p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8).170D4, (C2×Q8).11Q8, (C2×C4).44SD16, C2.15(C8⋊5D4), (C22×C4).327D4, C23.943(C2×D4), C4.43(C22⋊Q8), C42⋊9C4.19C2, C2.10(Q8⋊Q8), C2.17(C8.2D4), C22.82(C4⋊1D4), (C2×C42).394C22, (C22×C8).332C22, C22.109(C2×SD16), C2.9(C23.4Q8), (C22×Q8).82C22, (C22×C4).1477C23, C4.37(C22.D4), C22.120(C22⋊Q8), C2.10(C23.47D4), C22.148(C8.C22), C22.7C42.37C2, C23.67C23.22C2, C22.130(C22.D4), (C2×C4).752(C2×D4), (C2×C4).294(C2×Q8), (C2×C4.Q8).29C2, (C2×C4).783(C4○D4), (C2×C4⋊C4).166C22, (C2×Q8⋊C4).28C2, SmallGroup(128,828)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8).170D4
G = < a,b,c,d | a2=b8=1, c4=b4, d2=ab4, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b3, dcd-1=c3 >
Subgroups: 256 in 127 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, 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×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C22.7C42, C42⋊9C4, C23.67C23, C2×Q8⋊C4, C2×C4.Q8, (C2×C8).170D4
Quotients: C1, C2, C22, D4, Q8, C23, SD16, C2×D4, C2×Q8, C4○D4, C22⋊Q8, C22.D4, C4⋊1D4, C2×SD16, C8.C22, C23.4Q8, Q8⋊Q8, C23.47D4, C8⋊5D4, C8.2D4, (C2×C8).170D4
►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 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(17 76)(18 77)(19 78)(20 79)(21 80)(22 73)(23 74)(24 75)(25 85)(26 86)(27 87)(28 88)(29 81)(30 82)(31 83)(32 84)(33 100)(34 101)(35 102)(36 103)(37 104)(38 97)(39 98)(40 99)(41 96)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 112)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(65 120)(66 113)(67 114)(68 115)(69 116)(70 117)(71 118)(72 119)
(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 37 65 47 5 33 69 43)(2 97 66 95 6 101 70 91)(3 39 67 41 7 35 71 45)(4 99 68 89 8 103 72 93)(9 84 110 78 13 88 106 74)(10 25 111 20 14 29 107 24)(11 86 112 80 15 82 108 76)(12 27 105 22 16 31 109 18)(17 62 26 49 21 58 30 53)(19 64 28 51 23 60 32 55)(34 117 44 123 38 113 48 127)(36 119 46 125 40 115 42 121)(50 73 59 83 54 77 63 87)(52 75 61 85 56 79 57 81)(90 122 104 120 94 126 100 116)(92 124 98 114 96 128 102 118)
(1 78 126 23)(2 73 127 18)(3 76 128 21)(4 79 121 24)(5 74 122 19)(6 77 123 22)(7 80 124 17)(8 75 125 20)(9 94 64 43)(10 89 57 46)(11 92 58 41)(12 95 59 44)(13 90 60 47)(14 93 61 42)(15 96 62 45)(16 91 63 48)(25 72 81 115)(26 67 82 118)(27 70 83 113)(28 65 84 116)(29 68 85 119)(30 71 86 114)(31 66 87 117)(32 69 88 120)(33 110 104 51)(34 105 97 54)(35 108 98 49)(36 111 99 52)(37 106 100 55)(38 109 101 50)(39 112 102 53)(40 107 103 56)
G:=sub<Sym(128)| (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,121)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,76)(18,77)(19,78)(20,79)(21,80)(22,73)(23,74)(24,75)(25,85)(26,86)(27,87)(28,88)(29,81)(30,82)(31,83)(32,84)(33,100)(34,101)(35,102)(36,103)(37,104)(38,97)(39,98)(40,99)(41,96)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,112)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(65,120)(66,113)(67,114)(68,115)(69,116)(70,117)(71,118)(72,119), (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), (1,37,65,47,5,33,69,43)(2,97,66,95,6,101,70,91)(3,39,67,41,7,35,71,45)(4,99,68,89,8,103,72,93)(9,84,110,78,13,88,106,74)(10,25,111,20,14,29,107,24)(11,86,112,80,15,82,108,76)(12,27,105,22,16,31,109,18)(17,62,26,49,21,58,30,53)(19,64,28,51,23,60,32,55)(34,117,44,123,38,113,48,127)(36,119,46,125,40,115,42,121)(50,73,59,83,54,77,63,87)(52,75,61,85,56,79,57,81)(90,122,104,120,94,126,100,116)(92,124,98,114,96,128,102,118), (1,78,126,23)(2,73,127,18)(3,76,128,21)(4,79,121,24)(5,74,122,19)(6,77,123,22)(7,80,124,17)(8,75,125,20)(9,94,64,43)(10,89,57,46)(11,92,58,41)(12,95,59,44)(13,90,60,47)(14,93,61,42)(15,96,62,45)(16,91,63,48)(25,72,81,115)(26,67,82,118)(27,70,83,113)(28,65,84,116)(29,68,85,119)(30,71,86,114)(31,66,87,117)(32,69,88,120)(33,110,104,51)(34,105,97,54)(35,108,98,49)(36,111,99,52)(37,106,100,55)(38,109,101,50)(39,112,102,53)(40,107,103,56)>;
G:=Group( (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,121)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,76)(18,77)(19,78)(20,79)(21,80)(22,73)(23,74)(24,75)(25,85)(26,86)(27,87)(28,88)(29,81)(30,82)(31,83)(32,84)(33,100)(34,101)(35,102)(36,103)(37,104)(38,97)(39,98)(40,99)(41,96)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,112)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(65,120)(66,113)(67,114)(68,115)(69,116)(70,117)(71,118)(72,119), (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), (1,37,65,47,5,33,69,43)(2,97,66,95,6,101,70,91)(3,39,67,41,7,35,71,45)(4,99,68,89,8,103,72,93)(9,84,110,78,13,88,106,74)(10,25,111,20,14,29,107,24)(11,86,112,80,15,82,108,76)(12,27,105,22,16,31,109,18)(17,62,26,49,21,58,30,53)(19,64,28,51,23,60,32,55)(34,117,44,123,38,113,48,127)(36,119,46,125,40,115,42,121)(50,73,59,83,54,77,63,87)(52,75,61,85,56,79,57,81)(90,122,104,120,94,126,100,116)(92,124,98,114,96,128,102,118), (1,78,126,23)(2,73,127,18)(3,76,128,21)(4,79,121,24)(5,74,122,19)(6,77,123,22)(7,80,124,17)(8,75,125,20)(9,94,64,43)(10,89,57,46)(11,92,58,41)(12,95,59,44)(13,90,60,47)(14,93,61,42)(15,96,62,45)(16,91,63,48)(25,72,81,115)(26,67,82,118)(27,70,83,113)(28,65,84,116)(29,68,85,119)(30,71,86,114)(31,66,87,117)(32,69,88,120)(33,110,104,51)(34,105,97,54)(35,108,98,49)(36,111,99,52)(37,106,100,55)(38,109,101,50)(39,112,102,53)(40,107,103,56) );
G=PermutationGroup([[(1,122),(2,123),(3,124),(4,125),(5,126),(6,127),(7,128),(8,121),(9,60),(10,61),(11,62),(12,63),(13,64),(14,57),(15,58),(16,59),(17,76),(18,77),(19,78),(20,79),(21,80),(22,73),(23,74),(24,75),(25,85),(26,86),(27,87),(28,88),(29,81),(30,82),(31,83),(32,84),(33,100),(34,101),(35,102),(36,103),(37,104),(38,97),(39,98),(40,99),(41,96),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,112),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(65,120),(66,113),(67,114),(68,115),(69,116),(70,117),(71,118),(72,119)], [(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)], [(1,37,65,47,5,33,69,43),(2,97,66,95,6,101,70,91),(3,39,67,41,7,35,71,45),(4,99,68,89,8,103,72,93),(9,84,110,78,13,88,106,74),(10,25,111,20,14,29,107,24),(11,86,112,80,15,82,108,76),(12,27,105,22,16,31,109,18),(17,62,26,49,21,58,30,53),(19,64,28,51,23,60,32,55),(34,117,44,123,38,113,48,127),(36,119,46,125,40,115,42,121),(50,73,59,83,54,77,63,87),(52,75,61,85,56,79,57,81),(90,122,104,120,94,126,100,116),(92,124,98,114,96,128,102,118)], [(1,78,126,23),(2,73,127,18),(3,76,128,21),(4,79,121,24),(5,74,122,19),(6,77,123,22),(7,80,124,17),(8,75,125,20),(9,94,64,43),(10,89,57,46),(11,92,58,41),(12,95,59,44),(13,90,60,47),(14,93,61,42),(15,96,62,45),(16,91,63,48),(25,72,81,115),(26,67,82,118),(27,70,83,113),(28,65,84,116),(29,68,85,119),(30,71,86,114),(31,66,87,117),(32,69,88,120),(33,110,104,51),(34,105,97,54),(35,108,98,49),(36,111,99,52),(37,106,100,55),(38,109,101,50),(39,112,102,53),(40,107,103,56)]])
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 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | SD16 | C4○D4 | C8.C22 |
| kernel | (C2×C8).170D4 | C22.7C42 | C42⋊9C4 | C23.67C23 | C2×Q8⋊C4 | C2×C4.Q8 | C2×C8 | C22×C4 | C2×Q8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 8 | 6 | 2 |
Matrix representation of (C2×C8).170D4 ►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 |
| 3 | 12 | 0 | 0 | 0 | 0 |
| 5 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 3 | 0 | 0 |
| 0 | 0 | 14 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 14 | 12 | 0 | 0 | 0 | 0 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 5 | 10 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 12 | 0 | 0 |
| 0 | 0 | 5 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
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],[3,5,0,0,0,0,12,14,0,0,0,0,0,0,5,14,0,0,0,0,3,12,0,0,0,0,0,0,7,5,0,0,0,0,7,0],[14,5,0,0,0,0,12,3,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,5,0,0,0,0,7,10],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,14,5,0,0,0,0,12,3,0,0,0,0,0,0,13,4,0,0,0,0,0,4] >;
(C2×C8).170D4 in GAP, Magma, Sage, TeX
(C_2\times C_8)._{170}D_4 % in TeX
G:=Group("(C2xC8).170D4"); // GroupNames label
G:=SmallGroup(128,828);
// by ID
G=gap.SmallGroup(128,828);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,504,141,176,422,387,352,1018,521,248,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=a*b^4,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=c^3>;
// generators/relations