p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊C42, C42.22Q8, C4.Q8⋊6C4, C8⋊C4⋊7C4, C4.5(C4×Q8), C2.D8⋊11C4, C4.25(C2×C42), C22.97(C4×D4), C2.3(D8⋊C4), C42.134(C2×C4), C23.736(C2×D4), (C22×C4).673D4, C2.3(Q16⋊C4), C4.49(C42⋊C2), C2.4(SD16⋊C4), C22.56(C8⋊C22), C22.4Q16.46C2, (C2×C42).241C22, (C22×C8).210C22, C2.3(M4(2)⋊C4), (C22×C4).1317C23, C22.45(C8.C22), C2.15(C4×C4⋊C4), (C4×C4⋊C4).11C2, (C2×C8⋊C4).2C2, (C2×C4.Q8).3C2, C4⋊C4.148(C2×C4), (C2×C4).80(C4⋊C4), (C2×C8).140(C2×C4), C22.60(C2×C4⋊C4), (C2×C4).186(C2×Q8), (C2×C2.D8).32C2, (C2×C4).548(C4○D4), (C2×C4⋊C4).752C22, (C2×C4).524(C22×C4), SmallGroup(128,508)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊C42
G = < a,b,c | a8=b4=c4=1, bab-1=a-1, cac-1=a5, bc=cb >
Subgroups: 228 in 136 conjugacy classes, 84 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2.C42, C8⋊C4, C4.Q8, C2.D8, C2×C42, C2×C42, C2×C4⋊C4, C22×C8, C22.4Q16, C4×C4⋊C4, C2×C8⋊C4, C2×C4.Q8, C2×C2.D8, C8⋊C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C8⋊C22, C8.C22, C4×C4⋊C4, M4(2)⋊C4, SD16⋊C4, Q16⋊C4, D8⋊C4, C8⋊C42
►Regular action on 128 points
Generators in S128
(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 10 22 90)(2 9 23 89)(3 16 24 96)(4 15 17 95)(5 14 18 94)(6 13 19 93)(7 12 20 92)(8 11 21 91)(25 103 67 105)(26 102 68 112)(27 101 69 111)(28 100 70 110)(29 99 71 109)(30 98 72 108)(31 97 65 107)(32 104 66 106)(33 124 80 86)(34 123 73 85)(35 122 74 84)(36 121 75 83)(37 128 76 82)(38 127 77 81)(39 126 78 88)(40 125 79 87)(41 52 64 114)(42 51 57 113)(43 50 58 120)(44 49 59 119)(45 56 60 118)(46 55 61 117)(47 54 62 116)(48 53 63 115)
(1 36 51 97)(2 33 52 102)(3 38 53 99)(4 35 54 104)(5 40 55 101)(6 37 56 98)(7 34 49 103)(8 39 50 100)(9 124 64 68)(10 121 57 65)(11 126 58 70)(12 123 59 67)(13 128 60 72)(14 125 61 69)(15 122 62 66)(16 127 63 71)(17 74 116 106)(18 79 117 111)(19 76 118 108)(20 73 119 105)(21 78 120 110)(22 75 113 107)(23 80 114 112)(24 77 115 109)(25 92 85 44)(26 89 86 41)(27 94 87 46)(28 91 88 43)(29 96 81 48)(30 93 82 45)(31 90 83 42)(32 95 84 47)
G:=sub<Sym(128)| (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), (1,10,22,90)(2,9,23,89)(3,16,24,96)(4,15,17,95)(5,14,18,94)(6,13,19,93)(7,12,20,92)(8,11,21,91)(25,103,67,105)(26,102,68,112)(27,101,69,111)(28,100,70,110)(29,99,71,109)(30,98,72,108)(31,97,65,107)(32,104,66,106)(33,124,80,86)(34,123,73,85)(35,122,74,84)(36,121,75,83)(37,128,76,82)(38,127,77,81)(39,126,78,88)(40,125,79,87)(41,52,64,114)(42,51,57,113)(43,50,58,120)(44,49,59,119)(45,56,60,118)(46,55,61,117)(47,54,62,116)(48,53,63,115), (1,36,51,97)(2,33,52,102)(3,38,53,99)(4,35,54,104)(5,40,55,101)(6,37,56,98)(7,34,49,103)(8,39,50,100)(9,124,64,68)(10,121,57,65)(11,126,58,70)(12,123,59,67)(13,128,60,72)(14,125,61,69)(15,122,62,66)(16,127,63,71)(17,74,116,106)(18,79,117,111)(19,76,118,108)(20,73,119,105)(21,78,120,110)(22,75,113,107)(23,80,114,112)(24,77,115,109)(25,92,85,44)(26,89,86,41)(27,94,87,46)(28,91,88,43)(29,96,81,48)(30,93,82,45)(31,90,83,42)(32,95,84,47)>;
G:=Group( (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), (1,10,22,90)(2,9,23,89)(3,16,24,96)(4,15,17,95)(5,14,18,94)(6,13,19,93)(7,12,20,92)(8,11,21,91)(25,103,67,105)(26,102,68,112)(27,101,69,111)(28,100,70,110)(29,99,71,109)(30,98,72,108)(31,97,65,107)(32,104,66,106)(33,124,80,86)(34,123,73,85)(35,122,74,84)(36,121,75,83)(37,128,76,82)(38,127,77,81)(39,126,78,88)(40,125,79,87)(41,52,64,114)(42,51,57,113)(43,50,58,120)(44,49,59,119)(45,56,60,118)(46,55,61,117)(47,54,62,116)(48,53,63,115), (1,36,51,97)(2,33,52,102)(3,38,53,99)(4,35,54,104)(5,40,55,101)(6,37,56,98)(7,34,49,103)(8,39,50,100)(9,124,64,68)(10,121,57,65)(11,126,58,70)(12,123,59,67)(13,128,60,72)(14,125,61,69)(15,122,62,66)(16,127,63,71)(17,74,116,106)(18,79,117,111)(19,76,118,108)(20,73,119,105)(21,78,120,110)(22,75,113,107)(23,80,114,112)(24,77,115,109)(25,92,85,44)(26,89,86,41)(27,94,87,46)(28,91,88,43)(29,96,81,48)(30,93,82,45)(31,90,83,42)(32,95,84,47) );
G=PermutationGroup([[(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)], [(1,10,22,90),(2,9,23,89),(3,16,24,96),(4,15,17,95),(5,14,18,94),(6,13,19,93),(7,12,20,92),(8,11,21,91),(25,103,67,105),(26,102,68,112),(27,101,69,111),(28,100,70,110),(29,99,71,109),(30,98,72,108),(31,97,65,107),(32,104,66,106),(33,124,80,86),(34,123,73,85),(35,122,74,84),(36,121,75,83),(37,128,76,82),(38,127,77,81),(39,126,78,88),(40,125,79,87),(41,52,64,114),(42,51,57,113),(43,50,58,120),(44,49,59,119),(45,56,60,118),(46,55,61,117),(47,54,62,116),(48,53,63,115)], [(1,36,51,97),(2,33,52,102),(3,38,53,99),(4,35,54,104),(5,40,55,101),(6,37,56,98),(7,34,49,103),(8,39,50,100),(9,124,64,68),(10,121,57,65),(11,126,58,70),(12,123,59,67),(13,128,60,72),(14,125,61,69),(15,122,62,66),(16,127,63,71),(17,74,116,106),(18,79,117,111),(19,76,118,108),(20,73,119,105),(21,78,120,110),(22,75,113,107),(23,80,114,112),(24,77,115,109),(25,92,85,44),(26,89,86,41),(27,94,87,46),(28,91,88,43),(29,96,81,48),(30,93,82,45),(31,90,83,42),(32,95,84,47)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4AB | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C8⋊C42 | C22.4Q16 | C4×C4⋊C4 | C2×C8⋊C4 | C2×C4.Q8 | C2×C2.D8 | C8⋊C4 | C4.Q8 | C2.D8 | C42 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 8 | 8 | 2 | 2 | 4 | 2 | 2 |
Matrix representation of C8⋊C42 ►in GL8(𝔽17)
| 2 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 15 | 15 | 15 |
| 0 | 0 | 0 | 0 | 2 | 2 | 2 | 15 |
| 0 | 0 | 0 | 0 | 2 | 2 | 15 | 2 |
| 0 | 0 | 0 | 0 | 15 | 2 | 15 | 15 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 | 16 | 12 |
| 0 | 0 | 0 | 0 | 1 | 5 | 12 | 1 |
| 0 | 0 | 0 | 0 | 16 | 12 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 | 1 | 5 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(17))| [2,5,0,0,0,0,0,0,16,15,0,0,0,0,0,0,0,0,6,7,0,0,0,0,0,0,2,11,0,0,0,0,0,0,0,0,2,2,2,15,0,0,0,0,15,2,2,2,0,0,0,0,15,2,15,15,0,0,0,0,15,15,2,15],[13,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,12,1,16,12,0,0,0,0,1,5,12,1,0,0,0,0,16,12,12,1,0,0,0,0,12,1,1,5],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C8⋊C42 in GAP, Magma, Sage, TeX
C_8\rtimes C_4^2
% in TeX
G:=Group("C8:C4^2"); // GroupNames label
G:=SmallGroup(128,508);
// by ID
G=gap.SmallGroup(128,508);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,142,1018,248,1411,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=c^4=1,b*a*b^-1=a^-1,c*a*c^-1=a^5,b*c=c*b>;
// generators/relations