direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8⋊2Q8, C42.363D4, C42.717C23, C8⋊5(C2×Q8), (C2×C8)⋊12Q8, C4.20(C2×D8), (C2×C4).93D8, C4.14(C4⋊Q8), C4.14(C2×Q16), (C2×C4).44Q16, C4⋊C4.98C23, C2.12(C22×D8), C22.74(C2×D8), C4.10(C22×Q8), (C2×C8).563C23, (C2×C4).357C24, (C4×C8).409C22, (C22×C4).616D4, C23.884(C2×D4), C4⋊Q8.283C22, C22.46(C4⋊Q8), C2.12(C22×Q16), C22.51(C2×Q16), C2.D8.178C22, (C22×C8).538C22, C22.617(C22×D4), (C22×C4).1566C23, (C2×C42).1132C22, (C2×C4×C8).39C2, C2.27(C2×C4⋊Q8), (C2×C4⋊Q8).50C2, (C2×C4).861(C2×D4), (C2×C4).246(C2×Q8), (C2×C2.D8).30C2, (C2×C4⋊C4).630C22, SmallGroup(128,1891)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8⋊2Q8
G = < a,b,c,d | a2=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 372 in 212 conjugacy classes, 132 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C4×C8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C2×C4×C8, C2×C2.D8, C8⋊2Q8, C2×C4⋊Q8, C2×C8⋊2Q8
Quotients: C1, C2, C22, D4, Q8, C23, D8, Q16, C2×D4, C2×Q8, C24, C4⋊Q8, C2×D8, C2×Q16, C22×D4, C22×Q8, C8⋊2Q8, C2×C4⋊Q8, C22×D8, C22×Q16, C2×C8⋊2Q8
►Regular action on 128 points
Generators in S128
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(25 103)(26 104)(27 97)(28 98)(29 99)(30 100)(31 101)(32 102)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 73)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(65 110)(66 111)(67 112)(68 105)(69 106)(70 107)(71 108)(72 109)(81 96)(82 89)(83 90)(84 91)(85 92)(86 93)(87 94)(88 95)(113 128)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(120 127)
(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 59 43 36)(2 60 44 37)(3 61 45 38)(4 62 46 39)(5 63 47 40)(6 64 48 33)(7 57 41 34)(8 58 42 35)(9 78 24 53)(10 79 17 54)(11 80 18 55)(12 73 19 56)(13 74 20 49)(14 75 21 50)(15 76 22 51)(16 77 23 52)(25 67 115 94)(26 68 116 95)(27 69 117 96)(28 70 118 89)(29 71 119 90)(30 72 120 91)(31 65 113 92)(32 66 114 93)(81 97 106 124)(82 98 107 125)(83 99 108 126)(84 100 109 127)(85 101 110 128)(86 102 111 121)(87 103 112 122)(88 104 105 123)
(1 93 43 66)(2 92 44 65)(3 91 45 72)(4 90 46 71)(5 89 47 70)(6 96 48 69)(7 95 41 68)(8 94 42 67)(9 110 24 85)(10 109 17 84)(11 108 18 83)(12 107 19 82)(13 106 20 81)(14 105 21 88)(15 112 22 87)(16 111 23 86)(25 58 115 35)(26 57 116 34)(27 64 117 33)(28 63 118 40)(29 62 119 39)(30 61 120 38)(31 60 113 37)(32 59 114 36)(49 124 74 97)(50 123 75 104)(51 122 76 103)(52 121 77 102)(53 128 78 101)(54 127 79 100)(55 126 80 99)(56 125 73 98)
G:=sub<Sym(128)| (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(25,103)(26,104)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,73)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(65,110)(66,111)(67,112)(68,105)(69,106)(70,107)(71,108)(72,109)(81,96)(82,89)(83,90)(84,91)(85,92)(86,93)(87,94)(88,95)(113,128)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(120,127), (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), (1,59,43,36)(2,60,44,37)(3,61,45,38)(4,62,46,39)(5,63,47,40)(6,64,48,33)(7,57,41,34)(8,58,42,35)(9,78,24,53)(10,79,17,54)(11,80,18,55)(12,73,19,56)(13,74,20,49)(14,75,21,50)(15,76,22,51)(16,77,23,52)(25,67,115,94)(26,68,116,95)(27,69,117,96)(28,70,118,89)(29,71,119,90)(30,72,120,91)(31,65,113,92)(32,66,114,93)(81,97,106,124)(82,98,107,125)(83,99,108,126)(84,100,109,127)(85,101,110,128)(86,102,111,121)(87,103,112,122)(88,104,105,123), (1,93,43,66)(2,92,44,65)(3,91,45,72)(4,90,46,71)(5,89,47,70)(6,96,48,69)(7,95,41,68)(8,94,42,67)(9,110,24,85)(10,109,17,84)(11,108,18,83)(12,107,19,82)(13,106,20,81)(14,105,21,88)(15,112,22,87)(16,111,23,86)(25,58,115,35)(26,57,116,34)(27,64,117,33)(28,63,118,40)(29,62,119,39)(30,61,120,38)(31,60,113,37)(32,59,114,36)(49,124,74,97)(50,123,75,104)(51,122,76,103)(52,121,77,102)(53,128,78,101)(54,127,79,100)(55,126,80,99)(56,125,73,98)>;
G:=Group( (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(25,103)(26,104)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,73)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(65,110)(66,111)(67,112)(68,105)(69,106)(70,107)(71,108)(72,109)(81,96)(82,89)(83,90)(84,91)(85,92)(86,93)(87,94)(88,95)(113,128)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(120,127), (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), (1,59,43,36)(2,60,44,37)(3,61,45,38)(4,62,46,39)(5,63,47,40)(6,64,48,33)(7,57,41,34)(8,58,42,35)(9,78,24,53)(10,79,17,54)(11,80,18,55)(12,73,19,56)(13,74,20,49)(14,75,21,50)(15,76,22,51)(16,77,23,52)(25,67,115,94)(26,68,116,95)(27,69,117,96)(28,70,118,89)(29,71,119,90)(30,72,120,91)(31,65,113,92)(32,66,114,93)(81,97,106,124)(82,98,107,125)(83,99,108,126)(84,100,109,127)(85,101,110,128)(86,102,111,121)(87,103,112,122)(88,104,105,123), (1,93,43,66)(2,92,44,65)(3,91,45,72)(4,90,46,71)(5,89,47,70)(6,96,48,69)(7,95,41,68)(8,94,42,67)(9,110,24,85)(10,109,17,84)(11,108,18,83)(12,107,19,82)(13,106,20,81)(14,105,21,88)(15,112,22,87)(16,111,23,86)(25,58,115,35)(26,57,116,34)(27,64,117,33)(28,63,118,40)(29,62,119,39)(30,61,120,38)(31,60,113,37)(32,59,114,36)(49,124,74,97)(50,123,75,104)(51,122,76,103)(52,121,77,102)(53,128,78,101)(54,127,79,100)(55,126,80,99)(56,125,73,98) );
G=PermutationGroup([[(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,44),(10,45),(11,46),(12,47),(13,48),(14,41),(15,42),(16,43),(25,103),(26,104),(27,97),(28,98),(29,99),(30,100),(31,101),(32,102),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,73),(49,64),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63),(65,110),(66,111),(67,112),(68,105),(69,106),(70,107),(71,108),(72,109),(81,96),(82,89),(83,90),(84,91),(85,92),(86,93),(87,94),(88,95),(113,128),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(120,127)], [(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)], [(1,59,43,36),(2,60,44,37),(3,61,45,38),(4,62,46,39),(5,63,47,40),(6,64,48,33),(7,57,41,34),(8,58,42,35),(9,78,24,53),(10,79,17,54),(11,80,18,55),(12,73,19,56),(13,74,20,49),(14,75,21,50),(15,76,22,51),(16,77,23,52),(25,67,115,94),(26,68,116,95),(27,69,117,96),(28,70,118,89),(29,71,119,90),(30,72,120,91),(31,65,113,92),(32,66,114,93),(81,97,106,124),(82,98,107,125),(83,99,108,126),(84,100,109,127),(85,101,110,128),(86,102,111,121),(87,103,112,122),(88,104,105,123)], [(1,93,43,66),(2,92,44,65),(3,91,45,72),(4,90,46,71),(5,89,47,70),(6,96,48,69),(7,95,41,68),(8,94,42,67),(9,110,24,85),(10,109,17,84),(11,108,18,83),(12,107,19,82),(13,106,20,81),(14,105,21,88),(15,112,22,87),(16,111,23,86),(25,58,115,35),(26,57,116,34),(27,64,117,33),(28,63,118,40),(29,62,119,39),(30,61,120,38),(31,60,113,37),(32,59,114,36),(49,124,74,97),(50,123,75,104),(51,122,76,103),(52,121,77,102),(53,128,78,101),(54,127,79,100),(55,126,80,99),(56,125,73,98)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | D8 | Q16 |
| kernel | C2×C8⋊2Q8 | C2×C4×C8 | C2×C2.D8 | C8⋊2Q8 | C2×C4⋊Q8 | C42 | C2×C8 | C22×C4 | C2×C4 | C2×C4 |
| # reps | 1 | 1 | 4 | 8 | 2 | 2 | 8 | 2 | 8 | 8 |
Matrix representation of C2×C8⋊2Q8 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 15 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 3 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 15 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 3 | 0 | 0 |
| 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 5 | 12 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,1,0,0,0,15,1,0,0,0,0,0,3,3,0,0,0,14,3],[1,0,0,0,0,0,16,1,0,0,0,15,1,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,9,1,0,0,0,3,8,0,0,0,0,0,5,5,0,0,0,5,12] >;
C2×C8⋊2Q8 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes_2Q_8
% in TeX
G:=Group("C2xC8:2Q8"); // GroupNames label
G:=SmallGroup(128,1891);
// by ID
G=gap.SmallGroup(128,1891);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,520,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations