direct product, p-group, abelian, monomial
Aliases: C42×C8, SmallGroup(128,456)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C42×C8 |
| C1 — C42×C8 |
| C1 — C42×C8 |
Generators and relations for C42×C8
G = < a,b,c | a4=b4=c8=1, ab=ba, ac=ca, bc=cb >
Subgroups: 204, all normal (6 characteristic)
C1, C2, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, C22×C4, C4×C8, C2×C42, C22×C8, C43, C2×C4×C8, C42×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, C4×C8, C2×C42, C22×C8, C43, C2×C4×C8, C42×C8
►Regular action on 128 points
Generators in S128
(1 125 109 37)(2 126 110 38)(3 127 111 39)(4 128 112 40)(5 121 105 33)(6 122 106 34)(7 123 107 35)(8 124 108 36)(9 119 47 31)(10 120 48 32)(11 113 41 25)(12 114 42 26)(13 115 43 27)(14 116 44 28)(15 117 45 29)(16 118 46 30)(17 93 79 61)(18 94 80 62)(19 95 73 63)(20 96 74 64)(21 89 75 57)(22 90 76 58)(23 91 77 59)(24 92 78 60)(49 97 81 67)(50 98 82 68)(51 99 83 69)(52 100 84 70)(53 101 85 71)(54 102 86 72)(55 103 87 65)(56 104 88 66)
(1 95 87 29)(2 96 88 30)(3 89 81 31)(4 90 82 32)(5 91 83 25)(6 92 84 26)(7 93 85 27)(8 94 86 28)(9 127 75 67)(10 128 76 68)(11 121 77 69)(12 122 78 70)(13 123 79 71)(14 124 80 72)(15 125 73 65)(16 126 74 66)(17 101 43 35)(18 102 44 36)(19 103 45 37)(20 104 46 38)(21 97 47 39)(22 98 48 40)(23 99 41 33)(24 100 42 34)(49 119 111 57)(50 120 112 58)(51 113 105 59)(52 114 106 60)(53 115 107 61)(54 116 108 62)(55 117 109 63)(56 118 110 64)
(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)| (1,125,109,37)(2,126,110,38)(3,127,111,39)(4,128,112,40)(5,121,105,33)(6,122,106,34)(7,123,107,35)(8,124,108,36)(9,119,47,31)(10,120,48,32)(11,113,41,25)(12,114,42,26)(13,115,43,27)(14,116,44,28)(15,117,45,29)(16,118,46,30)(17,93,79,61)(18,94,80,62)(19,95,73,63)(20,96,74,64)(21,89,75,57)(22,90,76,58)(23,91,77,59)(24,92,78,60)(49,97,81,67)(50,98,82,68)(51,99,83,69)(52,100,84,70)(53,101,85,71)(54,102,86,72)(55,103,87,65)(56,104,88,66), (1,95,87,29)(2,96,88,30)(3,89,81,31)(4,90,82,32)(5,91,83,25)(6,92,84,26)(7,93,85,27)(8,94,86,28)(9,127,75,67)(10,128,76,68)(11,121,77,69)(12,122,78,70)(13,123,79,71)(14,124,80,72)(15,125,73,65)(16,126,74,66)(17,101,43,35)(18,102,44,36)(19,103,45,37)(20,104,46,38)(21,97,47,39)(22,98,48,40)(23,99,41,33)(24,100,42,34)(49,119,111,57)(50,120,112,58)(51,113,105,59)(52,114,106,60)(53,115,107,61)(54,116,108,62)(55,117,109,63)(56,118,110,64), (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( (1,125,109,37)(2,126,110,38)(3,127,111,39)(4,128,112,40)(5,121,105,33)(6,122,106,34)(7,123,107,35)(8,124,108,36)(9,119,47,31)(10,120,48,32)(11,113,41,25)(12,114,42,26)(13,115,43,27)(14,116,44,28)(15,117,45,29)(16,118,46,30)(17,93,79,61)(18,94,80,62)(19,95,73,63)(20,96,74,64)(21,89,75,57)(22,90,76,58)(23,91,77,59)(24,92,78,60)(49,97,81,67)(50,98,82,68)(51,99,83,69)(52,100,84,70)(53,101,85,71)(54,102,86,72)(55,103,87,65)(56,104,88,66), (1,95,87,29)(2,96,88,30)(3,89,81,31)(4,90,82,32)(5,91,83,25)(6,92,84,26)(7,93,85,27)(8,94,86,28)(9,127,75,67)(10,128,76,68)(11,121,77,69)(12,122,78,70)(13,123,79,71)(14,124,80,72)(15,125,73,65)(16,126,74,66)(17,101,43,35)(18,102,44,36)(19,103,45,37)(20,104,46,38)(21,97,47,39)(22,98,48,40)(23,99,41,33)(24,100,42,34)(49,119,111,57)(50,120,112,58)(51,113,105,59)(52,114,106,60)(53,115,107,61)(54,116,108,62)(55,117,109,63)(56,118,110,64), (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([[(1,125,109,37),(2,126,110,38),(3,127,111,39),(4,128,112,40),(5,121,105,33),(6,122,106,34),(7,123,107,35),(8,124,108,36),(9,119,47,31),(10,120,48,32),(11,113,41,25),(12,114,42,26),(13,115,43,27),(14,116,44,28),(15,117,45,29),(16,118,46,30),(17,93,79,61),(18,94,80,62),(19,95,73,63),(20,96,74,64),(21,89,75,57),(22,90,76,58),(23,91,77,59),(24,92,78,60),(49,97,81,67),(50,98,82,68),(51,99,83,69),(52,100,84,70),(53,101,85,71),(54,102,86,72),(55,103,87,65),(56,104,88,66)], [(1,95,87,29),(2,96,88,30),(3,89,81,31),(4,90,82,32),(5,91,83,25),(6,92,84,26),(7,93,85,27),(8,94,86,28),(9,127,75,67),(10,128,76,68),(11,121,77,69),(12,122,78,70),(13,123,79,71),(14,124,80,72),(15,125,73,65),(16,126,74,66),(17,101,43,35),(18,102,44,36),(19,103,45,37),(20,104,46,38),(21,97,47,39),(22,98,48,40),(23,99,41,33),(24,100,42,34),(49,119,111,57),(50,120,112,58),(51,113,105,59),(52,114,106,60),(53,115,107,61),(54,116,108,62),(55,117,109,63),(56,118,110,64)], [(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)]])
128 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4BD | 8A | ··· | 8BL |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||
| image | C1 | C2 | C2 | C4 | C4 | C8 |
| kernel | C42×C8 | C43 | C2×C4×C8 | C4×C8 | C2×C42 | C42 |
| # reps | 1 | 1 | 6 | 48 | 8 | 64 |
Matrix representation of C42×C8 ►in GL3(𝔽17) generated by
| 4 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 13 |
| 1 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 16 |
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 2 |
G:=sub<GL(3,GF(17))| [4,0,0,0,16,0,0,0,13],[1,0,0,0,13,0,0,0,16],[16,0,0,0,1,0,0,0,2] >;
C42×C8 in GAP, Magma, Sage, TeX
C_4^2\times C_8
% in TeX
G:=Group("C4^2xC8"); // GroupNames label
G:=SmallGroup(128,456);
// by ID
G=gap.SmallGroup(128,456);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,120,184,248]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^8=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations