11,35,136)(12,36,129)(13,37,130)(14,38,131)(15,39,132)(16,40,133)(41,94,49)(42,95,50)(43,96,51)(44,89,52)(45,90,53)(46,91,54)(47,92,55)(48,93,56)(57,78,65)(58,79,66)(59,80,67)(60,73,68)(61,74,69)(62,75,70)(63,76,71)(64,77,72)(81,125,117)(82,126,118)(83,127,119)(84,128,120)(85,121,113)(86,122,114)(87,123,115)(88,124,116)(97,141,105)(98,142,106)(99,143,107)(100,144,108)(101,137,109)(102,138,110)(103,139,111)(104,140,112), (1,83,46)(2,84,47)(3,85,48)(4,86,41)(5,87,42)(6,88,43)(7,81,44)(8,82,45)(9,142,74)(10,143,75)(11,144,76)(12,137,77)(13,138,78)(14,139,79)(15,140,80)(16,141,73)(17,118,53)(18,119,54)(19,120,55)(20,113,56)(21,114,49)(22,115,50)(23,116,51)(24,117,52)(25,126,90)(26,127,91)(27,128,92)(28,121,93)(29,122,94)(30,123,95)(31,124,96)(32,125,89)(33,106,69)(34,107,70)(35,108,71)(36,109,72)(37,110,65)(38,111,66)(39,112,67)(40,105,68)(57,130,102)(58,131,103)(59,132,104)(60,133,97)(61,134,98)(62,135,99)(63,136,100)(64,129,101), (1,58,5,62)(2,59,6,63)(3,60,7,64)(4,61,8,57)(9,126,13,122)(10,127,14,123)(11,128,15,124)(12,121,16,125)(17,65,21,69)(18,66,22,70)(19,67,23,71)(20,68,24,72)(25,78,29,74)(26,79,30,75)(27,80,31,76)(28,73,32,77)(33,118,37,114)(34,119,38,115)(35,120,39,116)(36,113,40,117)(41,98,45,102)(42,99,46,103)(43,100,47,104)(44,101,48,97)(49,106,53,110)(50,107,54,111)(51,108,55,112)(52,109,56,105)(81,129,85,133)(82,130,86,134)(83,131,87,135)(84,132,88,136)(89,137,93,141)(90,138,94,142)(91,139,95,143)(92,140,96,144), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,26)(27,32)(28,31)(29,30)(33,34)(35,40)(36,39)(37,38)(41,42)(43,48)(44,47)(45,46)(49,50)(51,56)(52,55)(53,54)(57,58)(59,64)(60,63)(61,62)(65,66)(67,72)(68,71)(69,70)(73,76)(74,75)(77,80)(78,79)(81,84)(82,83)(85,88)(86,87)(89,92)(90,91)(93,96)(94,95)(97,100)(98,99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)>;
G:=Group( (1,26,18)(2,27,19)(3,28,20)(4,29,21)(5,30,22)(6,31,23)(7,32,24)(8,25,17)(9,33,134)(10,34,135)(11,35,136)(12,36,129)(13,37,130)(14,38,131)(15,39,132)(16,40,133)(41,94,49)(42,95,50)(43,96,51)(44,89,52)(45,90,53)(46,91,54)(47,92,55)(48,93,56)(57,78,65)(58,79,66)(59,80,67)(60,73,68)(61,74,69)(62,75,70)(63,76,71)(64,77,72)(81,125,117)(82,126,118)(83,127,119)(84,128,120)(85,121,113)(86,122,114)(87,123,115)(88,124,116)(97,141,105)(98,142,106)(99,143,107)(100,144,108)(101,137,109)(102,138,110)(103,139,111)(104,140,112), (1,83,46)(2,84,47)(3,85,48)(4,86,41)(5,87,42)(6,88,43)(7,81,44)(8,82,45)(9,142,74)(10,143,75)(11,144,76)(12,137,77)(13,138,78)(14,139,79)(15,140,80)(16,141,73)(17,118,53)(18,119,54)(19,120,55)(20,113,56)(21,114,49)(22,115,50)(23,116,51)(24,117,52)(25,126,90)(26,127,91)(27,128,92)(28,121,93)(29,122,94)(30,123,95)(31,124,96)(32,125,89)(33,106,69)(34,107,70)(35,108,71)(36,109,72)(37,110,65)(38,111,66)(39,112,67)(40,105,68)(57,130,102)(58,131,103)(59,132,104)(60,133,97)(61,134,98)(62,135,99)(63,136,100)(64,129,101), (1,58,5,62)(2,59,6,63)(3,60,7,64)(4,61,8,57)(9,126,13,122)(10,127,14,123)(11,128,15,124)(12,121,16,125)(17,65,21,69)(18,66,22,70)(19,67,23,71)(20,68,24,72)(25,78,29,74)(26,79,30,75)(27,80,31,76)(28,73,32,77)(33,118,37,114)(34,119,38,115)(35,120,39,116)(36,113,40,117)(41,98,45,102)(42,99,46,103)(43,100,47,104)(44,101,48,97)(49,106,53,110)(50,107,54,111)(51,108,55,112)(52,109,56,105)(81,129,85,133)(82,130,86,134)(83,131,87,135)(84,132,88,136)(89,137,93,141)(90,138,94,142)(91,139,95,143)(92,140,96,144), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,26)(27,32)(28,31)(29,30)(33,34)(35,40)(36,39)(37,38)(41,42)(43,48)(44,47)(45,46)(49,50)(51,56)(52,55)(53,54)(57,58)(59,64)(60,63)(61,62)(65,66)(67,72)(68,71)(69,70)(73,76)(74,75)(77,80)(78,79)(81,84)(82,83)(85,88)(86,87)(89,92)(90,91)(93,96)(94,95)(97,100)(98,99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143) );
G=PermutationGroup([[(1,26,18),(2,27,19),(3,28,20),(4,29,21),(5,30,22),(6,31,23),(7,32,24),(8,25,17),(9,33,134),(10,34,135),(11,35,136),(12,36,129),(13,37,130),(14,38,131),(15,39,132),(16,40,133),(41,94,49),(42,95,50),(43,96,51),(44,89,52),(45,90,53),(46,91,54),(47,92,55),(48,93,56),(57,78,65),(58,79,66),(59,80,67),(60,73,68),(61,74,69),(62,75,70),(63,76,71),(64,77,72),(81,125,117),(82,126,118),(83,127,119),(84,128,120),(85,121,113),(86,122,114),(87,123,115),(88,124,116),(97,141,105),(98,142,106),(99,143,107),(100,144,108),(101,137,109),(102,138,110),(103,139,111),(104,140,112)], [(1,83,46),(2,84,47),(3,85,48),(4,86,41),(5,87,42),(6,88,43),(7,81,44),(8,82,45),(9,142,74),(10,143,75),(11,144,76),(12,137,77),(13,138,78),(14,139,79),(15,140,80),(16,141,73),(17,118,53),(18,119,54),(19,120,55),(20,113,56),(21,114,49),(22,115,50),(23,116,51),(24,117,52),(25,126,90),(26,127,91),(27,128,92),(28,121,93),(29,122,94),(30,123,95),(31,124,96),(32,125,89),(33,106,69),(34,107,70),(35,108,71),(36,109,72),(37,110,65),(38,111,66),(39,112,67),(40,105,68),(57,130,102),(58,131,103),(59,132,104),(60,133,97),(61,134,98),(62,135,99),(63,136,100),(64,129,101)], [(1,58,5,62),(2,59,6,63),(3,60,7,64),(4,61,8,57),(9,126,13,122),(10,127,14,123),(11,128,15,124),(12,121,16,125),(17,65,21,69),(18,66,22,70),(19,67,23,71),(20,68,24,72),(25,78,29,74),(26,79,30,75),(27,80,31,76),(28,73,32,77),(33,118,37,114),(34,119,38,115),(35,120,39,116),(36,113,40,117),(41,98,45,102),(42,99,46,103),(43,100,47,104),(44,101,48,97),(49,106,53,110),(50,107,54,111),(51,108,55,112),(52,109,56,105),(81,129,85,133),(82,130,86,134),(83,131,87,135),(84,132,88,136),(89,137,93,141),(90,138,94,142),(91,139,95,143),(92,140,96,144)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,18),(19,24),(20,23),(21,22),(25,26),(27,32),(28,31),(29,30),(33,34),(35,40),(36,39),(37,38),(41,42),(43,48),(44,47),(45,46),(49,50),(51,56),(52,55),(53,54),(57,58),(59,64),(60,63),(61,62),(65,66),(67,72),(68,71),(69,70),(73,76),(74,75),(77,80),(78,79),(81,84),(82,83),(85,88),(86,87),(89,92),(90,91),(93,96),(94,95),(97,100),(98,99),(101,104),(102,103),(105,108),(106,107),(109,112),(110,111),(113,116),(114,115),(117,120),(118,119),(121,124),(122,123),(125,128),(126,127),(129,132),(130,131),(133,136),(134,135),(137,140),(138,139),(141,144),(142,143)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6H | 6I | ··· | 6P | 6Q | ··· | 6AF | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 12Q | ··· | 12X | 12Y | ··· | 12AN | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | ··· | 1 | 1 | 1 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C3×D4 | C3×D4 | C4○D8 | C3×C4○D8 |
| kernel | C32×C4○D8 | C6×C24 | C32×D8 | C32×SD16 | C32×Q16 | C32×C4○D4 | C3×C4○D8 | C2×C24 | C3×D8 | C3×SD16 | C3×Q16 | C3×C4○D4 | C3×C12 | C62 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 8 | 8 | 8 | 16 | 8 | 16 | 1 | 1 | 8 | 8 | 4 | 32 |
Matrix representation of C32×C4○D8 ►in GL3(𝔽73) generated by
| 64 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 72 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 46 |
| 72 | 0 | 0 |
| 0 | 57 | 16 |
| 0 | 57 | 57 |
| 1 | 0 | 0 |
| 0 | 57 | 16 |
| 0 | 16 | 16 |
G:=sub<GL(3,GF(73))| [64,0,0,0,8,0,0,0,8],[8,0,0,0,8,0,0,0,8],[72,0,0,0,46,0,0,0,46],[72,0,0,0,57,57,0,16,57],[1,0,0,0,57,16,0,16,16] >;
C32×C4○D8 in GAP, Magma, Sage, TeX
C_3^2\times C_4\circ D_8
% in TeX
G:=Group("C3^2xC4oD8"); // GroupNames label
G:=SmallGroup(288,832);
// by ID
G=gap.SmallGroup(288,832);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,772,9077,4548,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=e^2=1,d^4=c^2,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,c*e=e*c,e*d*e=c^2*d^3>;
// generators/relations