direct product, metabelian, nilpotent (class 2), monomial
Aliases: C32×C4⋊D4, C62⋊17D4, C23.2C62, C62.290C23, C12⋊9(C3×D4), (C6×D4)⋊11C6, (C3×C12)⋊26D4, C6.85(C6×D4), C4⋊2(D4×C32), (C2×C4).3C62, (C22×C12)⋊15C6, C22⋊2(D4×C32), (C6×C12).370C22, (C2×C62).88C22, C22.11(C2×C62), C2.5(D4×C3×C6), C4⋊C4⋊2(C3×C6), (C2×C6×C12)⋊18C2, (D4×C3×C6)⋊20C2, (C2×C6)⋊7(C3×D4), (C3×C4⋊C4)⋊11C6, (C2×D4)⋊2(C3×C6), C22⋊C4⋊3(C3×C6), (C22×C4)⋊6(C3×C6), C6.51(C3×C4○D4), (C3×C22⋊C4)⋊11C6, (C32×C4⋊C4)⋊20C2, (C2×C12).96(C2×C6), (C3×C6).302(C2×D4), C2.4(C32×C4○D4), (C2×C6).96(C22×C6), (C22×C6).12(C2×C6), (C3×C6).168(C4○D4), (C32×C22⋊C4)⋊19C2, SmallGroup(288,818)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C4⋊D4
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, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 444 in 282 conjugacy classes, 144 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C32, C12, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C3×C6, C3×C6, C2×C12, C2×C12, C3×D4, C22×C6, C4⋊D4, C3×C12, C3×C12, C62, C62, C62, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×C12, C6×C12, C6×C12, D4×C32, C2×C62, C2×C62, C3×C4⋊D4, C32×C22⋊C4, C32×C4⋊C4, C2×C6×C12, D4×C3×C6, D4×C3×C6, C32×C4⋊D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2×C6, C2×D4, C4○D4, C3×C6, C3×D4, C22×C6, C4⋊D4, C62, C6×D4, C3×C4○D4, D4×C32, C2×C62, C3×C4⋊D4, D4×C3×C6, C32×C4○D4, C32×C4⋊D4
►On 144 points
Generators in S144
(1 23 15)(2 24 16)(3 21 13)(4 22 14)(5 101 93)(6 102 94)(7 103 95)(8 104 96)(9 89 81)(10 90 82)(11 91 83)(12 92 84)(17 85 77)(18 86 78)(19 87 79)(20 88 80)(25 105 33)(26 106 34)(27 107 35)(28 108 36)(29 97 37)(30 98 38)(31 99 39)(32 100 40)(41 121 49)(42 122 50)(43 123 51)(44 124 52)(45 113 53)(46 114 54)(47 115 55)(48 116 56)(57 76 65)(58 73 66)(59 74 67)(60 75 68)(61 141 133)(62 142 134)(63 143 135)(64 144 136)(69 137 129)(70 138 130)(71 139 131)(72 140 132)(109 125 117)(110 126 118)(111 127 119)(112 128 120)
(1 11 79)(2 12 80)(3 9 77)(4 10 78)(5 37 105)(6 38 106)(7 39 107)(8 40 108)(13 81 85)(14 82 86)(15 83 87)(16 84 88)(17 21 89)(18 22 90)(19 23 91)(20 24 92)(25 93 97)(26 94 98)(27 95 99)(28 96 100)(29 33 101)(30 34 102)(31 35 103)(32 36 104)(41 109 113)(42 110 114)(43 111 115)(44 112 116)(45 49 117)(46 50 118)(47 51 119)(48 52 120)(53 121 125)(54 122 126)(55 123 127)(56 124 128)(57 61 129)(58 62 130)(59 63 131)(60 64 132)(65 133 137)(66 134 138)(67 135 139)(68 136 140)(69 76 141)(70 73 142)(71 74 143)(72 75 144)
(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 95 111 59)(2 94 112 58)(3 93 109 57)(4 96 110 60)(5 125 76 21)(6 128 73 24)(7 127 74 23)(8 126 75 22)(9 97 113 61)(10 100 114 64)(11 99 115 63)(12 98 116 62)(13 101 117 65)(14 104 118 68)(15 103 119 67)(16 102 120 66)(17 105 121 69)(18 108 122 72)(19 107 123 71)(20 106 124 70)(25 41 129 77)(26 44 130 80)(27 43 131 79)(28 42 132 78)(29 45 133 81)(30 48 134 84)(31 47 135 83)(32 46 136 82)(33 49 137 85)(34 52 138 88)(35 51 139 87)(36 50 140 86)(37 53 141 89)(38 56 142 92)(39 55 143 91)(40 54 144 90)
(2 4)(5 76)(6 75)(7 74)(8 73)(10 12)(14 16)(18 20)(22 24)(25 129)(26 132)(27 131)(28 130)(29 133)(30 136)(31 135)(32 134)(33 137)(34 140)(35 139)(36 138)(37 141)(38 144)(39 143)(40 142)(42 44)(46 48)(50 52)(54 56)(57 93)(58 96)(59 95)(60 94)(61 97)(62 100)(63 99)(64 98)(65 101)(66 104)(67 103)(68 102)(69 105)(70 108)(71 107)(72 106)(78 80)(82 84)(86 88)(90 92)(110 112)(114 116)(118 120)(122 124)(126 128)
G:=sub<Sym(144)| (1,23,15)(2,24,16)(3,21,13)(4,22,14)(5,101,93)(6,102,94)(7,103,95)(8,104,96)(9,89,81)(10,90,82)(11,91,83)(12,92,84)(17,85,77)(18,86,78)(19,87,79)(20,88,80)(25,105,33)(26,106,34)(27,107,35)(28,108,36)(29,97,37)(30,98,38)(31,99,39)(32,100,40)(41,121,49)(42,122,50)(43,123,51)(44,124,52)(45,113,53)(46,114,54)(47,115,55)(48,116,56)(57,76,65)(58,73,66)(59,74,67)(60,75,68)(61,141,133)(62,142,134)(63,143,135)(64,144,136)(69,137,129)(70,138,130)(71,139,131)(72,140,132)(109,125,117)(110,126,118)(111,127,119)(112,128,120), (1,11,79)(2,12,80)(3,9,77)(4,10,78)(5,37,105)(6,38,106)(7,39,107)(8,40,108)(13,81,85)(14,82,86)(15,83,87)(16,84,88)(17,21,89)(18,22,90)(19,23,91)(20,24,92)(25,93,97)(26,94,98)(27,95,99)(28,96,100)(29,33,101)(30,34,102)(31,35,103)(32,36,104)(41,109,113)(42,110,114)(43,111,115)(44,112,116)(45,49,117)(46,50,118)(47,51,119)(48,52,120)(53,121,125)(54,122,126)(55,123,127)(56,124,128)(57,61,129)(58,62,130)(59,63,131)(60,64,132)(65,133,137)(66,134,138)(67,135,139)(68,136,140)(69,76,141)(70,73,142)(71,74,143)(72,75,144), (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), (1,95,111,59)(2,94,112,58)(3,93,109,57)(4,96,110,60)(5,125,76,21)(6,128,73,24)(7,127,74,23)(8,126,75,22)(9,97,113,61)(10,100,114,64)(11,99,115,63)(12,98,116,62)(13,101,117,65)(14,104,118,68)(15,103,119,67)(16,102,120,66)(17,105,121,69)(18,108,122,72)(19,107,123,71)(20,106,124,70)(25,41,129,77)(26,44,130,80)(27,43,131,79)(28,42,132,78)(29,45,133,81)(30,48,134,84)(31,47,135,83)(32,46,136,82)(33,49,137,85)(34,52,138,88)(35,51,139,87)(36,50,140,86)(37,53,141,89)(38,56,142,92)(39,55,143,91)(40,54,144,90), (2,4)(5,76)(6,75)(7,74)(8,73)(10,12)(14,16)(18,20)(22,24)(25,129)(26,132)(27,131)(28,130)(29,133)(30,136)(31,135)(32,134)(33,137)(34,140)(35,139)(36,138)(37,141)(38,144)(39,143)(40,142)(42,44)(46,48)(50,52)(54,56)(57,93)(58,96)(59,95)(60,94)(61,97)(62,100)(63,99)(64,98)(65,101)(66,104)(67,103)(68,102)(69,105)(70,108)(71,107)(72,106)(78,80)(82,84)(86,88)(90,92)(110,112)(114,116)(118,120)(122,124)(126,128)>;
G:=Group( (1,23,15)(2,24,16)(3,21,13)(4,22,14)(5,101,93)(6,102,94)(7,103,95)(8,104,96)(9,89,81)(10,90,82)(11,91,83)(12,92,84)(17,85,77)(18,86,78)(19,87,79)(20,88,80)(25,105,33)(26,106,34)(27,107,35)(28,108,36)(29,97,37)(30,98,38)(31,99,39)(32,100,40)(41,121,49)(42,122,50)(43,123,51)(44,124,52)(45,113,53)(46,114,54)(47,115,55)(48,116,56)(57,76,65)(58,73,66)(59,74,67)(60,75,68)(61,141,133)(62,142,134)(63,143,135)(64,144,136)(69,137,129)(70,138,130)(71,139,131)(72,140,132)(109,125,117)(110,126,118)(111,127,119)(112,128,120), (1,11,79)(2,12,80)(3,9,77)(4,10,78)(5,37,105)(6,38,106)(7,39,107)(8,40,108)(13,81,85)(14,82,86)(15,83,87)(16,84,88)(17,21,89)(18,22,90)(19,23,91)(20,24,92)(25,93,97)(26,94,98)(27,95,99)(28,96,100)(29,33,101)(30,34,102)(31,35,103)(32,36,104)(41,109,113)(42,110,114)(43,111,115)(44,112,116)(45,49,117)(46,50,118)(47,51,119)(48,52,120)(53,121,125)(54,122,126)(55,123,127)(56,124,128)(57,61,129)(58,62,130)(59,63,131)(60,64,132)(65,133,137)(66,134,138)(67,135,139)(68,136,140)(69,76,141)(70,73,142)(71,74,143)(72,75,144), (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), (1,95,111,59)(2,94,112,58)(3,93,109,57)(4,96,110,60)(5,125,76,21)(6,128,73,24)(7,127,74,23)(8,126,75,22)(9,97,113,61)(10,100,114,64)(11,99,115,63)(12,98,116,62)(13,101,117,65)(14,104,118,68)(15,103,119,67)(16,102,120,66)(17,105,121,69)(18,108,122,72)(19,107,123,71)(20,106,124,70)(25,41,129,77)(26,44,130,80)(27,43,131,79)(28,42,132,78)(29,45,133,81)(30,48,134,84)(31,47,135,83)(32,46,136,82)(33,49,137,85)(34,52,138,88)(35,51,139,87)(36,50,140,86)(37,53,141,89)(38,56,142,92)(39,55,143,91)(40,54,144,90), (2,4)(5,76)(6,75)(7,74)(8,73)(10,12)(14,16)(18,20)(22,24)(25,129)(26,132)(27,131)(28,130)(29,133)(30,136)(31,135)(32,134)(33,137)(34,140)(35,139)(36,138)(37,141)(38,144)(39,143)(40,142)(42,44)(46,48)(50,52)(54,56)(57,93)(58,96)(59,95)(60,94)(61,97)(62,100)(63,99)(64,98)(65,101)(66,104)(67,103)(68,102)(69,105)(70,108)(71,107)(72,106)(78,80)(82,84)(86,88)(90,92)(110,112)(114,116)(118,120)(122,124)(126,128) );
G=PermutationGroup([[(1,23,15),(2,24,16),(3,21,13),(4,22,14),(5,101,93),(6,102,94),(7,103,95),(8,104,96),(9,89,81),(10,90,82),(11,91,83),(12,92,84),(17,85,77),(18,86,78),(19,87,79),(20,88,80),(25,105,33),(26,106,34),(27,107,35),(28,108,36),(29,97,37),(30,98,38),(31,99,39),(32,100,40),(41,121,49),(42,122,50),(43,123,51),(44,124,52),(45,113,53),(46,114,54),(47,115,55),(48,116,56),(57,76,65),(58,73,66),(59,74,67),(60,75,68),(61,141,133),(62,142,134),(63,143,135),(64,144,136),(69,137,129),(70,138,130),(71,139,131),(72,140,132),(109,125,117),(110,126,118),(111,127,119),(112,128,120)], [(1,11,79),(2,12,80),(3,9,77),(4,10,78),(5,37,105),(6,38,106),(7,39,107),(8,40,108),(13,81,85),(14,82,86),(15,83,87),(16,84,88),(17,21,89),(18,22,90),(19,23,91),(20,24,92),(25,93,97),(26,94,98),(27,95,99),(28,96,100),(29,33,101),(30,34,102),(31,35,103),(32,36,104),(41,109,113),(42,110,114),(43,111,115),(44,112,116),(45,49,117),(46,50,118),(47,51,119),(48,52,120),(53,121,125),(54,122,126),(55,123,127),(56,124,128),(57,61,129),(58,62,130),(59,63,131),(60,64,132),(65,133,137),(66,134,138),(67,135,139),(68,136,140),(69,76,141),(70,73,142),(71,74,143),(72,75,144)], [(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)], [(1,95,111,59),(2,94,112,58),(3,93,109,57),(4,96,110,60),(5,125,76,21),(6,128,73,24),(7,127,74,23),(8,126,75,22),(9,97,113,61),(10,100,114,64),(11,99,115,63),(12,98,116,62),(13,101,117,65),(14,104,118,68),(15,103,119,67),(16,102,120,66),(17,105,121,69),(18,108,122,72),(19,107,123,71),(20,106,124,70),(25,41,129,77),(26,44,130,80),(27,43,131,79),(28,42,132,78),(29,45,133,81),(30,48,134,84),(31,47,135,83),(32,46,136,82),(33,49,137,85),(34,52,138,88),(35,51,139,87),(36,50,140,86),(37,53,141,89),(38,56,142,92),(39,55,143,91),(40,54,144,90)], [(2,4),(5,76),(6,75),(7,74),(8,73),(10,12),(14,16),(18,20),(22,24),(25,129),(26,132),(27,131),(28,130),(29,133),(30,136),(31,135),(32,134),(33,137),(34,140),(35,139),(36,138),(37,141),(38,144),(39,143),(40,142),(42,44),(46,48),(50,52),(54,56),(57,93),(58,96),(59,95),(60,94),(61,97),(62,100),(63,99),(64,98),(65,101),(66,104),(67,103),(68,102),(69,105),(70,108),(71,107),(72,106),(78,80),(82,84),(86,88),(90,92),(110,112),(114,116),(118,120),(122,124),(126,128)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6X | 6Y | ··· | 6AN | 6AO | ··· | 6BD | 12A | ··· | 12AF | 12AG | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D4 | C4○D4 | C3×D4 | C3×D4 | C3×C4○D4 |
| kernel | C32×C4⋊D4 | C32×C22⋊C4 | C32×C4⋊C4 | C2×C6×C12 | D4×C3×C6 | C3×C4⋊D4 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C6×D4 | C3×C12 | C62 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 1 | 3 | 8 | 16 | 8 | 8 | 24 | 2 | 2 | 2 | 16 | 16 | 16 |
Matrix representation of C32×C4⋊D4 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 10 |
| 0 | 0 | 0 | 8 |
| 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 12 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,9,0,0,0,0,9],[3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,5,0,0,0,10,8],[0,1,0,0,12,0,0,0,0,0,1,12,0,0,2,12],[1,0,0,0,0,12,0,0,0,0,1,12,0,0,0,12] >;
C32×C4⋊D4 in GAP, Magma, Sage, TeX
C_3^2\times C_4\rtimes D_4
% in TeX
G:=Group("C3^2xC4:D4"); // GroupNames label
G:=SmallGroup(288,818);
// by ID
G=gap.SmallGroup(288,818);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,512,3110]);
// 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,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations