p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8).52D4, (C2×Q8).9Q8, C4⋊C4.109D4, (C2×C4).18Q16, C2.21(C8⋊D4), C2.8(C4.Q16), (C22×C4).157D4, C23.925(C2×D4), C2.11(Q8.Q8), C4.38(C22⋊Q8), C22.59(C2×Q16), C4.156(C4⋊D4), C2.15(C4⋊2Q16), (C22×C8).83C22, C4.36(C42⋊2C2), C2.21(D4.2D4), C2.15(C8.18D4), C22.119(C4○D8), C22.4Q16.24C2, (C2×C42).379C22, C2.9(C23.Q8), (C22×Q8).76C22, C22.246(C4⋊D4), C22.147(C8⋊C22), (C22×C4).1459C23, C22.112(C22⋊Q8), C22.136(C8.C22), C23.67C23.18C2, C23.65C23.19C2, (C2×C4⋊C8).38C2, (C2×C4).286(C2×Q8), (C2×C2.D8).14C2, (C2×C4).1058(C2×D4), (C2×C4).623(C4○D4), (C2×C4⋊C4).144C22, (C2×Q8⋊C4).15C2, SmallGroup(128,800)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8).52D4
G = < a,b,c,d | a2=b8=c4=1, d2=b2, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=b-1, dcd-1=b6c-1 >
Subgroups: 248 in 124 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C4⋊C8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C22.4Q16, C23.65C23, C23.67C23, C2×Q8⋊C4, C2×C4⋊C8, C2×C2.D8, (C2×C8).52D4
Quotients: C1, C2, C22, D4, Q8, C23, Q16, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C42⋊2C2, C2×Q16, C4○D8, C8⋊C22, C8.C22, C23.Q8, C4⋊2Q16, D4.2D4, C8.18D4, C8⋊D4, C4.Q16, Q8.Q8, (C2×C8).52D4
►Regular action on 128 points
Generators in S128
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 117)(10 118)(11 119)(12 120)(13 113)(14 114)(15 115)(16 116)(17 80)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 127)(34 128)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 112)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(65 97)(66 98)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)(81 93)(82 94)(83 95)(84 96)(85 89)(86 90)(87 91)(88 92)
(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)
(1 121 23 94)(2 128 24 93)(3 127 17 92)(4 126 18 91)(5 125 19 90)(6 124 20 89)(7 123 21 96)(8 122 22 95)(9 58 106 102)(10 57 107 101)(11 64 108 100)(12 63 109 99)(13 62 110 98)(14 61 111 97)(15 60 112 104)(16 59 105 103)(25 48 65 114)(26 47 66 113)(27 46 67 120)(28 45 68 119)(29 44 69 118)(30 43 70 117)(31 42 71 116)(32 41 72 115)(33 80 88 55)(34 79 81 54)(35 78 82 53)(36 77 83 52)(37 76 84 51)(38 75 85 50)(39 74 86 49)(40 73 87 56)
(1 14 3 16 5 10 7 12)(2 115 4 117 6 119 8 113)(9 50 11 52 13 54 15 56)(17 105 19 107 21 109 23 111)(18 43 20 45 22 47 24 41)(25 86 27 88 29 82 31 84)(26 91 28 93 30 95 32 89)(33 69 35 71 37 65 39 67)(34 102 36 104 38 98 40 100)(42 74 44 76 46 78 48 80)(49 118 51 120 53 114 55 116)(57 94 59 96 61 90 63 92)(58 83 60 85 62 87 64 81)(66 126 68 128 70 122 72 124)(73 106 75 108 77 110 79 112)(97 125 99 127 101 121 103 123)
G:=sub<Sym(128)| (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,117)(10,118)(11,119)(12,120)(13,113)(14,114)(15,115)(16,116)(17,80)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,61)(26,62)(27,63)(28,64)(29,57)(30,58)(31,59)(32,60)(33,127)(34,128)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,112)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(81,93)(82,94)(83,95)(84,96)(85,89)(86,90)(87,91)(88,92), (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), (1,121,23,94)(2,128,24,93)(3,127,17,92)(4,126,18,91)(5,125,19,90)(6,124,20,89)(7,123,21,96)(8,122,22,95)(9,58,106,102)(10,57,107,101)(11,64,108,100)(12,63,109,99)(13,62,110,98)(14,61,111,97)(15,60,112,104)(16,59,105,103)(25,48,65,114)(26,47,66,113)(27,46,67,120)(28,45,68,119)(29,44,69,118)(30,43,70,117)(31,42,71,116)(32,41,72,115)(33,80,88,55)(34,79,81,54)(35,78,82,53)(36,77,83,52)(37,76,84,51)(38,75,85,50)(39,74,86,49)(40,73,87,56), (1,14,3,16,5,10,7,12)(2,115,4,117,6,119,8,113)(9,50,11,52,13,54,15,56)(17,105,19,107,21,109,23,111)(18,43,20,45,22,47,24,41)(25,86,27,88,29,82,31,84)(26,91,28,93,30,95,32,89)(33,69,35,71,37,65,39,67)(34,102,36,104,38,98,40,100)(42,74,44,76,46,78,48,80)(49,118,51,120,53,114,55,116)(57,94,59,96,61,90,63,92)(58,83,60,85,62,87,64,81)(66,126,68,128,70,122,72,124)(73,106,75,108,77,110,79,112)(97,125,99,127,101,121,103,123)>;
