p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4⋊C4⋊C8, (C2×C4).99D8, C2.4(D4⋊C8), C2.3(Q8⋊C8), (C2×C4).45Q16, C22.29C4≀C2, (C2×C4).82SD16, (C2×C4).7M4(2), (C22×C4).632D4, C2.1(C4.10D8), C22.46(C22⋊C8), C22.31(C23⋊C4), (C2×C42).119C22, C2.2(C22.SD16), C22.36(D4⋊C4), C23.210(C22⋊C4), C22.25(Q8⋊C4), C2.1(C23.31D4), C22.7C42.1C2, C22.13(C4.10D4), C23.65C23.1C2, C2.1(C42.2C22), C2.3(C22.M4(2)), (C2×C4⋊C4).2C4, (C2×C4).7(C2×C8), (C22×C4).143(C2×C4), SmallGroup(128,3)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C4⋊C8
G = < a,b,c | a4=b4=c8=1, bab-1=a-1, cac-1=ab2, cbc-1=ab >
Subgroups: 168 in 79 conjugacy classes, 34 normal (32 characteristic)
C1, C2, C4, C22, C8, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.7C42, C23.65C23, C4⋊C4⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), D8, SD16, Q16, C22⋊C8, C23⋊C4, C4.10D4, D4⋊C4, Q8⋊C4, C4≀C2, C22.M4(2), D4⋊C8, Q8⋊C8, C22.SD16, C23.31D4, C42.2C22, C4.10D8, C4⋊C4⋊C8
►Regular action on 128 points
Generators in S128
(1 63 87 119)(2 34 88 30)(3 57 81 113)(4 36 82 32)(5 59 83 115)(6 38 84 26)(7 61 85 117)(8 40 86 28)(9 67 90 97)(10 42 91 128)(11 69 92 99)(12 44 93 122)(13 71 94 101)(14 46 95 124)(15 65 96 103)(16 48 89 126)(17 45 79 123)(18 72 80 102)(19 47 73 125)(20 66 74 104)(21 41 75 127)(22 68 76 98)(23 43 77 121)(24 70 78 100)(25 105 37 53)(27 107 39 55)(29 109 33 49)(31 111 35 51)(50 120 110 64)(52 114 112 58)(54 116 106 60)(56 118 108 62)
(1 43 109 69)(2 24 110 12)(3 101 111 123)(4 95 112 80)(5 47 105 65)(6 20 106 16)(7 97 107 127)(8 91 108 76)(9 27 21 117)(10 56 22 86)(11 63 23 33)(13 31 17 113)(14 52 18 82)(15 59 19 37)(25 96 115 73)(26 66 116 48)(28 128 118 98)(29 92 119 77)(30 70 120 44)(32 124 114 102)(34 100 64 122)(35 79 57 94)(36 46 58 72)(38 104 60 126)(39 75 61 90)(40 42 62 68)(41 85 67 55)(45 81 71 51)(49 99 87 121)(50 93 88 78)(53 103 83 125)(54 89 84 74)
(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)| (1,63,87,119)(2,34,88,30)(3,57,81,113)(4,36,82,32)(5,59,83,115)(6,38,84,26)(7,61,85,117)(8,40,86,28)(9,67,90,97)(10,42,91,128)(11,69,92,99)(12,44,93,122)(13,71,94,101)(14,46,95,124)(15,65,96,103)(16,48,89,126)(17,45,79,123)(18,72,80,102)(19,47,73,125)(20,66,74,104)(21,41,75,127)(22,68,76,98)(23,43,77,121)(24,70,78,100)(25,105,37,53)(27,107,39,55)(29,109,33,49)(31,111,35,51)(50,120,110,64)(52,114,112,58)(54,116,106,60)(56,118,108,62), (1,43,109,69)(2,24,110,12)(3,101,111,123)(4,95,112,80)(5,47,105,65)(6,20,106,16)(7,97,107,127)(8,91,108,76)(9,27,21,117)(10,56,22,86)(11,63,23,33)(13,31,17,113)(14,52,18,82)(15,59,19,37)(25,96,115,73)(26,66,116,48)(28,128,118,98)(29,92,119,77)(30,70,120,44)(32,124,114,102)(34,100,64,122)(35,79,57,94)(36,46,58,72)(38,104,60,126)(39,75,61,90)(40,42,62,68)(41,85,67,55)(45,81,71,51)(49,99,87,121)(50,93,88,78)(53,103,83,125)(54,89,84,74), (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( (1,63,87,119)(2,34,88,30)(3,57,81,113)(4,36,82,32)(5,59,83,115)(6,38,84,26)(7,61,85,117)(8,40,86,28)(9,67,90,97)(10,42,91,128)(11,69,92,99)(12,44,93,122)(13,71,94,101)(14,46,95,124)(15,65,96,103)(16,48,89,126)(17,45,79,123)(18,72,80,102)(19,47,73,125)(20,66,74,104)(21,41,75,127)(22,68,76,98)(23,43,77,121)(24,70,78,100)(25,105,37,53)(27,107,39,55)(29,109,33,49)(31,111,35,51)(50,120,110,64)(52,114,112,58)(54,116,106,60)(56,118,108,62), (1,43,109,69)(2,24,110,12)(3,101,111,123)(4,95,112,80)(5,47,105,65)(6,20,106,16)(7,97,107,127)(8,91,108,76)(9,27,21,117)(10,56,22,86)(11,63,23,33)(13,31,17,113)(14,52,18,82)(15,59,19,37)(25,96,115,73)(26,66,116,48)(28,128,118,98)(29,92,119,77)(30,70,120,44)(32,124,114,102)(34,100,64,122)(35,79,57,94)(36,46,58,72)(38,104,60,126)(39,75,61,90)(40,42,62,68)(41,85,67,55)(45,81,71,51)(49,99,87,121)(50,93,88,78)(53,103,83,125)(54,89,84,74), (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([[(1,63,87,119),(2,34,88,30),(3,57,81,113),(4,36,82,32),(5,59,83,115),(6,38,84,26),(7,61,85,117),(8,40,86,28),(9,67,90,97),(10,42,91,128),(11,69,92,99),(12,44,93,122),(13,71,94,101),(14,46,95,124),(15,65,96,103),(16,48,89,126),(17,45,79,123),(18,72,80,102),(19,47,73,125),(20,66,74,104),(21,41,75,127),(22,68,76,98),(23,43,77,121),(24,70,78,100),(25,105,37,53),(27,107,39,55),(29,109,33,49),(31,111,35,51),(50,120,110,64),(52,114,112,58),(54,116,106,60),(56,118,108,62)], [(1,43,109,69),(2,24,110,12),(3,101,111,123),(4,95,112,80),(5,47,105,65),(6,20,106,16),(7,97,107,127),(8,91,108,76),(9,27,21,117),(10,56,22,86),(11,63,23,33),(13,31,17,113),(14,52,18,82),(15,59,19,37),(25,96,115,73),(26,66,116,48),(28,128,118,98),(29,92,119,77),(30,70,120,44),(32,124,114,102),(34,100,64,122),(35,79,57,94),(36,46,58,72),(38,104,60,126),(39,75,61,90),(40,42,62,68),(41,85,67,55),(45,81,71,51),(49,99,87,121),(50,93,88,78),(53,103,83,125),(54,89,84,74)], [(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)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C4 | C8 | D4 | M4(2) | D8 | SD16 | Q16 | C4≀C2 | C23⋊C4 | C4.10D4 |
| kernel | C4⋊C4⋊C8 | C22.7C42 | C23.65C23 | C2×C4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 4 | 8 | 2 | 2 | 2 | 4 | 2 | 8 | 1 | 1 |
Matrix representation of C4⋊C4⋊C8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 7 | 16 | 0 | 0 | 0 | 0 |
| 14 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 10 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 16 |
| 0 | 0 | 0 | 0 | 16 | 7 |
| 14 | 15 | 0 | 0 | 0 | 0 |
| 15 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 13 |
| 0 | 0 | 0 | 0 | 13 | 6 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,4,13,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[7,14,0,0,0,0,16,10,0,0,0,0,0,0,14,6,0,0,0,0,10,3,0,0,0,0,0,0,10,16,0,0,0,0,16,7],[14,15,0,0,0,0,15,3,0,0,0,0,0,0,4,0,0,0,0,0,10,1,0,0,0,0,0,0,11,13,0,0,0,0,13,6] >;
C4⋊C4⋊C8 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes C_8
% in TeX
G:=Group("C4:C4:C8"); // GroupNames label
G:=SmallGroup(128,3);
// by ID
G=gap.SmallGroup(128,3);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,2,56,85,120,422,387,184,794,248]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^8=1,b*a*b^-1=a^-1,c*a*c^-1=a*b^2,c*b*c^-1=a*b>;
// generators/relations