direct product, metabelian, nilpotent (class 2), monomial
Aliases: C32×C4.4D4, C122⋊19C2, C23.3C62, C62.293C23, (C4×C12)⋊18C6, C42⋊8(C3×C6), (C6×Q8)⋊13C6, C6.88(C6×D4), (C6×D4).24C6, C12.48(C3×D4), C4.4(D4×C32), (C3×C12).145D4, (C2×C4).11C62, (C2×C62).3C22, (C6×C12).271C22, C22.14(C2×C62), C2.8(D4×C3×C6), (Q8×C3×C6)⋊16C2, (C2×Q8)⋊4(C3×C6), (D4×C3×C6).19C2, C22⋊C4⋊5(C3×C6), (C2×D4).5(C3×C6), C6.54(C3×C4○D4), (C3×C22⋊C4)⋊13C6, (C2×C12).74(C2×C6), (C3×C6).305(C2×D4), C2.7(C32×C4○D4), (C2×C6).99(C22×C6), (C22×C6).13(C2×C6), (C3×C6).171(C4○D4), (C32×C22⋊C4)⋊21C2, SmallGroup(288,821)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C4.4D4
G = < a,b,c,d,e | a3=b3=c4=d4=1, e2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=c2d-1 >
Subgroups: 348 in 228 conjugacy classes, 132 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C3×C6, C3×C6, C3×C6, C2×C12, C3×D4, C3×Q8, C22×C6, C4.4D4, C3×C12, C3×C12, C62, C62, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C6×C12, C6×C12, D4×C32, Q8×C32, C2×C62, C3×C4.4D4, C122, C32×C22⋊C4, D4×C3×C6, Q8×C3×C6, C32×C4.4D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2×C6, C2×D4, C4○D4, C3×C6, C3×D4, C22×C6, C4.4D4, C62, C6×D4, C3×C4○D4, D4×C32, C2×C62, C3×C4.4D4, D4×C3×C6, C32×C4○D4, C32×C4.4D4
►On 144 points
Generators in S144
(1 111 31)(2 112 32)(3 109 29)(4 110 30)(5 121 13)(6 122 14)(7 123 15)(8 124 16)(9 113 17)(10 114 18)(11 115 19)(12 116 20)(21 47 39)(22 48 40)(23 45 37)(24 46 38)(25 141 133)(26 142 134)(27 143 135)(28 144 136)(33 137 129)(34 138 130)(35 139 131)(36 140 132)(41 87 79)(42 88 80)(43 85 77)(44 86 78)(49 83 75)(50 84 76)(51 81 73)(52 82 74)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(57 103 95)(58 104 96)(59 101 93)(60 102 94)(65 99 91)(66 100 92)(67 97 89)(68 98 90)(105 125 117)(106 126 118)(107 127 119)(108 128 120)
(1 27 131)(2 28 132)(3 25 129)(4 26 130)(5 105 113)(6 106 114)(7 107 115)(8 108 116)(9 13 117)(10 14 118)(11 15 119)(12 16 120)(17 121 125)(18 122 126)(19 123 127)(20 124 128)(21 85 51)(22 86 52)(23 87 49)(24 88 50)(29 133 137)(30 134 138)(31 135 139)(32 136 140)(33 109 141)(34 110 142)(35 111 143)(36 112 144)(37 41 75)(38 42 76)(39 43 73)(40 44 74)(45 79 83)(46 80 84)(47 77 81)(48 78 82)(53 57 91)(54 58 92)(55 59 89)(56 60 90)(61 95 99)(62 96 100)(63 93 97)(64 94 98)(65 69 103)(66 70 104)(67 71 101)(68 72 102)
(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)
(1 106 37 53)(2 107 38 54)(3 108 39 55)(4 105 40 56)(5 74 90 130)(6 75 91 131)(7 76 92 132)(8 73 89 129)(9 78 94 134)(10 79 95 135)(11 80 96 136)(12 77 93 133)(13 82 98 138)(14 83 99 139)(15 84 100 140)(16 81 97 137)(17 86 102 142)(18 87 103 143)(19 88 104 144)(20 85 101 141)(21 71 109 128)(22 72 110 125)(23 69 111 126)(24 70 112 127)(25 116 43 59)(26 113 44 60)(27 114 41 57)(28 115 42 58)(29 120 47 63)(30 117 48 64)(31 118 45 61)(32 119 46 62)(33 124 51 67)(34 121 52 68)(35 122 49 65)(36 123 50 66)
