p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8).1Q8, (C2×C4).25D8, C2.9(C8⋊Q8), C4⋊C4.110D4, (C2×C4).20Q16, C2.5(C8⋊2Q8), C22.90(C2×D8), C2.17(C4⋊D8), (C22×C4).319D4, C23.933(C2×D4), C4.11(C22⋊Q8), C22.51(C4⋊Q8), C22.62(C2×Q16), C42⋊9C4.16C2, C2.17(C4⋊2Q16), C4.19(C42.C2), C22.4Q16.40C2, (C22×C8).118C22, (C2×C42).384C22, C2.9(C22.D8), C22.256(C4⋊D4), C22.153(C8⋊C22), (C22×C4).1467C23, C2.9(C23.48D4), C22.142(C8.C22), C22.7C42.25C2, C23.65C23.20C2, C2.5(C23.81C23), C22.123(C22.D4), (C2×C4).220(C2×Q8), (C2×C2.D8).15C2, (C2×C4).1063(C2×D4), (C2×C4).886(C4○D4), (C2×C4⋊C4).154C22, SmallGroup(128,815)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8).1Q8
G = < a,b,c,d | a2=b8=c4=1, d2=ac2, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=ac-1 >
Subgroups: 232 in 115 conjugacy classes, 54 normal (28 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.7C42, C22.4Q16, C42⋊9C4, C23.65C23, C2×C2.D8, (C2×C8).1Q8
Quotients: C1, C2, C22, D4, Q8, C23, D8, Q16, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C2×D8, C2×Q16, C8⋊C22, C8.C22, C23.81C23, C4⋊D8, C4⋊2Q16, C22.D8, C23.48D4, C8⋊2Q8, C8⋊Q8, (C2×C8).1Q8
►Regular action on 128 points
Generators in S128
(1 126)(2 127)(3 128)(4 121)(5 122)(6 123)(7 124)(8 125)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(57 111)(58 112)(59 105)(60 106)(61 107)(62 108)(63 109)(64 110)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 113)(72 114)(73 100)(74 101)(75 102)(76 103)(77 104)(78 97)(79 98)(80 99)(81 89)(82 90)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)
(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 68 9 106)(2 117 10 59)(3 66 11 112)(4 115 12 57)(5 72 13 110)(6 113 14 63)(7 70 15 108)(8 119 16 61)(17 85 78 28)(18 92 79 37)(19 83 80 26)(20 90 73 35)(21 81 74 32)(22 96 75 33)(23 87 76 30)(24 94 77 39)(25 48 82 100)(27 46 84 98)(29 44 86 104)(31 42 88 102)(34 41 89 101)(36 47 91 99)(38 45 93 97)(40 43 95 103)(49 58 128 116)(50 111 121 65)(51 64 122 114)(52 109 123 71)(53 62 124 120)(54 107 125 69)(55 60 126 118)(56 105 127 67)
(1 104 55 24)(2 103 56 23)(3 102 49 22)(4 101 50 21)(5 100 51 20)(6 99 52 19)(7 98 53 18)(8 97 54 17)(9 44 126 77)(10 43 127 76)(11 42 128 75)(12 41 121 74)(13 48 122 73)(14 47 123 80)(15 46 124 79)(16 45 125 78)(25 72 90 64)(26 71 91 63)(27 70 92 62)(28 69 93 61)(29 68 94 60)(30 67 95 59)(31 66 96 58)(32 65 89 57)(33 116 88 112)(34 115 81 111)(35 114 82 110)(36 113 83 109)(37 120 84 108)(38 119 85 107)(39 118 86 106)(40 117 87 105)
G:=sub<Sym(128)| (1,126)(2,127)(3,128)(4,121)(5,122)(6,123)(7,124)(8,125)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,45)(18,46)(19,47)(20,48)(21,41)(22,42)(23,43)(24,44)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34)(57,111)(58,112)(59,105)(60,106)(61,107)(62,108)(63,109)(64,110)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,113)(72,114)(73,100)(74,101)(75,102)(76,103)(77,104)(78,97)(79,98)(80,99)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96), (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), (1,68,9,106)(2,117,10,59)(3,66,11,112)(4,115,12,57)(5,72,13,110)(6,113,14,63)(7,70,15,108)(8,119,16,61)(17,85,78,28)(18,92,79,37)(19,83,80,26)(20,90,73,35)(21,81,74,32)(22,96,75,33)(23,87,76,30)(24,94,77,39)(25,48,82,100)(27,46,84,98)(29,44,86,104)(31,42,88,102)(34,41,89,101)(36,47,91,99)(38,45,93,97)(40,43,95,103)(49,58,128,116)(50,111,121,65)(51,64,122,114)(52,109,123,71)(53,62,124,120)(54,107,125,69)(55,60,126,118)(56,105,127,67), (1,104,55,24)(2,103,56,23)(3,102,49,22)(4,101,50,21)(5,100,51,20)(6,99,52,19)(7,98,53,18)(8,97,54,17)(9,44,126,77)(10,43,127,76)(11,42,128,75)(12,41,121,74)(13,48,122,73)(14,47,123,80)(15,46,124,79)(16,45,125,78)(25,72,90,64)(26,71,91,63)(27,70,92,62)(28,69,93,61)(29,68,94,60)(30,67,95,59)(31,66,96,58)(32,65,89,57)(33,116,88,112)(34,115,81,111)(35,114,82,110)(36,113,83,109)(37,120,84,108)(38,119,85,107)(39,118,86,106)(40,117,87,105)>;
