p-group, metabelian, nilpotent (class 4), monomial
Aliases: C8.7C42, C22.7D16, C23.55D8, C22.4Q32, C22.9SD32, C2.D8⋊3C4, (C2×C16)⋊11C4, C8.12(C4⋊C4), (C2×C8).41Q8, (C2×C8).337D4, (C2×C4).31Q16, C4.8(C4.Q8), (C22×C16).3C2, (C2×C4).64SD16, C2.2(C16⋊3C4), C2.2(C16⋊4C4), C2.2(C2.D16), C8.33(C22⋊C4), (C22×C4).569D4, C4.13(Q8⋊C4), C2.2(C2.Q32), C22.18(C2.D8), C4.1(C2.C42), (C22×C8).521C22, C22.44(D4⋊C4), C2.10(C22.4Q16), (C2×C2.D8).1C2, (C2×C8).167(C2×C4), (C2×C4).108(C4⋊C4), (C2×C4).227(C22⋊C4), SmallGroup(128,112)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.7C42
G = < a,b,c | a8=b4=1, c4=a6, bab-1=a-1, ac=ca, cbc-1=a-1b >
Subgroups: 168 in 76 conjugacy classes, 48 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.D8, C2.D8, C2×C16, C2×C16, C2×C4⋊C4, C22×C8, C2×C2.D8, C22×C16, C8.7C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, D16, SD32, Q32, C22.4Q16, C2.D16, C2.Q32, C16⋊3C4, C16⋊4C4, C8.7C42
►Regular action on 128 points
Generators in S128
(1 86 13 82 9 94 5 90)(2 87 14 83 10 95 6 91)(3 88 15 84 11 96 7 92)(4 89 16 85 12 81 8 93)(17 73 29 69 25 65 21 77)(18 74 30 70 26 66 22 78)(19 75 31 71 27 67 23 79)(20 76 32 72 28 68 24 80)(33 53 45 49 41 61 37 57)(34 54 46 50 42 62 38 58)(35 55 47 51 43 63 39 59)(36 56 48 52 44 64 40 60)(97 122 109 118 105 114 101 126)(98 123 110 119 106 115 102 127)(99 124 111 120 107 116 103 128)(100 125 112 121 108 117 104 113)
(1 101 70 59)(2 127 71 36)(3 99 72 57)(4 125 73 34)(5 97 74 55)(6 123 75 48)(7 111 76 53)(8 121 77 46)(9 109 78 51)(10 119 79 44)(11 107 80 49)(12 117 65 42)(13 105 66 63)(14 115 67 40)(15 103 68 61)(16 113 69 38)(17 54 93 112)(18 47 94 122)(19 52 95 110)(20 45 96 120)(21 50 81 108)(22 43 82 118)(23 64 83 106)(24 41 84 116)(25 62 85 104)(26 39 86 114)(27 60 87 102)(28 37 88 128)(29 58 89 100)(30 35 90 126)(31 56 91 98)(32 33 92 124)
(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)
G:=sub<Sym(128)| (1,86,13,82,9,94,5,90)(2,87,14,83,10,95,6,91)(3,88,15,84,11,96,7,92)(4,89,16,85,12,81,8,93)(17,73,29,69,25,65,21,77)(18,74,30,70,26,66,22,78)(19,75,31,71,27,67,23,79)(20,76,32,72,28,68,24,80)(33,53,45,49,41,61,37,57)(34,54,46,50,42,62,38,58)(35,55,47,51,43,63,39,59)(36,56,48,52,44,64,40,60)(97,122,109,118,105,114,101,126)(98,123,110,119,106,115,102,127)(99,124,111,120,107,116,103,128)(100,125,112,121,108,117,104,113), (1,101,70,59)(2,127,71,36)(3,99,72,57)(4,125,73,34)(5,97,74,55)(6,123,75,48)(7,111,76,53)(8,121,77,46)(9,109,78,51)(10,119,79,44)(11,107,80,49)(12,117,65,42)(13,105,66,63)(14,115,67,40)(15,103,68,61)(16,113,69,38)(17,54,93,112)(18,47,94,122)(19,52,95,110)(20,45,96,120)(21,50,81,108)(22,43,82,118)(23,64,83,106)(24,41,84,116)(25,62,85,104)(26,39,86,114)(27,60,87,102)(28,37,88,128)(29,58,89,100)(30,35,90,126)(31,56,91,98)(32,33,92,124), (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)>;
