p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8).171D4, (C2×Q8).12Q8, (C22×C4).165D4, C23.944(C2×D4), C2.14(Q8.Q8), C4.44(C22⋊Q8), C42⋊8C4.17C2, C2.18(C8.2D4), C2.19(C8.12D4), C22.130(C4○D8), C22.83(C4⋊1D4), (C2×C42).395C22, (C22×C8).333C22, (C22×Q8).83C22, (C22×C4).1478C23, C4.38(C22.D4), C22.121(C22⋊Q8), C2.10(C23.4Q8), C2.14(C23.20D4), C22.149(C8.C22), C22.7C42.29C2, C23.67C23.23C2, C22.131(C22.D4), (C2×C4).753(C2×D4), (C2×C4).295(C2×Q8), (C2×C4.Q8).30C2, (C2×C2.D8).21C2, (C2×C4).784(C4○D4), (C2×C4⋊C4).167C22, (C2×Q8⋊C4).22C2, SmallGroup(128,829)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8).171D4
G = < a,b,c,d | a2=b8=1, c4=b4, d2=a, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b3, dcd-1=b4c3 >
Subgroups: 240 in 119 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C22×C8, C22×Q8, C22.7C42, C42⋊8C4, C23.67C23, C2×Q8⋊C4, C2×C4.Q8, C2×C2.D8, (C2×C8).171D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C22⋊Q8, C22.D4, C4⋊1D4, C4○D8, C8.C22, C23.4Q8, Q8.Q8, C23.20D4, C8.12D4, C8.2D4, (C2×C8).171D4
►Regular action on 128 points
Generators in S128
(1 122)(2 123)(3 124)(4 125)(5 126)(6 127)(7 128)(8 121)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)(49 116)(50 117)(51 118)(52 119)(53 120)(54 113)(55 114)(56 115)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 105)(73 104)(74 97)(75 98)(76 99)(77 100)(78 101)(79 102)(80 103)(81 96)(82 89)(83 90)(84 91)(85 92)(86 93)(87 94)(88 95)
(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 36 58 87 5 40 62 83)(2 22 59 95 6 18 63 91)(3 38 60 81 7 34 64 85)(4 24 61 89 8 20 57 93)(9 84 123 37 13 88 127 33)(10 92 124 23 14 96 128 19)(11 86 125 39 15 82 121 35)(12 94 126 17 16 90 122 21)(25 111 102 115 29 107 98 119)(26 71 103 49 30 67 99 53)(27 105 104 117 31 109 100 113)(28 65 97 51 32 69 101 55)(41 66 75 52 45 70 79 56)(42 108 76 120 46 112 80 116)(43 68 77 54 47 72 73 50)(44 110 78 114 48 106 74 118)
(1 78 122 101)(2 73 123 104)(3 76 124 99)(4 79 125 102)(5 74 126 97)(6 77 127 100)(7 80 128 103)(8 75 121 98)(9 31 63 43)(10 26 64 46)(11 29 57 41)(12 32 58 44)(13 27 59 47)(14 30 60 42)(15 25 61 45)(16 28 62 48)(17 65 40 106)(18 68 33 109)(19 71 34 112)(20 66 35 107)(21 69 36 110)(22 72 37 105)(23 67 38 108)(24 70 39 111)(49 81 116 96)(50 84 117 91)(51 87 118 94)(52 82 119 89)(53 85 120 92)(54 88 113 95)(55 83 114 90)(56 86 115 93)
G:=sub<Sym(128)| (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,121)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(49,116)(50,117)(51,118)(52,119)(53,120)(54,113)(55,114)(56,115)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,105)(73,104)(74,97)(75,98)(76,99)(77,100)(78,101)(79,102)(80,103)(81,96)(82,89)(83,90)(84,91)(85,92)(86,93)(87,94)(88,95), (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), (1,36,58,87,5,40,62,83)(2,22,59,95,6,18,63,91)(3,38,60,81,7,34,64,85)(4,24,61,89,8,20,57,93)(9,84,123,37,13,88,127,33)(10,92,124,23,14,96,128,19)(11,86,125,39,15,82,121,35)(12,94,126,17,16,90,122,21)(25,111,102,115,29,107,98,119)(26,71,103,49,30,67,99,53)(27,105,104,117,31,109,100,113)(28,65,97,51,32,69,101,55)(41,66,75,52,45,70,79,56)(42,108,76,120,46,112,80,116)(43,68,77,54,47,72,73,50)(44,110,78,114,48,106,74,118), (1,78,122,101)(2,73,123,104)(3,76,124,99)(4,79,125,102)(5,74,126,97)(6,77,127,100)(7,80,128,103)(8,75,121,98)(9,31,63,43)(10,26,64,46)(11,29,57,41)(12,32,58,44)(13,27,59,47)(14,30,60,42)(15,25,61,45)(16,28,62,48)(17,65,40,106)(18,68,33,109)(19,71,34,112)(20,66,35,107)(21,69,36,110)(22,72,37,105)(23,67,38,108)(24,70,39,111)(49,81,116,96)(50,84,117,91)(51,87,118,94)(52,82,119,89)(53,85,120,92)(54,88,113,95)(55,83,114,90)(56,86,115,93)>;
G:=Group( (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,121)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(49,116)(50,117)(51,118)(52,119)(53,120)(54,113)(55,114)(56,115)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,105)(73,104)(74,97)(75,98)(76,99)(77,100)(78,101)(79,102)(80,103)(81,96)(82,89)(83,90)(84,91)(85,92)(86,93)(87,94)(88,95), (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), (1,36,58,87,5,40,62,83)(2,22,59,95,6,18,63,91)(3,38,60,81,7,34,64,85)(4,24,61,89,8,20,57,93)(9,84,123,37,13,88,127,33)(10,92,124,23,14,96,128,19)(11,86,125,39,15,82,121,35)(12,94,126,17,16,90,122,21)(25,111,102,115,29,107,98,119)(26,71,103,49,30,67,99,53)(27,105,104,117,31,109,100,113)(28,65,97,51,32,69,101,55)(41,66,75,52,45,70,79,56)(42,108,76,120,46,112,80,116)(43,68,77,54,47,72,73,50)(44,110,78,114,48,106,74,118), (1,78,122,101)(2,73,123,104)(3,76,124,99)(4,79,125,102)(5,74,126,97)(6,77,127,100)(7,80,128,103)(8,75,121,98)(9,31,63,43)(10,26,64,46)(11,29,57,41)(12,32,58,44)(13,27,59,47)(14,30,60,42)(15,25,61,45)(16,28,62,48)(17,65,40,106)(18,68,33,109)(19,71,34,112)(20,66,35,107)(21,69,36,110)(22,72,37,105)(23,67,38,108)(24,70,39,111)(49,81,116,96)(50,84,117,91)(51,87,118,94)(52,82,119,89)(53,85,120,92)(54,88,113,95)(55,83,114,90)(56,86,115,93) );
G=PermutationGroup([[(1,122),(2,123),(3,124),(4,125),(5,126),(6,127),(7,128),(8,121),(9,63),(10,64),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,40),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(25,45),(26,46),(27,47),(28,48),(29,41),(30,42),(31,43),(32,44),(49,116),(50,117),(51,118),(52,119),(53,120),(54,113),(55,114),(56,115),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,105),(73,104),(74,97),(75,98),(76,99),(77,100),(78,101),(79,102),(80,103),(81,96),(82,89),(83,90),(84,91),(85,92),(86,93),(87,94),(88,95)], [(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)], [(1,36,58,87,5,40,62,83),(2,22,59,95,6,18,63,91),(3,38,60,81,7,34,64,85),(4,24,61,89,8,20,57,93),(9,84,123,37,13,88,127,33),(10,92,124,23,14,96,128,19),(11,86,125,39,15,82,121,35),(12,94,126,17,16,90,122,21),(25,111,102,115,29,107,98,119),(26,71,103,49,30,67,99,53),(27,105,104,117,31,109,100,113),(28,65,97,51,32,69,101,55),(41,66,75,52,45,70,79,56),(42,108,76,120,46,112,80,116),(43,68,77,54,47,72,73,50),(44,110,78,114,48,106,74,118)], [(1,78,122,101),(2,73,123,104),(3,76,124,99),(4,79,125,102),(5,74,126,97),(6,77,127,100),(7,80,128,103),(8,75,121,98),(9,31,63,43),(10,26,64,46),(11,29,57,41),(12,32,58,44),(13,27,59,47),(14,30,60,42),(15,25,61,45),(16,28,62,48),(17,65,40,106),(18,68,33,109),(19,71,34,112),(20,66,35,107),(21,69,36,110),(22,72,37,105),(23,67,38,108),(24,70,39,111),(49,81,116,96),(50,84,117,91),(51,87,118,94),(52,82,119,89),(53,85,120,92),(54,88,113,95),(55,83,114,90),(56,86,115,93)]])
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 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | C4○D8 | C8.C22 |
| kernel | (C2×C8).171D4 | C22.7C42 | C42⋊8C4 | C23.67C23 | C2×Q8⋊C4 | C2×C4.Q8 | C2×C2.D8 | C2×C8 | C22×C4 | C2×Q8 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 4 | 2 | 2 | 6 | 8 | 2 |
Matrix representation of (C2×C8).171D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 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 | 1 |
| 4 | 7 | 0 | 0 | 0 | 0 |
| 10 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 4 | 0 | 0 |
| 0 | 0 | 13 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 5 | 7 |
| 6 | 1 | 0 | 0 | 0 | 0 |
| 16 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 6 |
| 0 | 0 | 0 | 0 | 14 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 10 | 0 | 0 |
| 0 | 0 | 7 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 6 |
| 0 | 0 | 0 | 0 | 14 | 6 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,10,0,0,0,0,7,13,0,0,0,0,0,0,10,13,0,0,0,0,4,7,0,0,0,0,0,0,0,5,0,0,0,0,7,7],[6,16,0,0,0,0,1,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,14,0,0,0,0,6,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,7,0,0,0,0,10,13,0,0,0,0,0,0,11,14,0,0,0,0,6,6] >;
(C2×C8).171D4 in GAP, Magma, Sage, TeX
(C_2\times C_8)._{171}D_4 % in TeX
G:=Group("(C2xC8).171D4"); // GroupNames label
G:=SmallGroup(128,829);
// by ID
G=gap.SmallGroup(128,829);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,624,422,387,352,1018,521,248,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=a,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b^3,d*c*d^-1=b^4*c^3>;
// generators/relations