p-group, metabelian, nilpotent (class 4), monomial
Aliases: C4.3Q32, C4.4SD32, C8.19SD16, C42.334D4, (C4×C16).6C2, (C2×C4).82D8, C2.9(C2×Q32), (C2×C8).251D4, C8.51(C4○D4), C8⋊2Q8.13C2, C4⋊Q16.9C2, C2.15(C2×SD32), C4.15(C2×SD16), (C2×C16).73C22, (C2×C8).539C23, (C4×C8).400C22, C2.Q32.1C2, C4.8(C4.4D4), C22.125(C2×D8), C2.D8.24C22, C2.10(C4.4D8), (C2×Q16).13C22, (C2×C4).807(C2×D4), SmallGroup(128,973)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.SD32
G = < a,b,c | a4=b16=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b7 >
Subgroups: 168 in 70 conjugacy classes, 36 normal (18 characteristic)
C1, C2, C4, C4, C4, C22, C8, C2×C4, C2×C4, Q8, C16, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C4×C8, C2.D8, C2.D8, C2×C16, C4⋊Q8, C2×Q16, C2×Q16, C4×C16, C2.Q32, C4⋊Q16, C8⋊2Q8, C4.SD32
Quotients: C1, C2, C22, D4, C23, D8, SD16, C2×D4, C4○D4, SD32, Q32, C4.4D4, C2×D8, C2×SD16, C4.4D8, C2×SD32, C2×Q32, C4.SD32
►Regular action on 128 points
Generators in S128
(1 82 53 41)(2 83 54 42)(3 84 55 43)(4 85 56 44)(5 86 57 45)(6 87 58 46)(7 88 59 47)(8 89 60 48)(9 90 61 33)(10 91 62 34)(11 92 63 35)(12 93 64 36)(13 94 49 37)(14 95 50 38)(15 96 51 39)(16 81 52 40)(17 70 114 112)(18 71 115 97)(19 72 116 98)(20 73 117 99)(21 74 118 100)(22 75 119 101)(23 76 120 102)(24 77 121 103)(25 78 122 104)(26 79 123 105)(27 80 124 106)(28 65 125 107)(29 66 126 108)(30 67 127 109)(31 68 128 110)(32 69 113 111)
(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 53 109)(2 100 54 74)(3 65 55 107)(4 98 56 72)(5 79 57 105)(6 112 58 70)(7 77 59 103)(8 110 60 68)(9 75 61 101)(10 108 62 66)(11 73 63 99)(12 106 64 80)(13 71 49 97)(14 104 50 78)(15 69 51 111)(16 102 52 76)(17 87 114 46)(18 37 115 94)(19 85 116 44)(20 35 117 92)(21 83 118 42)(22 33 119 90)(23 81 120 40)(24 47 121 88)(25 95 122 38)(26 45 123 86)(27 93 124 36)(28 43 125 84)(29 91 126 34)(30 41 127 82)(31 89 128 48)(32 39 113 96)
G:=sub<Sym(128)| (1,82,53,41)(2,83,54,42)(3,84,55,43)(4,85,56,44)(5,86,57,45)(6,87,58,46)(7,88,59,47)(8,89,60,48)(9,90,61,33)(10,91,62,34)(11,92,63,35)(12,93,64,36)(13,94,49,37)(14,95,50,38)(15,96,51,39)(16,81,52,40)(17,70,114,112)(18,71,115,97)(19,72,116,98)(20,73,117,99)(21,74,118,100)(22,75,119,101)(23,76,120,102)(24,77,121,103)(25,78,122,104)(26,79,123,105)(27,80,124,106)(28,65,125,107)(29,66,126,108)(30,67,127,109)(31,68,128,110)(32,69,113,111), (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), (1,67,53,109)(2,100,54,74)(3,65,55,107)(4,98,56,72)(5,79,57,105)(6,112,58,70)(7,77,59,103)(8,110,60,68)(9,75,61,101)(10,108,62,66)(11,73,63,99)(12,106,64,80)(13,71,49,97)(14,104,50,78)(15,69,51,111)(16,102,52,76)(17,87,114,46)(18,37,115,94)(19,85,116,44)(20,35,117,92)(21,83,118,42)(22,33,119,90)(23,81,120,40)(24,47,121,88)(25,95,122,38)(26,45,123,86)(27,93,124,36)(28,43,125,84)(29,91,126,34)(30,41,127,82)(31,89,128,48)(32,39,113,96)>;