(1 55 3 53)(2 54 4 56)(5 76 7 74)(6 75 8 73)(9 80 11 78)(10 79 12 77)(13 84 15 82)(14 83 16 81)(17 88 19 86)(18 87 20 85)(21 126 23 128)(22 125 24 127)(25 57 27 59)(26 60 28 58)(29 61 31 63)(30 64 32 62)(33 65 35 67)(34 68 36 66)(37 108 39 106)(38 107 40 105)(41 116 43 114)(42 115 44 113)(45 120 47 118)(46 119 48 117)(49 124 51 122)(50 123 52 121)(69 111 71 109)(70 110 72 112)(89 129 91 131)(90 132 92 130)(93 133 95 135)(94 136 96 134)(97 137 99 139)(98 140 100 138)(101 141 103 143)(102 144 104 142)
G:=sub<Sym(144)| (1,111,31)(2,112,32)(3,109,29)(4,110,30)(5,121,13)(6,122,14)(7,123,15)(8,124,16)(9,113,17)(10,114,18)(11,115,19)(12,116,20)(21,47,39)(22,48,40)(23,45,37)(24,46,38)(25,141,133)(26,142,134)(27,143,135)(28,144,136)(33,137,129)(34,138,130)(35,139,131)(36,140,132)(41,87,79)(42,88,80)(43,85,77)(44,86,78)(49,83,75)(50,84,76)(51,81,73)(52,82,74)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,103,95)(58,104,96)(59,101,93)(60,102,94)(65,99,91)(66,100,92)(67,97,89)(68,98,90)(105,125,117)(106,126,118)(107,127,119)(108,128,120), (1,27,131)(2,28,132)(3,25,129)(4,26,130)(5,105,113)(6,106,114)(7,107,115)(8,108,116)(9,13,117)(10,14,118)(11,15,119)(12,16,120)(17,121,125)(18,122,126)(19,123,127)(20,124,128)(21,85,51)(22,86,52)(23,87,49)(24,88,50)(29,133,137)(30,134,138)(31,135,139)(32,136,140)(33,109,141)(34,110,142)(35,111,143)(36,112,144)(37,41,75)(38,42,76)(39,43,73)(40,44,74)(45,79,83)(46,80,84)(47,77,81)(48,78,82)(53,57,91)(54,58,92)(55,59,89)(56,60,90)(61,95,99)(62,96,100)(63,93,97)(64,94,98)(65,69,103)(66,70,104)(67,71,101)(68,72,102), (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), (1,106,37,53)(2,107,38,54)(3,108,39,55)(4,105,40,56)(5,74,90,130)(6,75,91,131)(7,76,92,132)(8,73,89,129)(9,78,94,134)(10,79,95,135)(11,80,96,136)(12,77,93,133)(13,82,98,138)(14,83,99,139)(15,84,100,140)(16,81,97,137)(17,86,102,142)(18,87,103,143)(19,88,104,144)(20,85,101,141)(21,71,109,128)(22,72,110,125)(23,69,111,126)(24,70,112,127)(25,116,43,59)(26,113,44,60)(27,114,41,57)(28,115,42,58)(29,120,47,63)(30,117,48,64)(31,118,45,61)(32,119,46,62)(33,124,51,67)(34,121,52,68)(35,122,49,65)(36,123,50,66), (1,55,3,53)(2,54,4,56)(5,76,7,74)(6,75,8,73)(9,80,11,78)(10,79,12,77)(13,84,15,82)(14,83,16,81)(17,88,19,86)(18,87,20,85)(21,126,23,128)(22,125,24,127)(25,57,27,59)(26,60,28,58)(29,61,31,63)(30,64,32,62)(33,65,35,67)(34,68,36,66)(37,108,39,106)(38,107,40,105)(41,116,43,114)(42,115,44,113)(45,120,47,118)(46,119,48,117)(49,124,51,122)(50,123,52,121)(69,111,71,109)(70,110,72,112)(89,129,91,131)(90,132,92,130)(93,133,95,135)(94,136,96,134)(97,137,99,139)(98,140,100,138)(101,141,103,143)(102,144,104,142)>;
G:=Group( (1,111,31)(2,112,32)(3,109,29)(4,110,30)(5,121,13)(6,122,14)(7,123,15)(8,124,16)(9,113,17)(10,114,18)(11,115,19)(12,116,20)(21,47,39)(22,48,40)(23,45,37)(24,46,38)(25,141,133)(26,142,134)(27,143,135)(28,144,136)(33,137,129)(34,138,130)(35,139,131)(36,140,132)(41,87,79)(42,88,80)(43,85,77)(44,86,78)(49,83,75)(50,84,76)(51,81,73)(52,82,74)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,103,95)(58,104,96)(59,101,93)(60,102,94)(65,99,91)(66,100,92)(67,97,89)(68,98,90)(105,125,117)(106,126,118)(107,127,119)(108,128,120), (1,27,131)(2,28,132)(3,25,129)(4,26,130)(5,105,113)(6,106,114)(7,107,115)(8,108,116)(9,13,117)(10,14,118)(11,15,119)(12,16,120)(17,121,125)(18,122,126)(19,123,127)(20,124,128)(21,85,51)(22,86,52)(23,87,49)(24,88,50)(29,133,137)(30,134,138)(31,135,139)(32,136,140)(33,109,141)(34,110,142)(35,111,143)(36,112,144)(37,41,75)(38,42,76)(39,43,73)(40,44,74)(45,79,83)(46,80,84)(47,77,81)(48,78,82)(53,57,91)(54,58,92)(55,59,89)(56,60,90)(61,95,99)(62,96,100)(63,93,97)(64,94,98)(65,69,103)(66,70,104)(67,71,101)(68,72,102), (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), (1,106,37,53)(2,107,38,54)(3,108,39,55)(4,105,40,56)(5,74,90,130)(6,75,91,131)(7,76,92,132)(8,73,89,129)(9,78,94,134)(10,79,95,135)(11,80,96,136)(12,77,93,133)(13,82,98,138)(14,83,99,139)(15,84,100,140)(16,81,97,137)(17,86,102,142)(18,87,103,143)(19,88,104,144)(20,85,101,141)(21,71,109,128)(22,72,110,125)(23,69,111,126)(24,70,112,127)(25,116,43,59)(26,113,44,60)(27,114,41,57)(28,115,42,58)(29,120,47,63)(30,117,48,64)(31,118,45,61)(32,119,46,62)(33,124,51,67)(34,121,52,68)(35,122,49,65)(36,123,50,66), (1,55,3,53)(2,54,4,56)(5,76,7,74)(6,75,8,73)(9,80,11,78)(10,79,12,77)(13,84,15,82)(14,83,16,81)(17,88,19,86)(18,87,20,85)(21,126,23,128)(22,125,24,127)(25,57,27,59)(26,60,28,58)(29,61,31,63)(30,64,32,62)(33,65,35,67)(34,68,36,66)(37,108,39,106)(38,107,40,105)(41,116,43,114)(42,115,44,113)(45,120,47,118)(46,119,48,117)(49,124,51,122)(50,123,52,121)(69,111,71,109)(70,110,72,112)(89,129,91,131)(90,132,92,130)(93,133,95,135)(94,136,96,134)(97,137,99,139)(98,140,100,138)(101,141,103,143)(102,144,104,142) );
G=PermutationGroup([[(1,111,31),(2,112,32),(3,109,29),(4,110,30),(5,121,13),(6,122,14),(7,123,15),(8,124,16),(9,113,17),(10,114,18),(11,115,19),(12,116,20),(21,47,39),(22,48,40),(23,45,37),(24,46,38),(25,141,133),(26,142,134),(27,143,135),(28,144,136),(33,137,129),(34,138,130),(35,139,131),(36,140,132),(41,87,79),(42,88,80),(43,85,77),(44,86,78),(49,83,75),(50,84,76),(51,81,73),(52,82,74),(53,69,61),(54,70,62),(55,71,63),(56,72,64),(57,103,95),(58,104,96),(59,101,93),(60,102,94),(65,99,91),(66,100,92),(67,97,89),(68,98,90),(105,125,117),(106,126,118),(107,127,119),(108,128,120)], [(1,27,131),(2,28,132),(3,25,129),(4,26,130),(5,105,113),(6,106,114),(7,107,115),(8,108,116),(9,13,117),(10,14,118),(11,15,119),(12,16,120),(17,121,125),(18,122,126),(19,123,127),(20,124,128),(21,85,51),(22,86,52),(23,87,49),(24,88,50),(29,133