G:=Group( (1,86,13,82,9,94,5,90)(2,87,14,83,10,95,6,91)(3,88,15,84,11,96,7,92)(4,89,16,85,12,81,8,93)(17,73,29,69,25,65,21,77)(18,74,30,70,26,66,22,78)(19,75,31,71,27,67,23,79)(20,76,32,72,28,68,24,80)(33,53,45,49,41,61,37,57)(34,54,46,50,42,62,38,58)(35,55,47,51,43,63,39,59)(36,56,48,52,44,64,40,60)(97,122,109,118,105,114,101,126)(98,123,110,119,106,115,102,127)(99,124,111,120,107,116,103,128)(100,125,112,121,108,117,104,113), (1,101,70,59)(2,127,71,36)(3,99,72,57)(4,125,73,34)(5,97,74,55)(6,123,75,48)(7,111,76,53)(8,121,77,46)(9,109,78,51)(10,119,79,44)(11,107,80,49)(12,117,65,42)(13,105,66,63)(14,115,67,40)(15,103,68,61)(16,113,69,38)(17,54,93,112)(18,47,94,122)(19,52,95,110)(20,45,96,120)(21,50,81,108)(22,43,82,118)(23,64,83,106)(24,41,84,116)(25,62,85,104)(26,39,86,114)(27,60,87,102)(28,37,88,128)(29,58,89,100)(30,35,90,126)(31,56,91,98)(32,33,92,124), (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) );
G=PermutationGroup([[(1,86,13,82,9,94,5,90),(2,87,14,83,10,95,6,91),(3,88,15,84,11,96,7,92),(4,89,16,85,12,81,8,93),(17,73,29,69,25,65,21,77),(18,74,30,70,26,66,22,78),(19,75,31,71,27,67,23,79),(20,76,32,72,28,68,24,80),(33,53,45,49,41,61,37,57),(34,54,46,50,42,62,38,58),(35,55,47,51,43,63,39,59),(36,56,48,52,44,64,40,60),(97,122,109,118,105,114,101,126),(98,123,110,119,106,115,102,127),(99,124,111,120,107,116,103,128),(100,125,112,121,108,117,104,113)], [(1,101,70,59),(2,127,71,36),(3,99,72,57),(4,125,73,34),(5,97,74,55),(6,123,75,48),(7,111,76,53),(8,121,77,46),(9,109,78,51),(10,119,79,44),(11,107,80,49),(12,117,65,42),(13,105,66,63),(14,115,67,40),(15,103,68,61),(16,113,69,38),(17,54,93,112),(18,47,94,122),(19,52,95,110),(20,45,96,120),(21,50,81,108),(22,43,82,118),(23,64,83,106),(24,41,84,116),(25,62,85,104),(26,39,86,114),(27,60,87,102),(28,37,88,128),(29,58,89,100),(30,35,90,126),(31,56,91,98),(32,33,92,124)], [(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)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | SD16 | Q16 | D8 | D16 | SD32 | Q32 |
| kernel | C8.7C42 | C2×C2.D8 | C22×C16 | C2.D8 | C2×C16 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 8 | 4 | 2 | 1 | 1 | 4 | 2 | 2 | 4 | 8 | 4 |
Matrix representation of C8.7C42 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 3 | 11 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 14 | 0 | 0 |
| 0 | 12 | 16 | 0 | 0 |
| 0 | 0 | 0 | 8 | 15 |
| 0 | 0 | 0 | 7 | 9 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 11 | 4 | 0 | 0 |
| 0 | 12 | 6 | 0 | 0 |
| 0 | 0 | 0 | 7 | 12 |
| 0 | 0 | 0 | 11 | 2 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,3,0,0,0,11,11],[4,0,0,0,0,0,1,12,0,0,0,14,16,0,0,0,0,0,8,7,0,0,0,15,9],[13,0,0,0,0,0,11,12,0,0,0,4,6,0,0,0,0,0,7,11,0,0,0,12,2] >;
C8.7C42 in GAP, Magma, Sage, TeX
C_8._7C_4^2
% in TeX
G:=Group("C8.7C4^2"); // GroupNames label
G:=SmallGroup(128,112);
// by ID
G=gap.SmallGroup(128,112);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,520,3924,102,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=1,c^4=a^6,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b>;
// generators/relations