G:=Group( (1,126)(2,127)(3,128)(4,121)(5,122)(6,123)(7,124)(8,125)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,45)(18,46)(19,47)(20,48)(21,41)(22,42)(23,43)(24,44)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34)(57,111)(58,112)(59,105)(60,106)(61,107)(62,108)(63,109)(64,110)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,113)(72,114)(73,100)(74,101)(75,102)(76,103)(77,104)(78,97)(79,98)(80,99)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96), (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), (1,68,9,106)(2,117,10,59)(3,66,11,112)(4,115,12,57)(5,72,13,110)(6,113,14,63)(7,70,15,108)(8,119,16,61)(17,85,78,28)(18,92,79,37)(19,83,80,26)(20,90,73,35)(21,81,74,32)(22,96,75,33)(23,87,76,30)(24,94,77,39)(25,48,82,100)(27,46,84,98)(29,44,86,104)(31,42,88,102)(34,41,89,101)(36,47,91,99)(38,45,93,97)(40,43,95,103)(49,58,128,116)(50,111,121,65)(51,64,122,114)(52,109,123,71)(53,62,124,120)(54,107,125,69)(55,60,126,118)(56,105,127,67), (1,104,55,24)(2,103,56,23)(3,102,49,22)(4,101,50,21)(5,100,51,20)(6,99,52,19)(7,98,53,18)(8,97,54,17)(9,44,126,77)(10,43,127,76)(11,42,128,75)(12,41,121,74)(13,48,122,73)(14,47,123,80)(15,46,124,79)(16,45,125,78)(25,72,90,64)(26,71,91,63)(27,70,92,62)(28,69,93,61)(29,68,94,60)(30,67,95,59)(31,66,96,58)(32,65,89,57)(33,116,88,112)(34,115,81,111)(35,114,82,110)(36,113,83,109)(37,120,84,108)(38,119,85,107)(39,118,86,106)(40,117,87,105) );
G=PermutationGroup([[(1,126),(2,127),(3,128),(4,121),(5,122),(6,123),(7,124),(8,125),(9,55),(10,56),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,45),(18,46),(19,47),(20,48),(21,41),(22,42),(23,43),(24,44),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(31,33),(32,34),(57,111),(58,112),(59,105),(60,106),(61,107),(62,108),(63,109),(64,110),(65,115),(66,116),(67,117),(68,118),(69,119),(70,120),(71,113),(72,114),(73,100),(74,101),(75,102),(76,103),(77,104),(78,97),(79,98),(80,99),(81,89),(82,90),(83,91),(84,92),(85,93),(86,94),(87,95),(88,96)], [(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)], [(1,68,9,106),(2,117,10,59),(3,66,11,112),(4,115,12,57),(5,72,13,110),(6,113,14,63),(7,70,15,108),(8,119,16,61),(17,85,78,28),(18,92,79,37),(19,83,80,26),(20,90,73,35),(21,81,74,32),(22,96,75,33),(23,87,76,30),(24,94,77,39),(25,48,82,100),(27,46,84,98),(29,44,86,104),(31,42,88,102),(34,41,89,101),(36,47,91,99),(38,45,93,97),(40,43,95,103),(49,58,128,116),(50,111,121,65),(51,64,122,114),(52,109,123,71),(53,62,124,120),(54,107,125,69),(55,60,126,118),(56,105,127,67)], [(1,104,55,24),(2,103,56,23),(3,102,49,22),(4,101,50,21),(5,100,51,20),(6,99,52,19),(7,98,53,18),(8,97,54,17),(9,44,126,77),(10,43,127,76),(11,42,128,75),(12,41,121,74),(13,48,122,73),(14,47,123,80),(15,46,124,79),(16,45,125,78),(25,72,90,64),(26,71,91,63),(27,70,92,62),(28,69,93,61),(29,68,94,60),(30,67,95,59),(31,66,96,58),(32,65,89,57),(33,116,88,112),(34,115,81,111),(35,114,82,110),(36,113,83,109),(37,120,84,108),(38,119,85,107),(39,118,86,106),(40,117,87,105)]])
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 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | D8 | Q16 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | (C2×C8).1Q8 | C22.7C42 | C22.4Q16 | C42⋊9C4 | C23.65C23 | C2×C2.D8 | C4⋊C4 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 6 | 1 | 1 |
Matrix representation of (C2×C8).1Q8 ►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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 10 | 9 | 0 | 0 | 0 | 0 |
| 6 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 6 | 12 | 0 | 0 | 0 | 0 |
| 7 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 6 | 2 | 0 | 0 | 0 | 0 |
| 7 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[10,6,0,0,0,0,9,7,0,0,0,0,0,0,15,0,0,0,0,0,0,8,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[6,7,0,0,0,0,12,11,0,0,0,0,0,0,0,9,0,0,0,0,15,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[6,7,0,0,0,0,2,11,0,0,0,0,0,0,0,15,0,0,0,0,9,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;
(C2×C8).1Q8 in GAP, Magma, Sage, TeX
(C_2\times C_8)._1Q_8
% in TeX
G:=Group("(C2xC8).1Q8"); // GroupNames label
G:=SmallGroup(128,815);
// by ID
G=gap.SmallGroup(128,815);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,64,422,387,58,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=a*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=b^-1,d*c*d^-1=a*c^-1>;
// generators/relations