p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.385D4, (C2×C8)⋊4C8, C4⋊C8.6C4, C4.2(C4×C8), (C2×C4).156D8, (C2×C4).65Q16, C2.1(C8⋊2C8), C2.1(C8⋊1C8), C4.2(C8⋊C4), (C22×C8).22C4, (C2×C4).50C42, (C22×C4).67Q8, C4.17(C22⋊C8), C22.15(C4⋊C8), C23.78(C4⋊C4), C42.251(C2×C4), (C2×C4).123SD16, (C22×C4).806D4, (C2×C4).66M4(2), C4.41(D4⋊C4), C4.27(Q8⋊C4), C22.15(C2.D8), C22.11(C4.Q8), C2.1(C22.4Q16), C22.9(C8.C4), C2.1(C4.C42), (C2×C42).1025C22, C2.6(C22.7C42), C22.18(C2.C42), (C2×C4×C8).2C2, (C2×C4⋊C8).1C2, (C2×C4).69(C2×C8), (C2×C4).64(C4⋊C4), (C22×C4).463(C2×C4), (C2×C4).369(C22⋊C4), SmallGroup(128,9)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.385D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=a-1bc3 >
Subgroups: 136 in 90 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, C22×C4, C4×C8, C4⋊C8, C4⋊C8, C2×C42, C22×C8, C22×C8, C2×C4×C8, C2×C4⋊C8, C42.385D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), D8, SD16, Q16, C2.C42, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C8.C4, C8⋊2C8, C8⋊1C8, C22.7C42, C22.4Q16, C4.C42, C42.385D4
►Regular action on 128 points
Generators in S128
(1 41 5 45)(2 42 6 46)(3 43 7 47)(4 44 8 48)(9 94 13 90)(10 95 14 91)(11 96 15 92)(12 89 16 93)(17 108 21 112)(18 109 22 105)(19 110 23 106)(20 111 24 107)(25 125 29 121)(26 126 30 122)(27 127 31 123)(28 128 32 124)(33 61 37 57)(34 62 38 58)(35 63 39 59)(36 64 40 60)(49 80 53 76)(50 73 54 77)(51 74 55 78)(52 75 56 79)(65 85 69 81)(66 86 70 82)(67 87 71 83)(68 88 72 84)(97 119 101 115)(98 120 102 116)(99 113 103 117)(100 114 104 118)
(1 117 125 80)(2 73 126 118)(3 119 127 74)(4 75 128 120)(5 113 121 76)(6 77 122 114)(7 115 123 78)(8 79 124 116)(9 35 86 111)(10 112 87 36)(11 37 88 105)(12 106 81 38)(13 39 82 107)(14 108 83 40)(15 33 84 109)(16 110 85 34)(17 71 64 95)(18 96 57 72)(19 65 58 89)(20 90 59 66)(21 67 60 91)(22 92 61 68)(23 69 62 93)(24 94 63 70)(25 49 45 103)(26 104 46 50)(27 51 47 97)(28 98 48 52)(29 53 41 99)(30 100 42 54)(31 55 43 101)(32 102 44 56)
(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 41 57 5 33 45 61)(2 71 42 83 6 67 46 87)(3 39 43 59 7 35 47 63)(4 65 44 85 8 69 48 81)(9 97 94 119 13 101 90 115)(10 126 95 30 14 122 91 26)(11 99 96 113 15 103 92 117)(12 128 89 32 16 124 93 28)(17 54 108 77 21 50 112 73)(18 121 109 25 22 125 105 29)(19 56 110 79 23 52 106 75)(20 123 111 27 24 127 107 31)(34 116 62 98 38 120 58 102)(36 118 64 100 40 114 60 104)(49 68 80 88 53 72 76 84)(51 70 74 82 55 66 78 86)
G:=sub<Sym(128)| (1,41,5,45)(2,42,6,46)(3,43,7,47)(4,44,8,48)(9,94,13,90)(10,95,14,91)(11,96,15,92)(12,89,16,93)(17,108,21,112)(18,109,22,105)(19,110,23,106)(20,111,24,107)(25,125,29,121)(26,126,30,122)(27,127,31,123)(28,128,32,124)(33,61,37,57)(34,62,38,58)(35,63,39,59)(36,64,40,60)(49,80,53,76)(50,73,54,77)(51,74,55,78)(52,75,56,79)(65,85,69,81)(66,86,70,82)(67,87,71,83)(68,88,72,84)(97,119,101,115)(98,120,102,116)(99,113,103,117)(100,114,104,118), (1,117,125,80)(2,73,126,118)(3,119,127,74)(4,75,128,120)(5,113,121,76)(6,77,122,114)(7,115,123,78)(8,79,124,116)(9,35,86,111)(10,112,87,36)(11,37,88,105)(12,106,81,38)(13,39,82,107)(14,108,83,40)(15,33,84,109)(16,110,85,34)(17,71,64,95)(18,96,57,72)(19,65,58,89)(20,90,59,66)(21,67,60,91)(22,92,61,68)(23,69,62,93)(24,94,63,70)(25,49,45,103)(26,104,46,50)(27,51,47,97)(28,98,48,52)(29,53,41,99)(30,100,42,54)(31,55,43,101)(32,102,44,56), (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), (1,37,41,57,5,33,45,61)(2,71,42,83,6,67,46,87)(3,39,43,59,7,35,47,63)(4,65,44,85,8,69,48,81)(9,97,94,119,13,101,90,115)(10,126,95,30,14,122,91,26)(11,99,96,113,15,103,92,117)(12,128,89,32,16,124,93,28)(17,54,108,77,21,50,112,73)(18,121,109,25,22,125,105,29)(19,56,110,79,23,52,106,75)(20,123,111,27,24,127,107,31)(34,116,62,98,38,120,58,102)(36,118,64,100,40,114,60,104)(49,68,80,88,53,72,76,84)(51,70,74,82,55,66,78,86)>;
G:=Group( (1,41,5,45)(2,42,6,46)(3,43,7,47)(4,44,8,48)(9,94,13,90)(10,95,14,91)(11,96,15,92)(12,89,16,93)(17,108,21,112)(18,109,22,105)(19,110,23,106)(20,111,24,107)(25,125,29,121)(26,126,30,122)(27,127,31,123)(28,128,32,124)(33,61,37,57)(34,62,38,58)(35,63,39,59)(36,64,40,60)(49,80,53,76)(50,73,54,77)(51,74,55,78)(52,75,56,79)(65,85,69,81)(66,86,70,82)(67,87,71,83)(68,88,72,84)(97,119,101,115)(98,120,102,116)(99,113,103,117)(100,114,104,118), (1,117,125,80)(2,73,126,118)(3,119,127,74)(4,75,128,120)(5,113,121,76)(6,77,122,114)(7,115,123,78)(8,79,124,116)(9,35,86,111)(10,112,87,36)(11,37,88,105)(12,106,81,38)(13,39,82,107)(14,108,83,40)(15,33,84,109)(16,110,85,34)(17,71,64,95)(18,96,57,72)(19,65,58,89)(20,90,59,66)(21,67,60,91)(22,92,61,68)(23,69,62,93)(24,94,63,70)(25,49,45,103)(26,104,46,50)(27,51,47,97)(28,98,48,52)(29,53,41,99)(30,100,42,54)(31,55,43,101)(32,102,44,56), (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), (1,37,41,57,5,33,45,61)(2,71,42,83,6,67,46,87)(3,39,43,59,7,35,47,63)(4,65,44,85,8,69,48,81)(9,97,94,119,13,101,90,115)(10,126,95,30,14,122,91,26)(11,99,96,113,15,103,92,117)(12,128,89,32,16,124,93,28)(17,54,108,77,21,50,112,73)(18,121,109,25,22,125,105,29)(19,56,110,79,23,52,106,75)(20,123,111,27,24,127,107,31)(34,116,62,98,38,120,58,102)(36,118,64,100,40,114,60,104)(49,68,80,88,53,72,76,84)(51,70,74,82,55,66,78,86) );
G=PermutationGroup([[(1,41,5,45),(2,42,6,46),(3,43,7,47),(4,44,8,48),(9,94,13,90),(10,95,14,91),(11,96,15,92),(12,89,16,93),(17,108,21,112),(18,109,22,105),(19,110,23,106),(20,111,24,107),(25,125,29,121),(26,126,30,122),(27,127,31,123),(28,128,32,124),(33,61,37,57),(34,62,38,58),(35,63,39,59),(36,64,40,60),(49,80,53,76),(50,73,54,77),(51,74,55,78),(52,75,56,79),(65,85,69,81),(66,86,70,82),(67,87,71,83),(68,88,72,84),(97,119,101,115),(98,120,102,116),(99,113,103,117),(100,114,104,118)], [(1,117,125,80),(2,73,126,118),(3,119,127,74),(4,75,128,120),(5,113,121,76),(6,77,122,114),(7,115,123,78),(8,79,124,116),(9,35,86,111),(10,112,87,36),(11,37,88,105),(12,106,81,38),(13,39,82,107),(14,108,83,40),(15,33,84,109),(16,110,85,34),(17,71,64,95),(18,96,57,72),(19,65,58,89),(20,90,59,66),(21,67,60,91),(22,92,61,68),(23,69,62,93),(24,94,63,70),(25,49,45,103),(26,104,46,50),(27,51,47,97),(28,98,48,52),(29,53,41,99),(30,100,42,54),(31,55,43,101),(32,102,44,56)], [(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)], [(1,37,41,57,5,33,45,61),(2,71,42,83,6,67,46,87),(3,39,43,59,7,35,47,63),(4,65,44,85,8,69,48,81),(9,97,94,119,13,101,90,115),(10,126,95,30,14,122,91,26),(11,99,96,113,15,103,92,117),(12,128,89,32,16,124,93,28),(17,54,108,77,21,50,112,73),(18,121,109,25,22,125,105,29),(19,56,110,79,23,52,106,75),(20,123,111,27,24,127,107,31),(34,116,62,98,38,120,58,102),(36,118,64,100,40,114,60,104),(49,68,80,88,53,72,76,84),(51,70,74,82,55,66,78,86)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 8A | ··· | 8P | 8Q | ··· | 8AF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | D4 | Q8 | M4(2) | D8 | SD16 | Q16 | C8.C4 |
| kernel | C42.385D4 | C2×C4×C8 | C2×C4⋊C8 | C4⋊C8 | C22×C8 | C2×C8 | C42 | C22×C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 8 | 4 | 16 | 2 | 1 | 1 | 4 | 2 | 4 | 2 | 8 |
Matrix representation of C42.385D4 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 3 | 15 |
| 0 | 0 | 15 | 14 |
| 9 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 2 | 14 |
| 0 | 0 | 14 | 15 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[9,0,0,0,0,4,0,0,0,0,3,15,0,0,15,14],[9,0,0,0,0,16,0,0,0,0,2,14,0,0,14,15] >;
C42.385D4 in GAP, Magma, Sage, TeX
C_4^2._{385}D_4 % in TeX
G:=Group("C4^2.385D4"); // GroupNames label
G:=SmallGroup(128,9);
// by ID
G=gap.SmallGroup(128,9);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,184,248,1684,102]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^-1*b*c^3>;
// generators/relations