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 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22 [×3], C22 [×4], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], C23, C16 [×2], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×4], C22×C4, C22×C4 [×2], C2.D8 [×4], C2.D8 [×2], C2×C16 [×2], C2×C16 [×2], C2×C4⋊C4 [×2], C22×C8, C2×C2.D8 [×2], C22×C16, C8.7C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, D16, SD32 [×2], Q32, C22.4Q16, C2.D16 [×2], C2.Q32 [×2], C16⋊3C4, C16⋊4C4, C8.7C42
►Regular action on 128 points
Generators in S128
(1 61 13 57 9 53 5 49)(2 62 14 58 10 54 6 50)(3 63 15 59 11 55 7 51)(4 64 16 60 12 56 8 52)(17 65 29 77 25 73 21 69)(18 66 30 78 26 74 22 70)(19 67 31 79 27 75 23 71)(20 68 32 80 28 76 24 72)(33 104 45 100 41 112 37 108)(34 105 46 101 42 97 38 109)(35 106 47 102 43 98 39 110)(36 107 48 103 44 99 40 111)(81 117 93 113 89 125 85 121)(82 118 94 114 90 126 86 122)(83 119 95 115 91 127 87 123)(84 120 96 116 92 128 88 124)
(1 90 68 98)(2 127 69 40)(3 88 70 112)(4 125 71 38)(5 86 72 110)(6 123 73 36)(7 84 74 108)(8 121 75 34)(9 82 76 106)(10 119 77 48)(11 96 78 104)(12 117 79 46)(13 94 80 102)(14 115 65 44)(15 92 66 100)(16 113 67 42)(17 99 62 91)(18 41 63 128)(19 97 64 89)(20 39 49 126)(21 111 50 87)(22 37 51 124)(23 109 52 85)(24 35 53 122)(25 107 54 83)(26 33 55 120)(27 105 56 81)(28 47 57 118)(29 103 58 95)(30 45 59 116)(31 101 60 93)(32 43 61 114)
(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)| (1,61,13,57,9,53,5,49)(2,62,14,58,10,54,6,50)(3,63,15,59,11,55,7,51)(4,64,16,60,12,56,8,52)(17,65,29,77,25,73,21,69)(18,66,30,78,26,74,22,70)(19,67,31,79,27,75,23,71)(20,68,32,80,28,76,24,72)(33,104,45,100,41,112,37,108)(34,105,46,101,42,97,38,109)(35,106,47,102,43,98,39,110)(36,107,48,103,44,99,40,111)(81,117,93,113,89,125,85,121)(82,118,94,114,90,126,86,122)(83,119,95,115,91,127,87,123)(84,120,96,116,92,128,88,124), (1,90,68,98)(2,127,69,40)(3,88,70,112)(4,125,71,38)(5,86,72,110)(6,123,73,36)(7,84,74,108)(8,121,75,34)(9,82,76,106)(10,119,77,48)(11,96,78,104)(12,117,79,46)(13,94,80,102)(14,115,65,44)(15,92,66,100)(16,113,67,42)(17,99,62,91)(18,41,63,128)(19,97,64,89)(20,39,49,126)(21,111,50,87)(22,37,51,124)(23,109,52,85)(24,35,53,122)(25,107,54,83)(26,33,55,120)(27,105,56,81)(28,47,57,118)(29,103,58,95)(30,45,59,116)(31,101,60,93)(32,43,61,114), (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( (1,61,13,57,9,53,5,49)(2,62,14,58,10,54,6,50)(3,63,15,59,11,55,7,51)(4,64,16,60,12,56,8,52)(17,65,29,77,25,73,21,69)(18,66,30,78,26,74,22,70)(19,67,31,79,27,75,23,71)(20,68,32,80,28,76,24,72)(33,104,45,100,41,112,37,108)(34,105,46,101,42,97,38,109)(35,106,47,102,43,98,39,110)(36,107,48,103,44,99,40,111)(81,117,93,113,89,125,85,121)(82,118,94,114,90,126,86,122)(83,119,95,115,91,127,87,123)(84,120,96,116,92,128,88,124), (1,90,68,98)(2,127,69,40)(3,88,70,112)(4,125,71,38)(5,86,72,110)(6,123,73,36)(7,84,74,108)(8,121,75,34)(9,82,76,106)(10,119,77,48)(11,96,78,104)(12,117,79,46)(13,94,80,102)(14,115,65,44)(15,92,66,100)(16,113,67,42)(17,99,62,91)(18,41,63,128)(19,97,64,89)(20,39,49,126)(21,111,50,87)(22,37,51,124)(23,109,52,85)(24,35,53,122)(25,107,54,83)(26,33,55,120)(27,105,56,81)(28,47,57,118)(29,103,58,95)(30,45,59,116)(31,101,60,93)(32,43,61,114), (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([(1,61,13,57,9,53,5,49),(2,62,14,58,10,54,6,50),(3,63,15,59,11,55,7,51),(4,64,16,60,12,56,8,52),(17,65,29,77,25,73,21,69),(18,66,30,78,26,74,22,70),(19,67,31,79,27,75,23,71),(20,68,32,80,28,76,24,72),(33,104,45,100,41,112,37,108),(34,105,46,101,42,97,38,109),(35,106,47,102,43,98,39,110),(36,107,48,103,44,99,40,111),(81,117,93,113,89,125,85,121),(82,118,94,114,90,126,86,122),(83,119,95,115,91,127,87,123),(84,120,96,116,92,128,88,124)], [(1,90,68,98),(2,127,69,40),(3,88,70,112),(4,125,71,38),(5,86,72,110),(6,123,73,36),(7,84,74,108),(8,121,75,34),(9,82,76,106),(10,119,77,48),(11,96,78,104),(12,117,79,46),(13,94,80,102),(14,115,65,44),(15,92,66,100),(16,113,67,42),(17,99,62,91),(18,41,63,128),(19,97,64,89),(20,39,49,126),(21,111,50,87),(22,37,51,124),(23,109,52,85),(24,35,53,122),(25,107,54,83),(26,33,55,120),(27,105,56,81),(28,47,57,118),(29,103,58,95),(30,45,59,116),(31,101,60,93),(32,43,61,114)], [(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