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 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×2], C22 [×3], C22 [×4], C8 [×8], C2×C4 [×4], C2×C4 [×10], C2×C4 [×2], C23, C42 [×4], C2×C8 [×4], C2×C8 [×12], C22×C4 [×3], C4×C8 [×2], C4⋊C8 [×4], C4⋊C8 [×2], C2×C42, C22×C8 [×2], C22×C8 [×2], C2×C4×C8, C2×C4⋊C8 [×2], C42.385D4
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], D8, SD16 [×2], Q16, C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8 [×2], C4.Q8, C2.D8, C8.C4 [×2], C8⋊2C8 [×2], C8⋊1C8 [×2], C22.7C42, C22.4Q16, C4.C42, C42.385D4
►Regular action on 128 points
Generators in S128
(1 49 5 53)(2 50 6 54)(3 51 7 55)(4 52 8 56)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)(25 115 29 119)(26 116 30 120)(27 117 31 113)(28 118 32 114)(33 59 37 63)(34 60 38 64)(35 61 39 57)(36 62 40 58)(41 74 45 78)(42 75 46 79)(43 76 47 80)(44 77 48 73)(65 85 69 81)(66 86 70 82)(67 87 71 83)(68 88 72 84)(89 112 93 108)(90 105 94 109)(91 106 95 110)(92 107 96 111)(97 122 101 126)(98 123 102 127)(99 124 103 128)(100 125 104 121)
(1 117 125 77)(2 78 126 118)(3 119 127 79)(4 80 128 120)(5 113 121 73)(6 74 122 114)(7 115 123 75)(8 76 124 116)(9 34 66 106)(10 107 67 35)(11 36 68 108)(12 109 69 37)(13 38 70 110)(14 111 71 39)(15 40 72 112)(16 105 65 33)(17 60 86 95)(18 96 87 61)(19 62 88 89)(20 90 81 63)(21 64 82 91)(22 92 83 57)(23 58 84 93)(24 94 85 59)(25 98 42 51)(26 52 43 99)(27 100 44 53)(28 54 45 101)(29 102 46 55)(30 56 47 103)(31 104 48 49)(32 50 41 97)
(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 67 49 87 5 71 53 83)(2 60 50 38 6 64 54 34)(3 69 51 81 7 65 55 85)(4 62 52 40 8 58 56 36)(9 78 17 41 13 74 21 45)(10 104 18 121 14 100 22 125)(11 80 19 43 15 76 23 47)(12 98 20 123 16 102 24 127)(25 90 115 105 29 94 119 109)(26 72 116 84 30 68 120 88)(27 92 117 107 31 96 113 111)(28 66 118 86 32 70 114 82)(33 46 59 79 37 42 63 75)(35 48 61 73 39 44 57 77)(89 99 112 124 93 103 108 128)(91 101 106 126 95 97 110 122)
G:=sub<Sym(128)| (1,49,5,53)(2,50,6,54)(3,51,7,55)(4,52,8,56)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)(25,115,29,119)(26,116,30,120)(27,117,31,113)(28,118,32,114)(33,59,37,63)(34,60,38,64)(35,61,39,57)(36,62,40,58)(41,74,45,78)(42,75,46,79)(43,76,47,80)(44,77,48,73)(65,85,69,81)(66,86,70,82)(67,87,71,83)(68,88,72,84)(89,112,93,108)(90,105,94,109)(91,106,95,110)(92,107,96,111)(97,122,101,126)(98,123,102,127)(99,124,103,128)(100,125,104,121), (1,117,125,77)(2,78,126,118)(3,119,127,79)(4,80,128,120)(5,113,121,73)(6,74,122,114)(7,115,123,75)(8,76,124,116)(9,34,66,106)(10,107,67,35)(11,36,68,108)(12,109,69,37)(13,38,70,110)(14,111,71,39)(15,40,72,112)(16,105,65,33)(17,60,86,95)(18,96,87,61)(19,62,88,89)(20,90,81,63)(21,64,82,91)(22,92,83,57)(23,58,84,93)(24,94,85,59)(25,98,42,51)(26,52,43,99)(27,100,44,53)(28,54,45,101)(29,102,46,55)(30,56,47,103)(31,104,48,49)(32,50,41,97), (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), (1,67,49,87,5,71,53,83)(2,60,50,38,6,64,54,34)(3,69,51,81,7,65,55,85)(4,62,52,40,8,58,56,36)(9,78,17,41,13,74,21,45)(10,104,18,121,14,100,22,125)(11,80,19,43,15,76,23,47)(12,98,20,123,16,102,24,127)(25,90,115,105,29,94,119,109)(26,72,116,84,30,68,120,88)(27,92,117,107,31,96,113,111)(28,66,118,86,32,70,114,82)(33,46,59,79,37,42,63,75)(35,48,61,73,39,44,57,77)(89,99,112,124,93,103,108,128)(91,101,106,126,95,97,110,122)>;
