p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4)⋊3Q16, (C2×C8).50D4, C4.78C22≀C2, (C2×Q8).107D4, C23.918(C2×D4), (C22×C4).152D4, (C22×Q16).5C2, C22.57(C2×Q16), C2.14(C4⋊2Q16), C4.74(C4.4D4), C2.14(C8.D4), (C22×C8).79C22, C2.14(C8.18D4), C22.114(C4○D8), C22.4Q16.37C2, (C2×C42).370C22, C2.19(Q8.D4), (C22×Q8).73C22, C22.240(C4⋊D4), (C22×C4).1452C23, C4.78(C22.D4), C2.29(C23.10D4), C22.130(C8.C22), C23.67C23.16C2, (C2×C4⋊C8).37C2, (C2×C4).1047(C2×D4), (C2×C4).883(C4○D4), (C2×C4⋊C4).133C22, (C2×Q8⋊C4).13C2, SmallGroup(128,788)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4)⋊3Q16
G = < a,b,c,d | a2=b4=c8=1, d2=c4, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=b-1, dcd-1=c-1 >
Subgroups: 288 in 144 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C4⋊C8, C2×C42, C2×C4⋊C4, C22×C8, C2×Q16, C22×Q8, C22.4Q16, C23.67C23, C2×Q8⋊C4, C2×C4⋊C8, C22×Q16, (C2×C4)⋊3Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C2×Q16, C4○D8, C8.C22, C23.10D4, C4⋊2Q16, Q8.D4, C8.18D4, C8.D4, (C2×C4)⋊3Q16
►Regular action on 128 points
Generators in S128
(1 37)(2 38)(3 39)(4 40)(5 33)(6 34)(7 35)(8 36)(9 107)(10 108)(11 109)(12 110)(13 111)(14 112)(15 105)(16 106)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 57)(41 70)(42 71)(43 72)(44 65)(45 66)(46 67)(47 68)(48 69)(73 96)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(97 125)(98 126)(99 127)(100 128)(101 121)(102 122)(103 123)(104 124)
(1 106 125 70)(2 71 126 107)(3 108 127 72)(4 65 128 109)(5 110 121 66)(6 67 122 111)(7 112 123 68)(8 69 124 105)(9 38 42 98)(10 99 43 39)(11 40 44 100)(12 101 45 33)(13 34 46 102)(14 103 47 35)(15 36 48 104)(16 97 41 37)(17 120 89 32)(18 25 90 113)(19 114 91 26)(20 27 92 115)(21 116 93 28)(22 29 94 117)(23 118 95 30)(24 31 96 119)(49 88 74 57)(50 58 75 81)(51 82 76 59)(52 60 77 83)(53 84 78 61)(54 62 79 85)(55 86 80 63)(56 64 73 87)
(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)
(1 80 5 76)(2 79 6 75)(3 78 7 74)(4 77 8 73)(9 62 13 58)(10 61 14 57)(11 60 15 64)(12 59 16 63)(17 99 21 103)(18 98 22 102)(19 97 23 101)(20 104 24 100)(25 107 29 111)(26 106 30 110)(27 105 31 109)(28 112 32 108)(33 91 37 95)(34 90 38 94)(35 89 39 93)(36 96 40 92)(41 86 45 82)(42 85 46 81)(43 84 47 88)(44 83 48 87)(49 127 53 123)(50 126 54 122)(51 125 55 121)(52 124 56 128)(65 115 69 119)(66 114 70 118)(67 113 71 117)(68 120 72 116)
G:=sub<Sym(128)| (1,37)(2,38)(3,39)(4,40)(5,33)(6,34)(7,35)(8,36)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,105)(16,106)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(41,70)(42,71)(43,72)(44,65)(45,66)(46,67)(47,68)(48,69)(73,96)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(97,125)(98,126)(99,127)(100,128)(101,121)(102,122)(103,123)(104,124), (1,106,125,70)(2,71,126,107)(3,108,127,72)(4,65,128,109)(5,110,121,66)(6,67,122,111)(7,112,123,68)(8,69,124,105)(9,38,42,98)(10,99,43,39)(11,40,44,100)(12,101,45,33)(13,34,46,102)(14,103,47,35)(15,36,48,104)(16,97,41,37)(17,120,89,32)(18,25,90,113)(19,114,91,26)(20,27,92,115)(21,116,93,28)(22,29,94,117)(23,118,95,30)(24,31,96,119)(49,88,74,57)(50,58,75,81)(51,82,76,59)(52,60,77,83)(53,84,78,61)(54,62,79,85)(55,86,80,63)(56,64,73,87), (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), (1,80,5,76)(2,79,6,75)(3,78,7,74)(4,77,8,73)(9,62,13,58)(10,61,14,57)(11,60,15,64)(12,59,16,63)(17,99,21,103)(18,98,22,102)(19,97,23,101)(20,104,24,100)(25,107,29,111)(26,106,30,110)(27,105,31,109)(28,112,32,108)(33,91,37,95)(34,90,38,94)(35,89,39,93)(36,96,40,92)(41,86,45,82)(42,85,46,81)(43,84,47,88)(44,83,48,87)(49,127,53,123)(50,126,54,122)(51,125,55,121)(52,124,56,128)(65,115,69,119)(66,114,70,118)(67,113,71,117)(68,120,72,116)>;
