p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊3(C4⋊C4), C4.Q8⋊7C4, C4.9(C4×Q8), C2.3(C8⋊Q8), (C2×C8).13Q8, (C2×C8).111D4, C2.4(C8⋊2D4), C2.4(C8.D4), C23.791(C2×D4), C22.177(C4×D4), (C22×C4).133D4, C4.77(C22⋊Q8), C22.34(C4⋊Q8), C4.7(C42.C2), C22.90(C8⋊C22), C22.4Q16.50C2, (C2×C42).308C22, (C22×C8).401C22, C2.18(SD16⋊C4), C22.137(C4⋊D4), (C22×C4).1393C23, C22.79(C8.C22), C23.65C23.13C2, C2.11(C23.65C23), C4.42(C2×C4⋊C4), C4⋊C4.90(C2×C4), (C2×C8).66(C2×C4), (C2×C8⋊C4).7C2, (C2×C4.Q8).6C2, (C2×C4).205(C2×Q8), (C2×C2.D8).35C2, (C2×C4).1350(C2×D4), (C2×C4⋊C4).77C22, (C2×C4).588(C4○D4), (C2×C4).411(C22×C4), SmallGroup(128,676)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊(C4⋊C4)
G = < a,b,c | a8=b4=c4=1, bab-1=a3, cac-1=a5, cbc-1=b-1 >
Subgroups: 228 in 120 conjugacy classes, 64 normal (28 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, 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, C8⋊(C4⋊C4)
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, SD16⋊C4, C8⋊2D4, C8.D4, C8⋊Q8, C8⋊(C4⋊C4)
►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)
(1 103 19 118)(2 98 20 113)(3 101 21 116)(4 104 22 119)(5 99 23 114)(6 102 24 117)(7 97 17 120)(8 100 18 115)(9 27 45 124)(10 30 46 127)(11 25 47 122)(12 28 48 125)(13 31 41 128)(14 26 42 123)(15 29 43 126)(16 32 44 121)(33 84 52 70)(34 87 53 65)(35 82 54 68)(36 85 55 71)(37 88 56 66)(38 83 49 69)(39 86 50 72)(40 81 51 67)(57 108 75 91)(58 111 76 94)(59 106 77 89)(60 109 78 92)(61 112 79 95)(62 107 80 90)(63 110 73 93)(64 105 74 96)
(1 50 44 73)(2 55 45 78)(3 52 46 75)(4 49 47 80)(5 54 48 77)(6 51 41 74)(7 56 42 79)(8 53 43 76)(9 60 20 36)(10 57 21 33)(11 62 22 38)(12 59 23 35)(13 64 24 40)(14 61 17 37)(15 58 18 34)(16 63 19 39)(25 90 119 69)(26 95 120 66)(27 92 113 71)(28 89 114 68)(29 94 115 65)(30 91 116 70)(31 96 117 67)(32 93 118 72)(81 128 105 102)(82 125 106 99)(83 122 107 104)(84 127 108 101)(85 124 109 98)(86 121 110 103)(87 126 111 100)(88 123 112 97)
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), (1,103,19,118)(2,98,20,113)(3,101,21,116)(4,104,22,119)(5,99,23,114)(6,102,24,117)(7,97,17,120)(8,100,18,115)(9,27,45,124)(10,30,46,127)(11,25,47,122)(12,28,48,125)(13,31,41,128)(14,26,42,123)(15,29,43,126)(16,32,44,121)(33,84,52,70)(34,87,53,65)(35,82,54,68)(36,85,55,71)(37,88,56,66)(38,83,49,69)(39,86,50,72)(40,81,51,67)(57,108,75,91)(58,111,76,94)(59,106,77,89)(60,109,78,92)(61,112,79,95)(62,107,80,90)(63,110,73,93)(64,105,74,96), (1,50,44,73)(2,55,45,78)(3,52,46,75)(4,49,47,80)(5,54,48,77)(6,51,41,74)(7,56,42,79)(8,53,43,76)(9,60,20,36)(10,57,21,33)(11,62,22,38)(12,59,23,35)(13,64,24,40)(14,61,17,37)(15,58,18,34)(16,63,19,39)(25,90,119,69)(26,95,120,66)(27,92,113,71)(28,89,114,68)(29,94,115,65)(30,91,116,70)(31,96,117,67)(32,93,118,72)(81,128,105,102)(82,125,106,99)(83,122,107,104)(84,127,108,101)(85,124,109,98)(86,121,110,103)(87,126,111,100)(88,123,112,97)>;
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), (1,103,19,118)(2,98,20,113)(3,101,21,116)(4,104,22,119)(5,99,23,114)(6,102,24,117)(7,97,17,120)(8,100,18,115)(9,27,45,124)(10,30,46,127)(11,25,47,122)(12,28,48,125)(13,31,41,128)(14,26,42,123)(15,29,43,126)(16,32,44,121)(33,84,52,70)(34,87,53,65)(35,82,54,68)(36,85,55,71)(37,88,56,66)(38,83,49,69)(39,86,50,72)(40,81,51,67)(57,108,75,91)(58,111,76,94)(59,106,77,89)(60,109,78,92)(61,112,79,95)(62,107,80,90)(63,110,73,93)(64,105,74,96), (1,50,44,73)(2,55,45,78)(3,52,46,75)(4,49,47,80)(5,54,48,77)(6,51,41,74)(7,56,42,79)(8,53,43,76)(9,60,20,36)(10,57,21,33)(11,62,22,38)(12,59,23,35)(13,64,24,40)(14,61,17,37)(15,58,18,34)(16,63,19,39)(25,90,119,69)(26,95,120,66)(27,92,113,71)(28,89,114,68)(29,94,115,65)(30,91,116,70)(31,96,117,67)(32,93,118,72)(81,128,105,102)(82,125,106,99)(83,122,107,104)(84,127,108,101)(85,124,109,98)(86,121,110,103)(87,126,111,100)(88,123,112,97) );
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)], [(1,103,19,118),(2,98,20,113),(3,101,21,116),(4,104,22,119),(5,99,23,114),(6,102,24,117),(7,97,17,120),(8,100,18,115),(9,27,45,124),(10,30,46,127),(11,25,47,122),(12,28,48,125),(13,31,41,128),(14,26,42,123),(15,29,43,126),(16,32,44,121),(33,84,52,70),(34,87,53,65),(35,82,54,68),(36,85,55,71),(37,88,56,66),(38,83,49,69),(39,86,50,72),(40,81,51,67),(57,108,75,91),(58,111,76,94),(59,106,77,89),(60,109,78,92),(61,112,79,95),(62,107,80,90),(63,110,73,93),(64,105,74,96)], [(1,50,44,73),(2,55,45,78),(3,52,46,75),(4,49,47,80),(5,54,48,77),(6,51,41,74),(7,56,42,79),(8,53,43,76),(9,60,20,36),(10,57,21,33),(11,62,22,38),(12,59,23,35),(13,64,24,40),(14,61,17,37),(15,58,18,34),(16,63,19,39),(25,90,119,69),(26,95,120,66),(27,92,113,71),(28,89,114,68),(29,94,115,65),(30,91,116,70),(31,96,117,67),(32,93,118,72),(81,128,105,102),(82,125,106,99),(83,122,107,104),(84,127,108,101),(85,124,109,98),(86,121,110,103),(87,126,111,100),(88,123,112,97)]])
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 | C8⋊(C4⋊C4) | C22.4Q16 | C23.65C23 | C2×C8⋊C4 | C2×C4.Q8 | C2×C2.D8 | C4.Q8 | 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 C8⋊(C4⋊C4) ►in GL8(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 8 | 4 | 8 |
| 0 | 0 | 0 | 0 | 8 | 10 | 8 | 13 |
| 0 | 0 | 0 | 0 | 13 | 9 | 8 | 10 |
| 0 | 0 | 0 | 0 | 9 | 4 | 10 | 9 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 6 | 16 | 2 |
| 0 | 0 | 0 | 0 | 11 | 13 | 15 | 16 |
| 0 | 0 | 0 | 0 | 15 | 16 | 4 | 11 |
| 0 | 0 | 0 | 0 | 1 | 15 | 6 | 4 |
G:=sub<GL(8,GF(17))| [0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[3,3,0,0,0,0,0,0,3,14,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,7,8,13,9,0,0,0,0,8,10,9,4,0,0,0,0,4,8,8,10,0,0,0,0,8,13,10,9],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,15,5,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,0,13,11,15,1,0,0,0,0,6,13,16,15,0,0,0,0,16,15,4,6,0,0,0,0,2,16,11,4] >;
C8⋊(C4⋊C4) in GAP, Magma, Sage, TeX
C_8\rtimes (C_4\rtimes C_4)
% in TeX
G:=Group("C8:(C4:C4)"); // GroupNames label
G:=SmallGroup(128,676);
// by ID
G=gap.SmallGroup(128,676);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,723,436,2804,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=c^4=1,b*a*b^-1=a^3,c*a*c^-1=a^5,c*b*c^-1=b^-1>;
// generators/relations