direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8.5Q8, C42.362D4, C42.716C23, C8.22(C2×Q8), (C2×C8).47Q8, C4.13(C4⋊Q8), C4.9(C22×Q8), C4⋊C4.97C23, (C2×C4).356C24, (C2×C8).562C23, (C4×C8).417C22, (C22×C4).567D4, C23.883(C2×D4), C22.45(C4⋊Q8), C22.99(C4○D8), C4.Q8.158C22, C2.D8.177C22, (C22×C8).568C22, C22.616(C22×D4), (C2×C42).1131C22, (C22×C4).1565C23, C42.C2.114C22, (C2×C4×C8).47C2, C2.26(C2×C4⋊Q8), C2.32(C2×C4○D8), (C2×C4).696(C2×D4), (C2×C4).245(C2×Q8), (C2×C2.D8).29C2, (C2×C4.Q8).34C2, (C2×C4⋊C4).629C22, (C2×C42.C2).33C2, SmallGroup(128,1890)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8.5Q8
G = < a,b,c,d | a2=b8=c4=1, d2=b4c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b4c-1 >
Subgroups: 276 in 180 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C4×C8, C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C42.C2, C22×C8, C2×C4×C8, C2×C4.Q8, C2×C2.D8, C8.5Q8, C2×C42.C2, C2×C8.5Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C4⋊Q8, C4○D8, C22×D4, C22×Q8, C8.5Q8, C2×C4⋊Q8, C2×C4○D8, C2×C8.5Q8
►Regular action on 128 points
Generators in S128
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 104)(26 97)(27 98)(28 99)(29 100)(30 101)(31 102)(32 103)(33 80)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)(65 108)(66 109)(67 110)(68 111)(69 112)(70 105)(71 106)(72 107)(81 94)(82 95)(83 96)(84 89)(85 90)(86 91)(87 92)(88 93)(113 126)(114 127)(115 128)(116 121)(117 122)(118 123)(119 124)(120 125)
(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 34)(2 60 44 35)(3 61 45 36)(4 62 46 37)(5 63 47 38)(6 64 48 39)(7 57 41 40)(8 58 42 33)(9 80 18 53)(10 73 19 54)(11 74 20 55)(12 75 21 56)(13 76 22 49)(14 77 23 50)(15 78 24 51)(16 79 17 52)(25 68 120 89)(26 69 113 90)(27 70 114 91)(28 71 115 92)(29 72 116 93)(30 65 117 94)(31 66 118 95)(32 67 119 96)(81 101 108 122)(82 102 109 123)(83 103 110 124)(84 104 111 125)(85 97 112 126)(86 98 105 127)(87 99 106 128)(88 100 107 121)
(1 93 47 68)(2 96 48 71)(3 91 41 66)(4 94 42 69)(5 89 43 72)(6 92 44 67)(7 95 45 70)(8 90 46 65)(9 112 22 81)(10 107 23 84)(11 110 24 87)(12 105 17 82)(13 108 18 85)(14 111 19 88)(15 106 20 83)(16 109 21 86)(25 63 116 34)(26 58 117 37)(27 61 118 40)(28 64 119 35)(29 59 120 38)(30 62 113 33)(31 57 114 36)(32 60 115 39)(49 126 80 101)(50 121 73 104)(51 124 74 99)(52 127 75 102)(53 122 76 97)(54 125 77 100)(55 128 78 103)(56 123 79 98)
G:=sub<Sym(128)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,104)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,80)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61)(65,108)(66,109)(67,110)(68,111)(69,112)(70,105)(71,106)(72,107)(81,94)(82,95)(83,96)(84,89)(85,90)(86,91)(87,92)(88,93)(113,126)(114,127)(115,128)(116,121)(117,122)(118,123)(119,124)(120,125), (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), (1,59,43,34)(2,60,44,35)(3,61,45,36)(4,62,46,37)(5,63,47,38)(6,64,48,39)(7,57,41,40)(8,58,42,33)(9,80,18,53)(10,73,19,54)(11,74,20,55)(12,75,21,56)(13,76,22,49)(14,77,23,50)(15,78,24,51)(16,79,17,52)(25,68,120,89)(26,69,113,90)(27,70,114,91)(28,71,115,92)(29,72,116,93)(30,65,117,94)(31,66,118,95)(32,67,119,96)(81,101,108,122)(82,102,109,123)(83,103,110,124)(84,104,111,125)(85,97,112,126)(86,98,105,127)(87,99,106,128)(88,100,107,121), (1,93,47,68)(2,96,48,71)(3,91,41,66)(4,94,42,69)(5,89,43,72)(6,92,44,67)(7,95,45,70)(8,90,46,65)(9,112,22,81)(10,107,23,84)(11,110,24,87)(12,105,17,82)(13,108,18,85)(14,111,19,88)(15,106,20,83)(16,109,21,86)(25,63,116,34)(26,58,117,37)(27,61,118,40)(28,64,119,35)(29,59,120,38)(30,62,113,33)(31,57,114,36)(32,60,115,39)(49,126,80,101)(50,121,73,104)(51,124,74,99)(52,127,75,102)(53,122,76,97)(54,125,77,100)(55,128,78,103)(56,123,79,98)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,104)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,80)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61)(65,108)(66,109)(67,110)(68,111)(69,112)(70,105)(71,106)(72,107)(81,94)(82,95)(83,96)(84,89)(85,90)(86,91)(87,92)(88,93)(113,126)(114,127)(115,128)(116,121)(117,122)(118,123)(119,124)(120,125), (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), (1,59,43,34)(2,60,44,35)(3,61,45,36)(4,62,46,37)(5,63,47,38)(6,64,48,39)(7,57,41,40)(8,58,42,33)(9,80,18,53)(10,73,19,54)(11,74,20,55)(12,75,21,56)(13,76,22,49)(14,77,23,50)(15,78,24,51)(16,79,17,52)(25,68,120,89)(26,69,113,90)(27,70,114,91)(28,71,115,92)(29,72,116,93)(30,65,117,94)(31,66,118,95)(32,67,119,96)(81,101,108,122)(82,102,109,123)(83,103,110,124)(84,104,111,125)(85,97,112,126)(86,98,105,127)(87,99,106,128)(88,100,107,121), (1,93,47,68)(2,96,48,71)(3,91,41,66)(4,94,42,69)(5,89,43,72)(6,92,44,67)(7,95,45,70)(8,90,46,65)(9,112,22,81)(10,107,23,84)(11,110,24,87)(12,105,17,82)(13,108,18,85)(14,111,19,88)(15,106,20,83)(16,109,21,86)(25,63,116,34)(26,58,117,37)(27,61,118,40)(28,64,119,35)(29,59,120,38)(30,62,113,33)(31,57,114,36)(32,60,115,39)(49,126,80,101)(50,121,73,104)(51,124,74,99)(52,127,75,102)(53,122,76,97)(54,125,77,100)(55,128,78,103)(56,123,79,98) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(25,104),(26,97),(27,98),(28,99),(29,100),(30,101),(31,102),(32,103),(33,80),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79),(49,62),(50,63),(51,64),(52,57),(53,58),(54,59),(55,60),(56,61),(65,108),(66,109),(67,110),(68,111),(69,112),(70,105),(71,106),(72,107),(81,94),(82,95),(83,96),(84,89),(85,90),(86,91),(87,92),(88,93),(113,126),(114,127),(115,128),(116,121),(117,122),(118,123),(119,124),(120,125)], [(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)], [(1,59,43,34),(2,60,44,35),(3,61,45,36),(4,62,46,37),(5,63,47,38),(6,64,48,39),(7,57,41,40),(8,58,42,33),(9,80,18,53),(10,73,19,54),(11,74,20,55),(12,75,21,56),(13,76,22,49),(14,77,23,50),(15,78,24,51),(16,79,17,52),(25,68,120,89),(26,69,113,90),(27,70,114,91),(28,71,115,92),(29,72,116,93),(30,65,117,94),(31,66,118,95),(32,67,119,96),(81,101,108,122),(82,102,109,123),(83,103,110,124),(84,104,111,125),(85,97,112,126),(86,98,105,127),(87,99,106,128),(88,100,107,121)], [(1,93,47,68),(2,96,48,71),(3,91,41,66),(4,94,42,69),(5,89,43,72),(6,92,44,67),(7,95,45,70),(8,90,46,65),(9,112,22,81),(10,107,23,84),(11,110,24,87),(12,105,17,82),(13,108,18,85),(14,111,19,88),(15,106,20,83),(16,109,21,86),(25,63,116,34),(26,58,117,37),(27,61,118,40),(28,64,119,35),(29,59,120,38),(30,62,113,33),(31,57,114,36),(32,60,115,39),(49,126,80,101),(50,121,73,104),(51,124,74,99),(52,127,75,102),(53,122,76,97),(54,125,77,100),(55,128,78,103),(56,123,79,98)]])
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 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | C4○D8 |
| kernel | C2×C8.5Q8 | C2×C4×C8 | C2×C4.Q8 | C2×C2.D8 | C8.5Q8 | C2×C42.C2 | C42 | C2×C8 | C22×C4 | C22 |
| # reps | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 8 | 2 | 16 |
Matrix representation of C2×C8.5Q8 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 5 | 5 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,5,5,0,0,0,12,5],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,13],[16,0,0,0,0,0,12,12,0,0,0,12,5,0,0,0,0,0,1,0,0,0,0,0,16] >;
C2×C8.5Q8 in GAP, Magma, Sage, TeX
C_2\times C_8._5Q_8
% in TeX
G:=Group("C2xC8.5Q8"); // GroupNames label
G:=SmallGroup(128,1890);
// by ID
G=gap.SmallGroup(128,1890);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,568,758,184,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^4*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^4*c^-1>;
// generators/relations