G:=Group( (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,117)(10,118)(11,119)(12,120)(13,113)(14,114)(15,115)(16,116)(17,80)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,61)(26,62)(27,63)(28,64)(29,57)(30,58)(31,59)(32,60)(33,127)(34,128)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,112)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(81,93)(82,94)(83,95)(84,96)(85,89)(86,90)(87,91)(88,92), (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), (1,121,23,94)(2,128,24,93)(3,127,17,92)(4,126,18,91)(5,125,19,90)(6,124,20,89)(7,123,21,96)(8,122,22,95)(9,58,106,102)(10,57,107,101)(11,64,108,100)(12,63,109,99)(13,62,110,98)(14,61,111,97)(15,60,112,104)(16,59,105,103)(25,48,65,114)(26,47,66,113)(27,46,67,120)(28,45,68,119)(29,44,69,118)(30,43,70,117)(31,42,71,116)(32,41,72,115)(33,80,88,55)(34,79,81,54)(35,78,82,53)(36,77,83,52)(37,76,84,51)(38,75,85,50)(39,74,86,49)(40,73,87,56), (1,14,3,16,5,10,7,12)(2,115,4,117,6,119,8,113)(9,50,11,52,13,54,15,56)(17,105,19,107,21,109,23,111)(18,43,20,45,22,47,24,41)(25,86,27,88,29,82,31,84)(26,91,28,93,30,95,32,89)(33,69,35,71,37,65,39,67)(34,102,36,104,38,98,40,100)(42,74,44,76,46,78,48,80)(49,118,51,120,53,114,55,116)(57,94,59,96,61,90,63,92)(58,83,60,85,62,87,64,81)(66,126,68,128,70,122,72,124)(73,106,75,108,77,110,79,112)(97,125,99,127,101,121,103,123) );
G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,49),(6,50),(7,51),(8,52),(9,117),(10,118),(11,119),(12,120),(13,113),(14,114),(15,115),(16,116),(17,80),(18,73),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(25,61),(26,62),(27,63),(28,64),(29,57),(30,58),(31,59),(32,60),(33,127),(34,128),(35,121),(36,122),(37,123),(38,124),(39,125),(40,126),(41,112),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(65,97),(66,98),(67,99),(68,100),(69,101),(70,102),(71,103),(72,104),(81,93),(82,94),(83,95),(84,96),(85,89),(86,90),(87,91),(88,92)], [(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)], [(1,121,23,94),(2,128,24,93),(3,127,17,92),(4,126,18,91),(5,125,19,90),(6,124,20,89),(7,123,21,96),(8,122,22,95),(9,58,106,102),(10,57,107,101),(11,64,108,100),(12,63,109,99),(13,62,110,98),(14,61,111,97),(15,60,112,104),(16,59,105,103),(25,48,65,114),(26,47,66,113),(27,46,67,120),(28,45,68,119),(29,44,69,118),(30,43,70,117),(31,42,71,116),(32,41,72,115),(33,80,88,55),(34,79,81,54),(35,78,82,53),(36,77,83,52),(37,76,84,51),(38,75,85,50),(39,74,86,49),(40,73,87,56)], [(1,14,3,16,5,10,7,12),(2,115,4,117,6,119,8,113),(9,50,11,52,13,54,15,56),(17,105,19,107,21,109,23,111),(18,43,20,45,22,47,24,41),(25,86,27,88,29,82,31,84),(26,91,28,93,30,95,32,89),(33,69,35,71,37,65,39,67),(34,102,36,104,38,98,40,100),(42,74,44,76,46,78,48,80),(49,118,51,120,53,114,55,116),(57,94,59,96,61,90,63,92),(58,83,60,85,62,87,64,81),(66,126,68,128,70,122,72,124),(73,106,75,108,77,110,79,112),(97,125,99,127,101,121,103,123)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | Q8 | Q16 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | (C2×C8).52D4 | C22.4Q16 | C23.65C23 | C23.67C23 | C2×Q8⋊C4 | C2×C4⋊C8 | C2×C2.D8 | C4⋊C4 | C2×C8 | C22×C4 | C2×Q8 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 6 | 4 | 1 | 1 |
Matrix representation of (C2×C8).52D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 2 | 0 | 0 |
| 0 | 0 | 7 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 9 | 0 | 0 |
| 0 | 0 | 2 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 8 | 0 | 0 |
| 0 | 0 | 15 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[3,14,0,0,0,0,3,3,0,0,0,0,0,0,6,7,0,0,0,0,2,11,0,0,0,0,0,0,13,0,0,0,0,0,9,4],[14,14,0,0,0,0,14,3,0,0,0,0,0,0,10,2,0,0,0,0,9,7,0,0,0,0,0,0,16,1,0,0,0,0,0,1],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,7,15,0,0,0,0,8,10,0,0,0,0,0,0,16,1,0,0,0,0,15,1] >;
(C2×C8).52D4 in GAP, Magma, Sage, TeX
(C_2\times C_8)._{52}D_4 % in TeX
G:=Group("(C2xC8).52D4"); // GroupNames label
G:=SmallGroup(128,800);
// by ID
G=gap.SmallGroup(128,800);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,512,422,387,352,2019,521,248,2804,718,172,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^2,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=b^6*c^-1>;
// generators/relations