G:=Group( (1,37)(2,38)(3,39)(4,40)(5,33)(6,34)(7,35)(8,36)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,105)(16,106)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(41,70)(42,71)(43,72)(44,65)(45,66)(46,67)(47,68)(48,69)(73,96)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(97,125)(98,126)(99,127)(100,128)(101,121)(102,122)(103,123)(104,124), (1,106,125,70)(2,71,126,107)(3,108,127,72)(4,65,128,109)(5,110,121,66)(6,67,122,111)(7,112,123,68)(8,69,124,105)(9,38,42,98)(10,99,43,39)(11,40,44,100)(12,101,45,33)(13,34,46,102)(14,103,47,35)(15,36,48,104)(16,97,41,37)(17,120,89,32)(18,25,90,113)(19,114,91,26)(20,27,92,115)(21,116,93,28)(22,29,94,117)(23,118,95,30)(24,31,96,119)(49,88,74,57)(50,58,75,81)(51,82,76,59)(52,60,77,83)(53,84,78,61)(54,62,79,85)(55,86,80,63)(56,64,73,87), (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), (1,80,5,76)(2,79,6,75)(3,78,7,74)(4,77,8,73)(9,62,13,58)(10,61,14,57)(11,60,15,64)(12,59,16,63)(17,99,21,103)(18,98,22,102)(19,97,23,101)(20,104,24,100)(25,107,29,111)(26,106,30,110)(27,105,31,109)(28,112,32,108)(33,91,37,95)(34,90,38,94)(35,89,39,93)(36,96,40,92)(41,86,45,82)(42,85,46,81)(43,84,47,88)(44,83,48,87)(49,127,53,123)(50,126,54,122)(51,125,55,121)(52,124,56,128)(65,115,69,119)(66,114,70,118)(67,113,71,117)(68,120,72,116) );
G=PermutationGroup([[(1,37),(2,38),(3,39),(4,40),(5,33),(6,34),(7,35),(8,36),(9,107),(10,108),(11,109),(12,110),(13,111),(14,112),(15,105),(16,106),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,57),(41,70),(42,71),(43,72),(44,65),(45,66),(46,67),(47,68),(48,69),(73,96),(74,89),(75,90),(76,91),(77,92),(78,93),(79,94),(80,95),(81,113),(82,114),(83,115),(84,116),(85,117),(86,118),(87,119),(88,120),(97,125),(98,126),(99,127),(100,128),(101,121),(102,122),(103,123),(104,124)], [(1,106,125,70),(2,71,126,107),(3,108,127,72),(4,65,128,109),(5,110,121,66),(6,67,122,111),(7,112,123,68),(8,69,124,105),(9,38,42,98),(10,99,43,39),(11,40,44,100),(12,101,45,33),(13,34,46,102),(14,103,47,35),(15,36,48,104),(16,97,41,37),(17,120,89,32),(18,25,90,113),(19,114,91,26),(20,27,92,115),(21,116,93,28),(22,29,94,117),(23,118,95,30),(24,31,96,119),(49,88,74,57),(50,58,75,81),(51,82,76,59),(52,60,77,83),(53,84,78,61),(54,62,79,85),(55,86,80,63),(56,64,73,87)], [(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)], [(1,80,5,76),(2,79,6,75),(3,78,7,74),(4,77,8,73),(9,62,13,58),(10,61,14,57),(11,60,15,64),(12,59,16,63),(17,99,21,103),(18,98,22,102),(19,97,23,101),(20,104,24,100),(25,107,29,111),(26,106,30,110),(27,105,31,109),(28,112,32,108),(33,91,37,95),(34,90,38,94),(35,89,39,93),(36,96,40,92),(41,86,45,82),(42,85,46,81),(43,84,47,88),(44,83,48,87),(49,127,53,123),(50,126,54,122),(51,125,55,121),(52,124,56,128),(65,115,69,119),(66,114,70,118),(67,113,71,117),(68,120,72,116)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | Q16 | C4○D4 | C4○D8 | C8.C22 |
| kernel | (C2×C4)⋊3Q16 | C22.4Q16 | C23.67C23 | C2×Q8⋊C4 | C2×C4⋊C8 | C22×Q16 | C2×C8 | C22×C4 | C2×Q8 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 6 | 4 | 2 |
Matrix representation of (C2×C4)⋊3Q16 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 1 |
| 0 | 0 | 0 | 0 | 1 | 10 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 5 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 7 |
| 0 | 0 | 0 | 0 | 7 | 1 |
| 16 | 15 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 1 | 0 | 0 |
| 0 | 0 | 16 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,7,1,0,0,0,0,1,10],[9,5,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,7,0,0,0,0,7,1],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,11,16,0,0,0,0,1,6,0,0,0,0,0,0,0,13,0,0,0,0,4,0] >;
(C2×C4)⋊3Q16 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_3Q_{16} % in TeX
G:=Group("(C2xC4):3Q16"); // GroupNames label
G:=SmallGroup(128,788);
// by ID
G=gap.SmallGroup(128,788);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,456,422,387,58,2019,1018,248,2804,1411,172,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=c^4,d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations