metabelian, supersoluble, monomial
Aliases: C24.19D6, C32⋊8SD32, D8.(C3⋊S3), (C3×D8).5S3, (C3×C6).38D8, C3⋊3(D8.S3), (C3×C12).53D4, C24.S3⋊4C2, C32⋊5Q16⋊5C2, C6.24(D4⋊S3), (C32×D8).2C2, C12.35(C3⋊D4), (C3×C24).18C22, C2.5(C32⋊7D8), C4.2(C32⋊7D4), C8.5(C2×C3⋊S3), SmallGroup(288,302)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊8SD32
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=b-1, bd=db, dcd=c7 >
Subgroups: 312 in 78 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, D4, Q8, C32, Dic3, C12, C2×C6, C16, D8, Q16, C3×C6, C3×C6, C24, Dic6, C3×D4, SD32, C3⋊Dic3, C3×C12, C62, C3⋊C16, Dic12, C3×D8, C3×C24, C32⋊4Q8, D4×C32, D8.S3, C24.S3, C32⋊5Q16, C32×D8, C32⋊8SD32
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊S3, C3⋊D4, SD32, C2×C3⋊S3, D4⋊S3, C32⋊7D4, D8.S3, C32⋊7D8, C32⋊8SD32
►On 144 points
Generators in S144
(1 125 37)(2 38 126)(3 127 39)(4 40 128)(5 113 41)(6 42 114)(7 115 43)(8 44 116)(9 117 45)(10 46 118)(11 119 47)(12 48 120)(13 121 33)(14 34 122)(15 123 35)(16 36 124)(17 75 110)(18 111 76)(19 77 112)(20 97 78)(21 79 98)(22 99 80)(23 65 100)(24 101 66)(25 67 102)(26 103 68)(27 69 104)(28 105 70)(29 71 106)(30 107 72)(31 73 108)(32 109 74)(49 86 129)(50 130 87)(51 88 131)(52 132 89)(53 90 133)(54 134 91)(55 92 135)(56 136 93)(57 94 137)(58 138 95)(59 96 139)(60 140 81)(61 82 141)(62 142 83)(63 84 143)(64 144 85)
(1 110 90)(2 91 111)(3 112 92)(4 93 97)(5 98 94)(6 95 99)(7 100 96)(8 81 101)(9 102 82)(10 83 103)(11 104 84)(12 85 105)(13 106 86)(14 87 107)(15 108 88)(16 89 109)(17 133 125)(18 126 134)(19 135 127)(20 128 136)(21 137 113)(22 114 138)(23 139 115)(24 116 140)(25 141 117)(26 118 142)(27 143 119)(28 120 144)(29 129 121)(30 122 130)(31 131 123)(32 124 132)(33 71 49)(34 50 72)(35 73 51)(36 52 74)(37 75 53)(38 54 76)(39 77 55)(40 56 78)(41 79 57)(42 58 80)(43 65 59)(44 60 66)(45 67 61)(46 62 68)(47 69 63)(48 64 70)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(18 24)(19 31)(20 22)(21 29)(23 27)(26 32)(28 30)(33 41)(34 48)(35 39)(36 46)(38 44)(40 42)(43 47)(49 57)(50 64)(51 55)(52 62)(54 60)(56 58)(59 63)(65 69)(66 76)(68 74)(70 72)(71 79)(73 77)(78 80)(81 91)(83 89)(84 96)(85 87)(86 94)(88 92)(93 95)(97 99)(98 106)(100 104)(101 111)(103 109)(105 107)(108 112)(113 121)(114 128)(115 119)(116 126)(118 124)(120 122)(123 127)(129 137)(130 144)(131 135)(132 142)(134 140)(136 138)(139 143)
G:=sub<Sym(144)| (1,125,37)(2,38,126)(3,127,39)(4,40,128)(5,113,41)(6,42,114)(7,115,43)(8,44,116)(9,117,45)(10,46,118)(11,119,47)(12,48,120)(13,121,33)(14,34,122)(15,123,35)(16,36,124)(17,75,110)(18,111,76)(19,77,112)(20,97,78)(21,79,98)(22,99,80)(23,65,100)(24,101,66)(25,67,102)(26,103,68)(27,69,104)(28,105,70)(29,71,106)(30,107,72)(31,73,108)(32,109,74)(49,86,129)(50,130,87)(51,88,131)(52,132,89)(53,90,133)(54,134,91)(55,92,135)(56,136,93)(57,94,137)(58,138,95)(59,96,139)(60,140,81)(61,82,141)(62,142,83)(63,84,143)(64,144,85), (1,110,90)(2,91,111)(3,112,92)(4,93,97)(5,98,94)(6,95,99)(7,100,96)(8,81,101)(9,102,82)(10,83,103)(11,104,84)(12,85,105)(13,106,86)(14,87,107)(15,108,88)(16,89,109)(17,133,125)(18,126,134)(19,135,127)(20,128,136)(21,137,113)(22,114,138)(23,139,115)(24,116,140)(25,141,117)(26,118,142)(27,143,119)(28,120,144)(29,129,121)(30,122,130)(31,131,123)(32,124,132)(33,71,49)(34,50,72)(35,73,51)(36,52,74)(37,75,53)(38,54,76)(39,77,55)(40,56,78)(41,79,57)(42,58,80)(43,65,59)(44,60,66)(45,67,61)(46,62,68)(47,69,63)(48,64,70), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(18,24)(19,31)(20,22)(21,29)(23,27)(26,32)(28,30)(33,41)(34,48)(35,39)(36,46)(38,44)(40,42)(43,47)(49,57)(50,64)(51,55)(52,62)(54,60)(56,58)(59,63)(65,69)(66,76)(68,74)(70,72)(71,79)(73,77)(78,80)(81,91)(83,89)(84,96)(85,87)(86,94)(88,92)(93,95)(97,99)(98,106)(100,104)(101,111)(103,109)(105,107)(108,112)(113,121)(114,128)(115,119)(116,126)(118,124)(120,122)(123,127)(129,137)(130,144)(131,135)(132,142)(134,140)(136,138)(139,143)>;
G:=Group( (1,125,37)(2,38,126)(3,127,39)(4,40,128)(5,113,41)(6,42,114)(7,115,43)(8,44,116)(9,117,45)(10,46,118)(11,119,47)(12,48,120)(13,121,33)(14,34,122)(15,123,35)(16,36,124)(17,75,110)(18,111,76)(19,77,112)(20,97,78)(21,79,98)(22,99,80)(23,65,100)(24,101,66)(25,67,102)(26,103,68)(27,69,104)(28,105,70)(29,71,106)(30,107,72)(31,73,108)(32,109,74)(49,86,129)(50,130,87)(51,88,131)(52,132,89)(53,90,133)(54,134,91)(55,92,135)(56,136,93)(57,94,137)(58,138,95)(59,96,139)(60,140,81)(61,82,141)(62,142,83)(63,84,143)(64,144,85), (1,110,90)(2,91,111)(3,112,92)(4,93,97)(5,98,94)(6,95,99)(7,100,96)(8,81,101)(9,102,82)(10,83,103)(11,104,84)(12,85,105)(13,106,86)(14,87,107)(15,108,88)(16,89,109)(17,133,125)(18,126,134)(19,135,127)(20,128,136)(21,137,113)(22,114,138)(23,139,115)(24,116,140)(25,141,117)(26,118,142)(27,143,119)(28,120,144)(29,129,121)(30,122,130)(31,131,123)(32,124,132)(33,71,49)(34,50,72)(35,73,51)(36,52,74)(37,75,53)(38,54,76)(39,77,55)(40,56,78)(41,79,57)(42,58,80)(43,65,59)(44,60,66)(45,67,61)(46,62,68)(47,69,63)(48,64,70), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(18,24)(19,31)(20,22)(21,29)(23,27)(26,32)(28,30)(33,41)(34,48)(35,39)(36,46)(38,44)(40,42)(43,47)(49,57)(50,64)(51,55)(52,62)(54,60)(56,58)(59,63)(65,69)(66,76)(68,74)(70,72)(71,79)(73,77)(78,80)(81,91)(83,89)(84,96)(85,87)(86,94)(88,92)(93,95)(97,99)(98,106)(100,104)(101,111)(103,109)(105,107)(108,112)(113,121)(114,128)(115,119)(116,126)(118,124)(120,122)(123,127)(129,137)(130,144)(131,135)(132,142)(134,140)(136,138)(139,143) );
G=PermutationGroup([[(1,125,37),(2,38,126),(3,127,39),(4,40,128),(5,113,41),(6,42,114),(7,115,43),(8,44,116),(9,117,45),(10,46,118),(11,119,47),(12,48,120),(13,121,33),(14,34,122),(15,123,35),(16,36,124),(17,75,110),(18,111,76),(19,77,112),(20,97,78),(21,79,98),(22,99,80),(23,65,100),(24,101,66),(25,67,102),(26,103,68),(27,69,104),(28,105,70),(29,71,106),(30,107,72),(31,73,108),(32,109,74),(49,86,129),(50,130,87),(51,88,131),(52,132,89),(53,90,133),(54,134,91),(55,92,135),(56,136,93),(57,94,137),(58,138,95),(59,96,139),(60,140,81),(61,82,141),(62,142,83),(63,84,143),(64,144,85)], [(1,110,90),(2,91,111),(3,112,92),(4,93,97),(5,98,94),(6,95,99),(7,100,96),(8,81,101),(9,102,82),(10,83,103),(11,104,84),(12,85,105),(13,106,86),(14,87,107),(15,108,88),(16,89,109),(17,133,125),(18,126,134),(19,135,127),(20,128,136),(21,137,113),(22,114,138),(23,139,115),(24,116,140),(25,141,117),(26,118,142),(27,143,119),(28,120,144),(29,129,121),(30,122,130),(31,131,123),(32,124,132),(33,71,49),(34,50,72),(35,73,51),(36,52,74),(37,75,53),(38,54,76),(39,77,55),(40,56,78),(41,79,57),(42,58,80),(43,65,59),(44,60,66),(45,67,61),(46,62,68),(47,69,63),(48,64,70)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(18,24),(19,31),(20,22),(21,29),(23,27),(26,32),(28,30),(33,41),(34,48),(35,39),(36,46),(38,44),(40,42),(43,47),(49,57),(50,64),(51,55),(52,62),(54,60),(56,58),(59,63),(65,69),(66,76),(68,74),(70,72),(71,79),(73,77),(78,80),(81,91),(83,89),(84,96),(85,87),(86,94),(88,92),(93,95),(97,99),(98,106),(100,104),(101,111),(103,109),(105,107),(108,112),(113,121),(114,128),(115,119),(116,126),(118,124),(120,122),(123,127),(129,137),(130,144),(131,135),(132,142),(134,140),(136,138),(139,143)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 8A | 8B | 12A | 12B | 12C | 12D | 16A | 16B | 16C | 16D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 2 | 72 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D8 | C3⋊D4 | SD32 | D4⋊S3 | D8.S3 |
| kernel | C32⋊8SD32 | C24.S3 | C32⋊5Q16 | C32×D8 | C3×D8 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 8 | 4 | 4 | 8 |
Matrix representation of C32⋊8SD32 ►in GL6(𝔽97)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 1 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 |
| 10 | 44 | 0 | 0 | 0 | 0 |
| 53 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 83 | 0 | 0 |
| 0 | 0 | 50 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 66 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 96 |
G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,61,0,0,0,0,0,0,35],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0,0,0,0,0,0,0,61,0,0,0,0,0,0,35],[10,53,0,0,0,0,44,10,0,0,0,0,0,0,64,50,0,0,0,0,83,33,0,0,0,0,0,0,0,66,0,0,0,0,72,0],[1,0,0,0,0,0,0,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96] >;
C32⋊8SD32 in GAP, Magma, Sage, TeX
C_3^2\rtimes_8{\rm SD}_{32} % in TeX
G:=Group("C3^2:8SD32"); // GroupNames label
G:=SmallGroup(288,302);
// by ID
G=gap.SmallGroup(288,302);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,254,135,142,675,346,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^7>;
// generators/relations