G:=Group( (1,82,53,41)(2,83,54,42)(3,84,55,43)(4,85,56,44)(5,86,57,45)(6,87,58,46)(7,88,59,47)(8,89,60,48)(9,90,61,33)(10,91,62,34)(11,92,63,35)(12,93,64,36)(13,94,49,37)(14,95,50,38)(15,96,51,39)(16,81,52,40)(17,70,114,112)(18,71,115,97)(19,72,116,98)(20,73,117,99)(21,74,118,100)(22,75,119,101)(23,76,120,102)(24,77,121,103)(25,78,122,104)(26,79,123,105)(27,80,124,106)(28,65,125,107)(29,66,126,108)(30,67,127,109)(31,68,128,110)(32,69,113,111), (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), (1,67,53,109)(2,100,54,74)(3,65,55,107)(4,98,56,72)(5,79,57,105)(6,112,58,70)(7,77,59,103)(8,110,60,68)(9,75,61,101)(10,108,62,66)(11,73,63,99)(12,106,64,80)(13,71,49,97)(14,104,50,78)(15,69,51,111)(16,102,52,76)(17,87,114,46)(18,37,115,94)(19,85,116,44)(20,35,117,92)(21,83,118,42)(22,33,119,90)(23,81,120,40)(24,47,121,88)(25,95,122,38)(26,45,123,86)(27,93,124,36)(28,43,125,84)(29,91,126,34)(30,41,127,82)(31,89,128,48)(32,39,113,96) );
G=PermutationGroup([[(1,82,53,41),(2,83,54,42),(3,84,55,43),(4,85,56,44),(5,86,57,45),(6,87,58,46),(7,88,59,47),(8,89,60,48),(9,90,61,33),(10,91,62,34),(11,92,63,35),(12,93,64,36),(13,94,49,37),(14,95,50,38),(15,96,51,39),(16,81,52,40),(17,70,114,112),(18,71,115,97),(19,72,116,98),(20,73,117,99),(21,74,118,100),(22,75,119,101),(23,76,120,102),(24,77,121,103),(25,78,122,104),(26,79,123,105),(27,80,124,106),(28,65,125,107),(29,66,126,108),(30,67,127,109),(31,68,128,110),(32,69,113,111)], [(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)], [(1,67,53,109),(2,100,54,74),(3,65,55,107),(4,98,56,72),(5,79,57,105),(6,112,58,70),(7,77,59,103),(8,110,60,68),(9,75,61,101),(10,108,62,66),(11,73,63,99),(12,106,64,80),(13,71,49,97),(14,104,50,78),(15,69,51,111),(16,102,52,76),(17,87,114,46),(18,37,115,94),(19,85,116,44),(20,35,117,92),(21,83,118,42),(22,33,119,90),(23,81,120,40),(24,47,121,88),(25,95,122,38),(26,45,123,86),(27,93,124,36),(28,43,125,84),(29,91,126,34),(30,41,127,82),(31,89,128,48),(32,39,113,96)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 16 | 16 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | SD16 | C4○D4 | D8 | SD32 | Q32 |
| kernel | C4.SD32 | C4×C16 | C2.Q32 | C4⋊Q16 | C8⋊2Q8 | C42 | C2×C8 | C8 | C8 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C4.SD32 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 1 | 1 |
| 12 | 5 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 2 | 12 |
| 0 | 0 | 11 | 7 |
| 1 | 10 | 0 | 0 |
| 10 | 16 | 0 | 0 |
| 0 | 0 | 6 | 14 |
| 0 | 0 | 1 | 11 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,16,1,0,0,15,1],[12,12,0,0,5,12,0,0,0,0,2,11,0,0,12,7],[1,10,0,0,10,16,0,0,0,0,6,1,0,0,14,11] >;
C4.SD32 in GAP, Magma, Sage, TeX
C_4.{\rm SD}_{32} % in TeX
G:=Group("C4.SD32"); // GroupNames label
G:=SmallGroup(128,973);
// by ID
G=gap.SmallGroup(128,973);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,120,422,58,1123,360,3924,102,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^16=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^7>;
// generators/relations