G:=Group( (1,49,5,53)(2,50,6,54)(3,51,7,55)(4,52,8,56)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)(25,115,29,119)(26,116,30,120)(27,117,31,113)(28,118,32,114)(33,59,37,63)(34,60,38,64)(35,61,39,57)(36,62,40,58)(41,74,45,78)(42,75,46,79)(43,76,47,80)(44,77,48,73)(65,85,69,81)(66,86,70,82)(67,87,71,83)(68,88,72,84)(89,112,93,108)(90,105,94,109)(91,106,95,110)(92,107,96,111)(97,122,101,126)(98,123,102,127)(99,124,103,128)(100,125,104,121), (1,117,125,77)(2,78,126,118)(3,119,127,79)(4,80,128,120)(5,113,121,73)(6,74,122,114)(7,115,123,75)(8,76,124,116)(9,34,66,106)(10,107,67,35)(11,36,68,108)(12,109,69,37)(13,38,70,110)(14,111,71,39)(15,40,72,112)(16,105,65,33)(17,60,86,95)(18,96,87,61)(19,62,88,89)(20,90,81,63)(21,64,82,91)(22,92,83,57)(23,58,84,93)(24,94,85,59)(25,98,42,51)(26,52,43,99)(27,100,44,53)(28,54,45,101)(29,102,46,55)(30,56,47,103)(31,104,48,49)(32,50,41,97), (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), (1,67,49,87,5,71,53,83)(2,60,50,38,6,64,54,34)(3,69,51,81,7,65,55,85)(4,62,52,40,8,58,56,36)(9,78,17,41,13,74,21,45)(10,104,18,121,14,100,22,125)(11,80,19,43,15,76,23,47)(12,98,20,123,16,102,24,127)(25,90,115,105,29,94,119,109)(26,72,116,84,30,68,120,88)(27,92,117,107,31,96,113,111)(28,66,118,86,32,70,114,82)(33,46,59,79,37,42,63,75)(35,48,61,73,39,44,57,77)(89,99,112,124,93,103,108,128)(91,101,106,126,95,97,110,122) );
G=PermutationGroup([(1,49,5,53),(2,50,6,54),(3,51,7,55),(4,52,8,56),(9,17,13,21),(10,18,14,22),(11,19,15,23),(12,20,16,24),(25,115,29,119),(26,116,30,120),(27,117,31,113),(28,118,32,114),(33,59,37,63),(34,60,38,64),(35,61,39,57),(36,62,40,58),(41,74,45,78),(42,75,46,79),(43,76,47,80),(44,77,48,73),(65,85,69,81),(66,86,70,82),(67,87,71,83),(68,88,72,84),(89,112,93,108),(90,105,94,109),(91,106,95,110),(92,107,96,111),(97,122,101,126),(98,123,102,127),(99,124,103,128),(100,125,104,121)], [(1,117,125,77),(2,78,126,118),(3,119,127,79),(4,80,128,120),(5,113,121,73),(6,74,122,114),(7,115,123,75),(8,76,124,116),(9,34,66,106),(10,107,67,35),(11,36,68,108),(12,109,69,37),(13,38,70,110),(14,111,71,39),(15,40,72,112),(16,105,65,33),(17,60,86,95),(18,96,87,61),(19,62,88,89),(20,90,81,63),(21,64,82,91),(22,92,83,57),(23,58,84,93),(24,94,85,59),(25,98,42,51),(26,52,43,99),(27,100,44,53),(28,54,45,101),(29,102,46,55),(30,56,47,103),(31,104,48,49),(32,50,41,97)], [(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)], [(1,67,49,87,5,71,53,83),(2,60,50,38,6,64,54,34),(3,69,51,81,7,65,55,85),(4,62,52,40,8,58,56,36),(9,78,17,41,13,74,21,45),(10,104,18,121,14,100,22,125),(11,80,19,43,15,76,23,47),(12,98,20,123,16,102,24,127),(25,90,115,105,29,94,119,109),(26,72,116,84,30,68,120,88),(27,92,117,107,31,96,113,111),(28,66,118,86,32,70,114,82),(33,46,59,79,37,42,63,75),(35,48,61,73,39,44,57,77),(89,99,112,124,93,103,108,128),(91,101,106,126,95,97,110,122)])
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