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, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, C22×C4, C4×C8, C4⋊C8, C4⋊C8, C2×C42, C22×C8, C22×C8, C8⋊2C8, C2×C4×C8, C2×C4⋊C8, C2×C8⋊2C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×Q8, C4⋊C8, C4.Q8, C8.C4, C2×C4⋊C4, C22×C8, C2×M4(2), C2×SD16, C8⋊2C8, C2×C4⋊C8, C2×C4.Q8, C2×C8.C4, C2×C8⋊2C8
►Regular action on 128 points
Generators in S128
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 33)(9 117)(10 118)(11 119)(12 120)(13 113)(14 114)(15 115)(16 116)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(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)(89 102)(90 103)(91 104)(92 97)(93 98)(94 99)(95 100)(96 101)(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 65 103 118 21 49 86 123)(2 68 104 113 22 52 87 126)(3 71 97 116 23 55 88 121)(4 66 98 119 24 50 81 124)(5 69 99 114 17 53 82 127)(6 72 100 117 18 56 83 122)(7 67 101 120 19 51 84 125)(8 70 102 115 20 54 85 128)(9 30 76 42 107 39 60 95)(10 25 77 45 108 34 61 90)(11 28 78 48 109 37 62 93)(12 31 79 43 110 40 63 96)(13 26 80 46 111 35 64 91)(14 29 73 41 112 38 57 94)(15 32 74 44 105 33 58 89)(16 27 75 47 106 36 59 92)
G:=sub<Sym(128)| (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,33)(9,117)(10,118)(11,119)(12,120)(13,113)(14,114)(15,115)(16,116)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(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)(89,102)(90,103)(91,104)(92,97)(93,98)(94,99)(95,100)(96,101)(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), (1,65,103,118,21,49,86,123)(2,68,104,113,22,52,87,126)(3,71,97,116,23,55,88,121)(4,66,98,119,24,50,81,124)(5,69,99,114,17,53,82,127)(6,72,100,117,18,56,83,122)(7,67,101,120,19,51,84,125)(8,70,102,115,20,54,85,128)(9,30,76,42,107,39,60,95)(10,25,77,45,108,34,61,90)(11,28,78,48,109,37,62,93)(12,31,79,43,110,40,63,96)(13,26,80,46,111,35,64,91)(14,29,73,41,112,38,57,94)(15,32,74,44,105,33,58,89)(16,27,75,47,106,36,59,92)>;
G:=Group( (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,33)(9,117)(10,118)(11,119)(12,120)(13,113)(14,114)(15,115)(16,116)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(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)(89,102)(90,103)(91,104)(92,97)(93,98)(94,99)(95,100)(96,101)(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), (1,65,103,118,21,49,86,123)(2,68,104,113,22,52,87,126)(3,71,97,116,23,55,88,121)(4,66,98,119,24,50,81,124)(5,69,99,114,17,53,82,127)(6,72,100,117,18,56,83,122)(7,67,101,120,19,51,84,125)(8,70,102,115,20,54,85,128)(9,30,76,42,107,39,60,95)(10,25,77,45,108,34,61,90)(11,28,78,48,109,37,62,93)(12,31,79,43,110,40,63,96)(13,26,80,46,111,35,64,91)(14,29,73,41,112,38,57,94)(15,32,74,44,105,33,58,89)(16,27,75,47,106,36,59,92) );
G=PermutationGroup([[(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,33),(9,117),(10,118),(11,119),(12,120),(13,113),(14,114),(15,115),(16,116),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28),(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),(89,102),(90,103),(91,104),(92,97),(93,98),(94,99),(95,100),(96,101),(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)], [(1,65,103,118,21,49,86,123),(2,68,104,113,22,52,87,126),(3,71,97,116,23,55,88,121),(4,66,98,119,24,50,81,124),(5,69,99,114,17,53,82,127),(6,72,100,117,18,56,83,122),(7,67,101,120,19,51,84,125),(8,70,102,115,20,54,85,128),(9,30,76,42,107,39,60,95),(10,25,77,45,108,34,61,90),(11,28,78,48,109,37,62,93),(12,31,79,43,110,40,63,96),(13,26,80,46,111,35,64,91),(14,29,73,41,112,38,57,94),(15,32,74,44,105,33,58,89),(16,27,75,47,106,36,59,92)]])
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