direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C8⋊C8, C23.40C42, C42.740C23, C8⋊9(C2×C8), (C2×C8)⋊8C8, C4○(C8⋊C8), (C4×C8).15C4, C4.13(C4×C8), C42○(C8⋊C8), (C2×C4).85C42, C4.31(C22×C8), (C22×C8).42C4, C22.15(C4×C8), C4.16(C8⋊C4), (C4×C8).358C22, C42.337(C2×C4), C4.55(C2×M4(2)), (C2×C4).87M4(2), C22.20(C2×C42), C22.13(C8⋊C4), (C2×C42).1142C22, C2.1(C2×C4×C8), (C2×C4×C8).6C2, (C2×C4)○(C8⋊C8), C2.1(C2×C8⋊C4), (C2×C4).98(C2×C8), (C2×C8).199(C2×C4), (C2×C4).577(C22×C4), (C22×C4).502(C2×C4), SmallGroup(128,180)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8⋊C8
G = < a,b,c | a2=b8=c8=1, ab=ba, ac=ca, cbc-1=b5 >
Subgroups: 140 in 124 conjugacy classes, 108 normal (12 characteristic)
C1, C2, C2 [×6], C4 [×12], C22, C22 [×6], C8 [×8], C8 [×8], C2×C4 [×2], C2×C4 [×16], C23, C42 [×2], C42 [×2], C2×C8 [×20], C2×C8 [×8], C22×C4, C22×C4 [×2], C4×C8 [×12], C2×C42, C22×C8 [×6], C8⋊C8 [×4], C2×C4×C8, C2×C4×C8 [×2], C2×C8⋊C8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], M4(2) [×8], C22×C4 [×3], C4×C8 [×4], C8⋊C4 [×8], C2×C42, C22×C8 [×2], C2×M4(2) [×4], C8⋊C8 [×4], C2×C4×C8, C2×C8⋊C4 [×2], C2×C8⋊C8
►Regular action on 128 points
Generators in S128
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 81)(16 82)(17 70)(18 71)(19 72)(20 65)(21 66)(22 67)(23 68)(24 69)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 73)(32 74)(41 112)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 119)(50 120)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 97)(89 124)(90 125)(91 126)(92 127)(93 128)(94 121)(95 122)(96 123)
(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)
(1 116 79 87 42 57 18 125)(2 113 80 84 43 62 19 122)(3 118 73 81 44 59 20 127)(4 115 74 86 45 64 21 124)(5 120 75 83 46 61 22 121)(6 117 76 88 47 58 23 126)(7 114 77 85 48 63 24 123)(8 119 78 82 41 60 17 128)(9 109 102 67 94 35 50 25)(10 106 103 72 95 40 51 30)(11 111 104 69 96 37 52 27)(12 108 97 66 89 34 53 32)(13 105 98 71 90 39 54 29)(14 110 99 68 91 36 55 26)(15 107 100 65 92 33 56 31)(16 112 101 70 93 38 49 28)
G:=sub<Sym(128)| (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,81)(16,82)(17,70)(18,71)(19,72)(20,65)(21,66)(22,67)(23,68)(24,69)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(41,112)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,119)(50,120)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,97)(89,124)(90,125)(91,126)(92,127)(93,128)(94,121)(95,122)(96,123), (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), (1,116,79,87,42,57,18,125)(2,113,80,84,43,62,19,122)(3,118,73,81,44,59,20,127)(4,115,74,86,45,64,21,124)(5,120,75,83,46,61,22,121)(6,117,76,88,47,58,23,126)(7,114,77,85,48,63,24,123)(8,119,78,82,41,60,17,128)(9,109,102,67,94,35,50,25)(10,106,103,72,95,40,51,30)(11,111,104,69,96,37,52,27)(12,108,97,66,89,34,53,32)(13,105,98,71,90,39,54,29)(14,110,99,68,91,36,55,26)(15,107,100,65,92,33,56,31)(16,112,101,70,93,38,49,28)>;
G:=Group( (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,81)(16,82)(17,70)(18,71)(19,72)(20,65)(21,66)(22,67)(23,68)(24,69)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(41,112)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,119)(50,120)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,97)(89,124)(90,125)(91,126)(92,127)(93,128)(94,121)(95,122)(96,123), (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), (1,116,79,87,42,57,18,125)(2,113,80,84,43,62,19,122)(3,118,73,81,44,59,20,127)(4,115,74,86,45,64,21,124)(5,120,75,83,46,61,22,121)(6,117,76,88,47,58,23,126)(7,114,77,85,48,63,24,123)(8,119,78,82,41,60,17,128)(9,109,102,67,94,35,50,25)(10,106,103,72,95,40,51,30)(11,111,104,69,96,37,52,27)(12,108,97,66,89,34,53,32)(13,105,98,71,90,39,54,29)(14,110,99,68,91,36,55,26)(15,107,100,65,92,33,56,31)(16,112,101,70,93,38,49,28) );
G=PermutationGroup([(1,39),(2,40),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,83),(10,84),(11,85),(12,86),(13,87),(14,88),(15,81),(16,82),(17,70),(18,71),(19,72),(20,65),(21,66),(22,67),(23,68),(24,69),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,73),(32,74),(41,112),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,119),(50,120),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,98),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,97),(89,124),(90,125),(91,126),(92,127),(93,128),(94,121),(95,122),(96,123)], [(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)], [(1,116,79,87,42,57,18,125),(2,113,80,84,43,62,19,122),(3,118,73,81,44,59,20,127),(4,115,74,86,45,64,21,124),(5,120,75,83,46,61,22,121),(6,117,76,88,47,58,23,126),(7,114,77,85,48,63,24,123),(8,119,78,82,41,60,17,128),(9,109,102,67,94,35,50,25),(10,106,103,72,95,40,51,30),(11,111,104,69,96,37,52,27),(12,108,97,66,89,34,53,32),(13,105,98,71,90,39,54,29),(14,110,99,68,91,36,55,26),(15,107,100,65,92,33,56,31),(16,112,101,70,93,38,49,28)])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4X | 8A | ··· | 8AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | M4(2) |
| kernel | C2×C8⋊C8 | C8⋊C8 | C2×C4×C8 | C4×C8 | C22×C8 | C2×C8 | C2×C4 |
| # reps | 1 | 4 | 3 | 12 | 12 | 32 | 16 |
Matrix representation of C2×C8⋊C8 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 8 |
| 2 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 13 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,9,0,0,0,0,8],[2,0,0,0,0,4,0,0,0,0,0,13,0,0,16,0] >;
C2×C8⋊C8 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes C_8
% in TeX
G:=Group("C2xC8:C8"); // GroupNames label
G:=SmallGroup(128,180);
// by ID
G=gap.SmallGroup(128,180);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,477,120,136,172]);
// Polycyclic
G:=Group<a,b,c|a^2=b^8=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations