p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊2(C4⋊C4), C4.8(C4×Q8), C2.D8⋊12C4, C2.2(C8⋊Q8), (C2×C8).12Q8, (C2×C8).110D4, C2.6(C8⋊D4), C23.790(C2×D4), (C22×C4).132D4, C22.176(C4×D4), C22.33(C4⋊Q8), C4.76(C22⋊Q8), C4.6(C42.C2), C2.11(D8⋊C4), C2.11(Q16⋊C4), C22.89(C8⋊C22), C22.4Q16.49C2, (C2×C42).307C22, (C22×C8).400C22, C22.136(C4⋊D4), (C22×C4).1392C23, C22.78(C8.C22), C23.65C23.12C2, C2.10(C23.65C23), C4.41(C2×C4⋊C4), C4⋊C4.89(C2×C4), (C2×C8).65(C2×C4), (C2×C8⋊C4).6C2, (C2×C4.Q8).5C2, (C2×C4).204(C2×Q8), (C2×C2.D8).34C2, (C2×C4).1349(C2×D4), (C2×C4⋊C4).76C22, (C2×C4).587(C4○D4), (C2×C4).410(C22×C4), SmallGroup(128,675)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.(C4×Q8)
G = < a,b,c,d | a4=c4=1, b4=a2, d2=ac2, ab=ba, cac-1=a-1, ad=da, cbc-1=a-1b, dbd-1=a2b, dcd-1=ac-1 >
Subgroups: 228 in 120 conjugacy classes, 64 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, 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×C4⋊C4, C2×C4⋊C4, C22×C8, C22.4Q16, C23.65C23, C2×C8⋊C4, C2×C4.Q8, C2×C2.D8, C4.(C4×Q8)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C8⋊C22, C8.C22, C23.65C23, Q16⋊C4, D8⋊C4, C8⋊D4, C8⋊Q8, C4.(C4×Q8)
►Regular action on 128 points
Generators in S128
(1 59 5 63)(2 60 6 64)(3 61 7 57)(4 62 8 58)(9 33 13 37)(10 34 14 38)(11 35 15 39)(12 36 16 40)(17 94 21 90)(18 95 22 91)(19 96 23 92)(20 89 24 93)(25 43 29 47)(26 44 30 48)(27 45 31 41)(28 46 32 42)(49 84 53 88)(50 85 54 81)(51 86 55 82)(52 87 56 83)(65 78 69 74)(66 79 70 75)(67 80 71 76)(68 73 72 77)(97 122 101 126)(98 123 102 127)(99 124 103 128)(100 125 104 121)(105 116 109 120)(106 117 110 113)(107 118 111 114)(108 119 112 115)
(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)
(1 105 95 80)(2 113 96 68)(3 111 89 78)(4 119 90 66)(5 109 91 76)(6 117 92 72)(7 107 93 74)(8 115 94 70)(9 102 42 86)(10 124 43 52)(11 100 44 84)(12 122 45 50)(13 98 46 82)(14 128 47 56)(15 104 48 88)(16 126 41 54)(17 75 62 108)(18 71 63 116)(19 73 64 106)(20 69 57 114)(21 79 58 112)(22 67 59 120)(23 77 60 110)(24 65 61 118)(25 87 38 103)(26 53 39 125)(27 85 40 101)(28 51 33 123)(29 83 34 99)(30 49 35 121)(31 81 36 97)(32 55 37 127)
(1 43 22 34 5 47 18 38)(2 48 23 39 6 44 19 35)(3 45 24 36 7 41 20 40)(4 42 17 33 8 46 21 37)(9 62 28 94 13 58 32 90)(10 59 29 91 14 63 25 95)(11 64 30 96 15 60 26 92)(12 61 31 93 16 57 27 89)(49 106 100 72 53 110 104 68)(50 111 101 69 54 107 97 65)(51 108 102 66 55 112 98 70)(52 105 103 71 56 109 99 67)(73 84 117 125 77 88 113 121)(74 81 118 122 78 85 114 126)(75 86 119 127 79 82 115 123)(76 83 120 124 80 87 116 128)
G:=sub<Sym(128)| (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,33,13,37)(10,34,14,38)(11,35,15,39)(12,36,16,40)(17,94,21,90)(18,95,22,91)(19,96,23,92)(20,89,24,93)(25,43,29,47)(26,44,30,48)(27,45,31,41)(28,46,32,42)(49,84,53,88)(50,85,54,81)(51,86,55,82)(52,87,56,83)(65,78,69,74)(66,79,70,75)(67,80,71,76)(68,73,72,77)(97,122,101,126)(98,123,102,127)(99,124,103,128)(100,125,104,121)(105,116,109,120)(106,117,110,113)(107,118,111,114)(108,119,112,115), (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), (1,105,95,80)(2,113,96,68)(3,111,89,78)(4,119,90,66)(5,109,91,76)(6,117,92,72)(7,107,93,74)(8,115,94,70)(9,102,42,86)(10,124,43,52)(11,100,44,84)(12,122,45,50)(13,98,46,82)(14,128,47,56)(15,104,48,88)(16,126,41,54)(17,75,62,108)(18,71,63,116)(19,73,64,106)(20,69,57,114)(21,79,58,112)(22,67,59,120)(23,77,60,110)(24,65,61,118)(25,87,38,103)(26,53,39,125)(27,85,40,101)(28,51,33,123)(29,83,34,99)(30,49,35,121)(31,81,36,97)(32,55,37,127), (1,43,22,34,5,47,18,38)(2,48,23,39,6,44,19,35)(3,45,24,36,7,41,20,40)(4,42,17,33,8,46,21,37)(9,62,28,94,13,58,32,90)(10,59,29,91,14,63,25,95)(11,64,30,96,15,60,26,92)(12,61,31,93,16,57,27,89)(49,106,100,72,53,110,104,68)(50,111,101,69,54,107,97,65)(51,108,102,66,55,112,98,70)(52,105,103,71,56,109,99,67)(73,84,117,125,77,88,113,121)(74,81,118,122,78,85,114,126)(75,86,119,127,79,82,115,123)(76,83,120,124,80,87,116,128)>;
G:=Group( (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,33,13,37)(10,34,14,38)(11,35,15,39)(12,36,16,40)(17,94,21,90)(18,95,22,91)(19,96,23,92)(20,89,24,93)(25,43,29,47)(26,44,30,48)(27,45,31,41)(28,46,32,42)(49,84,53,88)(50,85,54,81)(51,86,55,82)(52,87,56,83)(65,78,69,74)(66,79,70,75)(67,80,71,76)(68,73,72,77)(97,122,101,126)(98,123,102,127)(99,124,103,128)(100,125,104,121)(105,116,109,120)(106,117,110,113)(107,118,111,114)(108,119,112,115), (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), (1,105,95,80)(2,113,96,68)(3,111,89,78)(4,119,90,66)(5,109,91,76)(6,117,92,72)(7,107,93,74)(8,115,94,70)(9,102,42,86)(10,124,43,52)(11,100,44,84)(12,122,45,50)(13,98,46,82)(14,128,47,56)(15,104,48,88)(16,126,41,54)(17,75,62,108)(18,71,63,116)(19,73,64,106)(20,69,57,114)(21,79,58,112)(22,67,59,120)(23,77,60,110)(24,65,61,118)(25,87,38,103)(26,53,39,125)(27,85,40,101)(28,51,33,123)(29,83,34,99)(30,49,35,121)(31,81,36,97)(32,55,37,127), (1,43,22,34,5,47,18,38)(2,48,23,39,6,44,19,35)(3,45,24,36,7,41,20,40)(4,42,17,33,8,46,21,37)(9,62,28,94,13,58,32,90)(10,59,29,91,14,63,25,95)(11,64,30,96,15,60,26,92)(12,61,31,93,16,57,27,89)(49,106,100,72,53,110,104,68)(50,111,101,69,54,107,97,65)(51,108,102,66,55,112,98,70)(52,105,103,71,56,109,99,67)(73,84,117,125,77,88,113,121)(74,81,118,122,78,85,114,126)(75,86,119,127,79,82,115,123)(76,83,120,124,80,87,116,128) );
G=PermutationGroup([[(1,59,5,63),(2,60,6,64),(3,61,7,57),(4,62,8,58),(9,33,13,37),(10,34,14,38),(11,35,15,39),(12,36,16,40),(17,94,21,90),(18,95,22,91),(19,96,23,92),(20,89,24,93),(25,43,29,47),(26,44,30,48),(27,45,31,41),(28,46,32,42),(49,84,53,88),(50,85,54,81),(51,86,55,82),(52,87,56,83),(65,78,69,74),(66,79,70,75),(67,80,71,76),(68,73,72,77),(97,122,101,126),(98,123,102,127),(99,124,103,128),(100,125,104,121),(105,116,109,120),(106,117,110,113),(107,118,111,114),(108,119,112,115)], [(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)], [(1,105,95,80),(2,113,96,68),(3,111,89,78),(4,119,90,66),(5,109,91,76),(6,117,92,72),(7,107,93,74),(8,115,94,70),(9,102,42,86),(10,124,43,52),(11,100,44,84),(12,122,45,50),(13,98,46,82),(14,128,47,56),(15,104,48,88),(16,126,41,54),(17,75,62,108),(18,71,63,116),(19,73,64,106),(20,69,57,114),(21,79,58,112),(22,67,59,120),(23,77,60,110),(24,65,61,118),(25,87,38,103),(26,53,39,125),(27,85,40,101),(28,51,33,123),(29,83,34,99),(30,49,35,121),(31,81,36,97),(32,55,37,127)], [(1,43,22,34,5,47,18,38),(2,48,23,39,6,44,19,35),(3,45,24,36,7,41,20,40),(4,42,17,33,8,46,21,37),(9,62,28,94,13,58,32,90),(10,59,29,91,14,63,25,95),(11,64,30,96,15,60,26,92),(12,61,31,93,16,57,27,89),(49,106,100,72,53,110,104,68),(50,111,101,69,54,107,97,65),(51,108,102,66,55,112,98,70),(52,105,103,71,56,109,99,67),(73,84,117,125,77,88,113,121),(74,81,118,122,78,85,114,126),(75,86,119,127,79,82,115,123),(76,83,120,124,80,87,116,128)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C4.(C4×Q8) | C22.4Q16 | C23.65C23 | C2×C8⋊C4 | C2×C4.Q8 | C2×C2.D8 | C2.D8 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 2 | 4 | 2 | 4 | 2 | 2 |
Matrix representation of C4.(C4×Q8) ►in GL8(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 | 16 | 15 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
| 13 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 3 | 15 | 7 |
| 0 | 0 | 0 | 0 | 6 | 14 | 9 | 11 |
| 0 | 0 | 0 | 0 | 13 | 5 | 8 | 16 |
| 0 | 0 | 0 | 0 | 0 | 12 | 14 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 5 | 7 | 11 |
| 0 | 0 | 0 | 0 | 1 | 3 | 13 | 3 |
| 0 | 0 | 0 | 0 | 0 | 10 | 15 | 14 |
| 0 | 0 | 0 | 0 | 9 | 9 | 16 | 12 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 16 |
G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,16,1,0,0,0,0,1,0,16,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[13,16,0,0,0,0,0,0,15,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,12,6,13,0,0,0,0,0,3,14,5,12,0,0,0,0,15,9,8,14,0,0,0,0,7,11,16,0],[13,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,1,0,9,0,0,0,0,5,3,10,9,0,0,0,0,7,13,15,16,0,0,0,0,11,3,14,12],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,16,0,0,0,0,0,1,16,0,0,0,0,0,16,1,0,0,0,0,0,0,0,2,0,16] >;
C4.(C4×Q8) in GAP, Magma, Sage, TeX
C_4.(C_4\times Q_8)
% in TeX
G:=Group("C4.(C4xQ8)"); // GroupNames label
G:=SmallGroup(128,675);
// by ID
G=gap.SmallGroup(128,675);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,723,100,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^4=1,b^4=a^2,d^2=a*c^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,d*b*d^-1=a^2*b,d*c*d^-1=a*c^-1>;
// generators/relations