p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2.D8⋊5C4, C4⋊C4.14Q8, C4.26(C4×Q8), C2.8(C4×Q16), (C2×C4).48Q16, C2.5(D4.Q8), C2.8(D8⋊C4), C2.3(C4.Q16), C23.779(C2×D4), C22.159(C4×D4), (C22×C4).694D4, C22.34(C2×Q16), C22.64(C4○D8), (C22×C8).45C22, C4.14(C42.C2), C4.16(C42⋊2C2), C4.25(C42⋊C2), C22.83(C8⋊C22), C22.4Q16.10C2, (C2×C42).294C22, C22.78(C22⋊Q8), (C22×C4).1377C23, C2.4(C23.48D4), C2.5(C23.19D4), C22.7C42.21C2, C23.65C23.7C2, C22.91(C22.D4), C2.12(C23.63C23), (C4×C4⋊C4).17C2, C4⋊C4.79(C2×C4), (C2×C8).39(C2×C4), (C2×C2.D8).6C2, (C2×C4).272(C2×Q8), (C2×C4).574(C4○D4), (C2×C4⋊C4).772C22, (C2×C4).395(C22×C4), SmallGroup(128,653)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2.D8⋊5C4
G = < a,b,c,d | a2=b8=d4=1, c2=a, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab5, dcd-1=ab4c >
Subgroups: 220 in 115 conjugacy classes, 58 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C2.D8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.7C42, C22.4Q16, C4×C4⋊C4, C23.65C23, C2×C2.D8, C2.D8⋊5C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C2×Q16, C4○D8, C8⋊C22, C23.63C23, C4×Q16, D8⋊C4, C4.Q16, D4.Q8, C23.19D4, C23.48D4, C2.D8⋊5C4
►Regular action on 128 points
Generators in S128
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 109)(10 110)(11 111)(12 112)(13 105)(14 106)(15 107)(16 108)(17 102)(18 103)(19 104)(20 97)(21 98)(22 99)(23 100)(24 101)(25 94)(26 95)(27 96)(28 89)(29 90)(30 91)(31 92)(32 93)(33 80)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 88)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 65)(56 66)(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)
(1 110 63 10)(2 109 64 9)(3 108 57 16)(4 107 58 15)(5 106 59 14)(6 105 60 13)(7 112 61 12)(8 111 62 11)(17 86 102 47)(18 85 103 46)(19 84 104 45)(20 83 97 44)(21 82 98 43)(22 81 99 42)(23 88 100 41)(24 87 101 48)(25 78 94 39)(26 77 95 38)(27 76 96 37)(28 75 89 36)(29 74 90 35)(30 73 91 34)(31 80 92 33)(32 79 93 40)(49 116 67 123)(50 115 68 122)(51 114 69 121)(52 113 70 128)(53 120 71 127)(54 119 72 126)(55 118 65 125)(56 117 66 124)
(1 78 127 103)(2 36 128 23)(3 80 121 97)(4 38 122 17)(5 74 123 99)(6 40 124 19)(7 76 125 101)(8 34 126 21)(9 32 70 84)(10 90 71 42)(11 26 72 86)(12 92 65 44)(13 28 66 88)(14 94 67 46)(15 30 68 82)(16 96 69 48)(18 63 39 120)(20 57 33 114)(22 59 35 116)(24 61 37 118)(25 49 85 106)(27 51 87 108)(29 53 81 110)(31 55 83 112)(41 105 89 56)(43 107 91 50)(45 109 93 52)(47 111 95 54)(58 77 115 102)(60 79 117 104)(62 73 119 98)(64 75 113 100)
G:=sub<Sym(128)| (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,109)(10,110)(11,111)(12,112)(13,105)(14,106)(15,107)(16,108)(17,102)(18,103)(19,104)(20,97)(21,98)(22,99)(23,100)(24,101)(25,94)(26,95)(27,96)(28,89)(29,90)(30,91)(31,92)(32,93)(33,80)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,88)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,65)(56,66)(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), (1,110,63,10)(2,109,64,9)(3,108,57,16)(4,107,58,15)(5,106,59,14)(6,105,60,13)(7,112,61,12)(8,111,62,11)(17,86,102,47)(18,85,103,46)(19,84,104,45)(20,83,97,44)(21,82,98,43)(22,81,99,42)(23,88,100,41)(24,87,101,48)(25,78,94,39)(26,77,95,38)(27,76,96,37)(28,75,89,36)(29,74,90,35)(30,73,91,34)(31,80,92,33)(32,79,93,40)(49,116,67,123)(50,115,68,122)(51,114,69,121)(52,113,70,128)(53,120,71,127)(54,119,72,126)(55,118,65,125)(56,117,66,124), (1,78,127,103)(2,36,128,23)(3,80,121,97)(4,38,122,17)(5,74,123,99)(6,40,124,19)(7,76,125,101)(8,34,126,21)(9,32,70,84)(10,90,71,42)(11,26,72,86)(12,92,65,44)(13,28,66,88)(14,94,67,46)(15,30,68,82)(16,96,69,48)(18,63,39,120)(20,57,33,114)(22,59,35,116)(24,61,37,118)(25,49,85,106)(27,51,87,108)(29,53,81,110)(31,55,83,112)(41,105,89,56)(43,107,91,50)(45,109,93,52)(47,111,95,54)(58,77,115,102)(60,79,117,104)(62,73,119,98)(64,75,113,100)>;
G:=Group( (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,109)(10,110)(11,111)(12,112)(13,105)(14,106)(15,107)(16,108)(17,102)(18,103)(19,104)(20,97)(21,98)(22,99)(23,100)(24,101)(25,94)(26,95)(27,96)(28,89)(29,90)(30,91)(31,92)(32,93)(33,80)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,88)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,65)(56,66)(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), (1,110,63,10)(2,109,64,9)(3,108,57,16)(4,107,58,15)(5,106,59,14)(6,105,60,13)(7,112,61,12)(8,111,62,11)(17,86,102,47)(18,85,103,46)(19,84,104,45)(20,83,97,44)(21,82,98,43)(22,81,99,42)(23,88,100,41)(24,87,101,48)(25,78,94,39)(26,77,95,38)(27,76,96,37)(28,75,89,36)(29,74,90,35)(30,73,91,34)(31,80,92,33)(32,79,93,40)(49,116,67,123)(50,115,68,122)(51,114,69,121)(52,113,70,128)(53,120,71,127)(54,119,72,126)(55,118,65,125)(56,117,66,124), (1,78,127,103)(2,36,128,23)(3,80,121,97)(4,38,122,17)(5,74,123,99)(6,40,124,19)(7,76,125,101)(8,34,126,21)(9,32,70,84)(10,90,71,42)(11,26,72,86)(12,92,65,44)(13,28,66,88)(14,94,67,46)(15,30,68,82)(16,96,69,48)(18,63,39,120)(20,57,33,114)(22,59,35,116)(24,61,37,118)(25,49,85,106)(27,51,87,108)(29,53,81,110)(31,55,83,112)(41,105,89,56)(43,107,91,50)(45,109,93,52)(47,111,95,54)(58,77,115,102)(60,79,117,104)(62,73,119,98)(64,75,113,100) );
G=PermutationGroup([[(1,63),(2,64),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,109),(10,110),(11,111),(12,112),(13,105),(14,106),(15,107),(16,108),(17,102),(18,103),(19,104),(20,97),(21,98),(22,99),(23,100),(24,101),(25,94),(26,95),(27,96),(28,89),(29,90),(30,91),(31,92),(32,93),(33,80),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79),(41,88),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,65),(56,66),(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)], [(1,110,63,10),(2,109,64,9),(3,108,57,16),(4,107,58,15),(5,106,59,14),(6,105,60,13),(7,112,61,12),(8,111,62,11),(17,86,102,47),(18,85,103,46),(19,84,104,45),(20,83,97,44),(21,82,98,43),(22,81,99,42),(23,88,100,41),(24,87,101,48),(25,78,94,39),(26,77,95,38),(27,76,96,37),(28,75,89,36),(29,74,90,35),(30,73,91,34),(31,80,92,33),(32,79,93,40),(49,116,67,123),(50,115,68,122),(51,114,69,121),(52,113,70,128),(53,120,71,127),(54,119,72,126),(55,118,65,125),(56,117,66,124)], [(1,78,127,103),(2,36,128,23),(3,80,121,97),(4,38,122,17),(5,74,123,99),(6,40,124,19),(7,76,125,101),(8,34,126,21),(9,32,70,84),(10,90,71,42),(11,26,72,86),(12,92,65,44),(13,28,66,88),(14,94,67,46),(15,30,68,82),(16,96,69,48),(18,63,39,120),(20,57,33,114),(22,59,35,116),(24,61,37,118),(25,49,85,106),(27,51,87,108),(29,53,81,110),(31,55,83,112),(41,105,89,56),(43,107,91,50),(45,109,93,52),(47,111,95,54),(58,77,115,102),(60,79,117,104),(62,73,119,98),(64,75,113,100)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 4U | 4V | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | Q16 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | C2.D8⋊5C4 | C22.7C42 | C22.4Q16 | C4×C4⋊C4 | C23.65C23 | C2×C2.D8 | C2.D8 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 3 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 8 | 4 | 2 |
Matrix representation of C2.D8⋊5C4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 12 | 0 | 0 |
| 0 | 0 | 7 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[3,3,0,0,0,0,3,14,0,0,0,0,0,0,0,11,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,9],[12,12,0,0,0,0,12,5,0,0,0,0,0,0,11,7,0,0,0,0,12,6,0,0,0,0,0,0,0,2,0,0,0,0,8,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;
C2.D8⋊5C4 in GAP, Magma, Sage, TeX
C_2.D_8\rtimes_5C_4
% in TeX
G:=Group("C2.D8:5C4"); // GroupNames label
G:=SmallGroup(128,653);
// by ID
G=gap.SmallGroup(128,653);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,680,422,394,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=d^4=1,c^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=a*b^5,d*c*d^-1=a*b^4*c>;
// generators/relations