direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8⋊2C8, C42.48Q8, C42.317D4, C42.619C23, C8⋊8(C2×C8), (C2×C8)⋊7C8, C4○(C8⋊2C8), (C4×C8).31C4, C4.10(C4⋊C8), (C22×C8).39C4, C4.24(C22×C8), C4.97(C2×SD16), C4.17(C4.Q8), (C22×C4).72Q8, C4⋊C8.263C22, C23.79(C4⋊C4), C22.20(C4⋊C8), (C4×C8).423C22, C42.307(C2×C4), (C2×C4).127SD16, (C22×C4).809D4, C4.39(C2×M4(2)), (C2×C4).73M4(2), C22.14(C4.Q8), C22.10(C8.C4), (C2×C42).1038C22, C2.4(C2×C4⋊C8), (C2×C4×C8).51C2, (C2×C4)○(C8⋊2C8), (C2×C4⋊C8).16C2, C2.1(C2×C4.Q8), (C2×C4).80(C2×C8), (C2×C8).230(C2×C4), C2.1(C2×C8.C4), C22.45(C2×C4⋊C4), (C2×C4).146(C2×Q8), (C2×C4).114(C4⋊C4), (C2×C4).1455(C2×D4), (C22×C4).471(C2×C4), (C2×C4).501(C22×C4), SmallGroup(128,294)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8⋊2C8
G = < a,b,c | a2=b8=c8=1, ab=ba, ac=ca, cbc-1=b3 >
Subgroups: 140 in 100 conjugacy classes, 76 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×8], C2×C4 [×2], C23, C42 [×4], C2×C8 [×8], C2×C8 [×10], C22×C4 [×3], C4×C8 [×4], C4⋊C8 [×4], C4⋊C8 [×2], C2×C42, C22×C8 [×2], C22×C8 [×2], C8⋊2C8 [×4], C2×C4×C8, C2×C4⋊C8 [×2], C2×C8⋊2C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8, C4⋊C8 [×4], C4.Q8 [×4], C8.C4 [×2], C2×C4⋊C4, C22×C8, C2×M4(2), C2×SD16 [×2], C8⋊2C8 [×4], C2×C4⋊C8, C2×C4.Q8, C2×C8.C4, C2×C8⋊2C8
►Regular action on 128 points
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 117)(10 118)(11 119)(12 120)(13 113)(14 114)(15 115)(16 116)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 81)(49 77)(50 78)(51 79)(52 80)(53 73)(54 74)(55 75)(56 76)(57 69)(58 70)(59 71)(60 72)(61 65)(62 66)(63 67)(64 68)(89 98)(90 99)(91 100)(92 101)(93 102)(94 103)(95 104)(96 97)(105 128)(106 121)(107 122)(108 123)(109 124)(110 125)(111 126)(112 127)
(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 65 96 127 39 16 86 75)(2 68 89 122 40 11 87 78)(3 71 90 125 33 14 88 73)(4 66 91 128 34 9 81 76)(5 69 92 123 35 12 82 79)(6 72 93 126 36 15 83 74)(7 67 94 121 37 10 84 77)(8 70 95 124 38 13 85 80)(17 57 101 108 25 120 41 51)(18 60 102 111 26 115 42 54)(19 63 103 106 27 118 43 49)(20 58 104 109 28 113 44 52)(21 61 97 112 29 116 45 55)(22 64 98 107 30 119 46 50)(23 59 99 110 31 114 47 53)(24 62 100 105 32 117 48 56)
G:=sub<Sym(128)| (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,117)(10,118)(11,119)(12,120)(13,113)(14,114)(15,115)(16,116)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,81)(49,77)(50,78)(51,79)(52,80)(53,73)(54,74)(55,75)(56,76)(57,69)(58,70)(59,71)(60,72)(61,65)(62,66)(63,67)(64,68)(89,98)(90,99)(91,100)(92,101)(93,102)(94,103)(95,104)(96,97)(105,128)(106,121)(107,122)(108,123)(109,124)(110,125)(111,126)(112,127), (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), (1,65,96,127,39,16,86,75)(2,68,89,122,40,11,87,78)(3,71,90,125,33,14,88,73)(4,66,91,128,34,9,81,76)(5,69,92,123,35,12,82,79)(6,72,93,126,36,15,83,74)(7,67,94,121,37,10,84,77)(8,70,95,124,38,13,85,80)(17,57,101,108,25,120,41,51)(18,60,102,111,26,115,42,54)(19,63,103,106,27,118,43,49)(20,58,104,109,28,113,44,52)(21,61,97,112,29,116,45,55)(22,64,98,107,30,119,46,50)(23,59,99,110,31,114,47,53)(24,62,100,105,32,117,48,56)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,117)(10,118)(11,119)(12,120)(13,113)(14,114)(15,115)(16,116)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,81)(49,77)(50,78)(51,79)(52,80)(53,73)(54,74)(55,75)(56,76)(57,69)(58,70)(59,71)(60,72)(61,65)(62,66)(63,67)(64,68)(89,98)(90,99)(91,100)(92,101)(93,102)(94,103)(95,104)(96,97)(105,128)(106,121)(107,122)(108,123)(109,124)(110,125)(111,126)(112,127), (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), (1,65,96,127,39,16,86,75)(2,68,89,122,40,11,87,78)(3,71,90,125,33,14,88,73)(4,66,91,128,34,9,81,76)(5,69,92,123,35,12,82,79)(6,72,93,126,36,15,83,74)(7,67,94,121,37,10,84,77)(8,70,95,124,38,13,85,80)(17,57,101,108,25,120,41,51)(18,60,102,111,26,115,42,54)(19,63,103,106,27,118,43,49)(20,58,104,109,28,113,44,52)(21,61,97,112,29,116,45,55)(22,64,98,107,30,119,46,50)(23,59,99,110,31,114,47,53)(24,62,100,105,32,117,48,56) );
G=PermutationGroup([(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,117),(10,118),(11,119),(12,120),(13,113),(14,114),(15,115),(16,116),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(31,33),(32,34),(41,82),(42,83),(43,84),(44,85),(45,86),(46,87),(47,88),(48,81),(49,77),(50,78),(51,79),(52,80),(53,73),(54,74),(55,75),(56,76),(57,69),(58,70),(59,71),(60,72),(61,65),(62,66),(63,67),(64,68),(89,98),(90,99),(91,100),(92,101),(93,102),(94,103),(95,104),(96,97),(105,128),(106,121),(107,122),(108,123),(109,124),(110,125),(111,126),(112,127)], [(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)], [(1,65,96,127,39,16,86,75),(2,68,89,122,40,11,87,78),(3,71,90,125,33,14,88,73),(4,66,91,128,34,9,81,76),(5,69,92,123,35,12,82,79),(6,72,93,126,36,15,83,74),(7,67,94,121,37,10,84,77),(8,70,95,124,38,13,85,80),(17,57,101,108,25,120,41,51),(18,60,102,111,26,115,42,54),(19,63,103,106,27,118,43,49),(20,58,104,109,28,113,44,52),(21,61,97,112,29,116,45,55),(22,64,98,107,30,119,46,50),(23,59,99,110,31,114,47,53),(24,62,100,105,32,117,48,56)])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 8A | ··· | 8P | 8Q | ··· | 8AF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D4 | Q8 | M4(2) | SD16 | C8.C4 |
| kernel | C2×C8⋊2C8 | C8⋊2C8 | C2×C4×C8 | C2×C4⋊C8 | C4×C8 | C22×C8 | C2×C8 | C42 | C42 | C22×C4 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 2 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 4 | 8 | 8 |
Matrix representation of C2×C8⋊2C8 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 8 |
| 2 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,2,0,0,0,0,8],[2,0,0,0,0,1,0,0,0,0,0,1,0,0,16,0] >;
C2×C8⋊2C8 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes_2C_8
% in TeX
G:=Group("C2xC8:2C8"); // GroupNames label
G:=SmallGroup(128,294);
// by ID
G=gap.SmallGroup(128,294);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,1123,136,172]);
// 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^3>;
// generators/relations