,137),(30,134,138),(31,135,139),(32,136,140),(33,109,141),(34,110,142),(35,111,143),(36,112,144),(37,41,75),(38,42,76),(39,43,73),(40,44,74),(45,79,83),(46,80,84),(47,77,81),(48,78,82),(53,57,91),(54,58,92),(55,59,89),(56,60,90),(61,95,99),(62,96,100),(63,93,97),(64,94,98),(65,69,103),(66,70,104),(67,71,101),(68,72,102)], [(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)], [(1,106,37,53),(2,107,38,54),(3,108,39,55),(4,105,40,56),(5,74,90,130),(6,75,91,131),(7,76,92,132),(8,73,89,129),(9,78,94,134),(10,79,95,135),(11,80,96,136),(12,77,93,133),(13,82,98,138),(14,83,99,139),(15,84,100,140),(16,81,97,137),(17,86,102,142),(18,87,103,143),(19,88,104,144),(20,85,101,141),(21,71,109,128),(22,72,110,125),(23,69,111,126),(24,70,112,127),(25,116,43,59),(26,113,44,60),(27,114,41,57),(28,115,42,58),(29,120,47,63),(30,117,48,64),(31,118,45,61),(32,119,46,62),(33,124,51,67),(34,121,52,68),(35,122,49,65),(36,123,50,66)], [(1,55,3,53),(2,54,4,56),(5,76,7,74),(6,75,8,73),(9,80,11,78),(10,79,12,77),(13,84,15,82),(14,83,16,81),(17,88,19,86),(18,87,20,85),(21,126,23,128),(22,125,24,127),(25,57,27,59),(26,60,28,58),(29,61,31,63),(30,64,32,62),(33,65,35,67),(34,68,36,66),(37,108,39,106),(38,107,40,105),(41,116,43,114),(42,115,44,113),(45,120,47,118),(46,119,48,117),(49,124,51,122),(50,123,52,121),(69,111,71,109),(70,110,72,112),(89,129,91,131),(90,132,92,130),(93,133,95,135),(94,136,96,134),(97,137,99,139),(98,140,100,138),(101,141,103,143),(102,144,104,142)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | ··· | 3H | 4A | ··· | 4F | 4G | 4H | 6A | ··· | 6X | 6Y | ··· | 6AN | 12A | ··· | 12AV | 12AW | ··· | 12BL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | C4○D4 | C3×D4 | C3×C4○D4 |
| kernel | C32×C4.4D4 | C122 | C32×C22⋊C4 | D4×C3×C6 | Q8×C3×C6 | C3×C4.4D4 | C4×C12 | C3×C22⋊C4 | C6×D4 | C6×Q8 | C3×C12 | C3×C6 | C12 | C6 |
| # reps | 1 | 1 | 4 | 1 | 1 | 8 | 8 | 32 | 8 | 8 | 2 | 4 | 16 | 32 |
Matrix representation of C32×C4.4D4 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 10 | 0 | 0 |
| 8 | 7 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 4 | 11 | 0 | 0 |
| 1 | 9 | 0 | 0 |
| 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 5 |
| 4 | 11 | 0 | 0 |
| 2 | 9 | 0 | 0 |
| 0 | 0 | 8 | 2 |
| 0 | 0 | 1 | 5 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[6,8,0,0,10,7,0,0,0,0,12,0,0,0,0,12],[4,1,0,0,11,9,0,0,0,0,8,0,0,0,2,5],[4,2,0,0,11,9,0,0,0,0,8,1,0,0,2,5] >;
C32×C4.4D4 in GAP, Magma, Sage, TeX
C_3^2\times C_4._4D_4
% in TeX
G:=Group("C3^2xC4.4D4"); // GroupNames label
G:=SmallGroup(288,821);
// by ID
G=gap.SmallGroup(288,821);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,1016,3110,394]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=d^4=1,e^2=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,e*c*e^-1=c^-1,e*d*e^-1=c^2*d^-1>;
// generators/relations