direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4⋊Q16, C42.358D4, C42.712C23, (C2×C4)⋊7Q16, C4⋊1(C2×Q16), C8.51(C2×D4), (C2×C8).260D4, C4.3(C22×D4), C4.13(C4⋊1D4), (C4×C8).408C22, (C2×C8).559C23, (C2×C4).343C24, C23.878(C2×D4), (C22×C4).613D4, C4⋊Q8.278C22, (C22×Q16).9C2, (C2×Q8).97C23, C2.11(C22×Q16), C22.50(C2×Q16), C22.49(C4⋊1D4), (C22×C8).536C22, (C2×Q16).125C22, C22.603(C22×D4), (C22×C4).1558C23, (C2×C42).1127C22, (C22×Q8).305C22, (C2×C4×C8).38C2, (C2×C4⋊Q8).48C2, (C2×C4).853(C2×D4), C2.22(C2×C4⋊1D4), SmallGroup(128,1877)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4⋊Q16
G = < a,b,c,d | a2=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 468 in 276 conjugacy classes, 132 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C2×C42, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×C8, C2×Q16, C2×Q16, C22×Q8, C2×C4×C8, C4⋊Q16, C2×C4⋊Q8, C22×Q16, C2×C4⋊Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C24, C4⋊1D4, C2×Q16, C22×D4, C4⋊Q16, C2×C4⋊1D4, C22×Q16, C2×C4⋊Q16
►Regular action on 128 points
Generators in S128
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 80)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(41 106)(42 107)(43 108)(44 109)(45 110)(46 111)(47 112)(48 105)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 81)(89 119)(90 120)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 122)(98 123)(99 124)(100 125)(101 126)(102 127)(103 128)(104 121)
(1 61 23 26)(2 62 24 27)(3 63 17 28)(4 64 18 29)(5 57 19 30)(6 58 20 31)(7 59 21 32)(8 60 22 25)(9 36 80 54)(10 37 73 55)(11 38 74 56)(12 39 75 49)(13 40 76 50)(14 33 77 51)(15 34 78 52)(16 35 79 53)(41 69 99 114)(42 70 100 115)(43 71 101 116)(44 72 102 117)(45 65 103 118)(46 66 104 119)(47 67 97 120)(48 68 98 113)(81 127 95 109)(82 128 96 110)(83 121 89 111)(84 122 90 112)(85 123 91 105)(86 124 92 106)(87 125 93 107)(88 126 94 108)
(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 72 5 68)(2 71 6 67)(3 70 7 66)(4 69 8 65)(9 110 13 106)(10 109 14 105)(11 108 15 112)(12 107 16 111)(17 115 21 119)(18 114 22 118)(19 113 23 117)(20 120 24 116)(25 103 29 99)(26 102 30 98)(27 101 31 97)(28 100 32 104)(33 91 37 95)(34 90 38 94)(35 89 39 93)(36 96 40 92)(41 60 45 64)(42 59 46 63)(43 58 47 62)(44 57 48 61)(49 87 53 83)(50 86 54 82)(51 85 55 81)(52 84 56 88)(73 127 77 123)(74 126 78 122)(75 125 79 121)(76 124 80 128)
G:=sub<Sym(128)| (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,80)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,105)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,81)(89,119)(90,120)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,122)(98,123)(99,124)(100,125)(101,126)(102,127)(103,128)(104,121), (1,61,23,26)(2,62,24,27)(3,63,17,28)(4,64,18,29)(5,57,19,30)(6,58,20,31)(7,59,21,32)(8,60,22,25)(9,36,80,54)(10,37,73,55)(11,38,74,56)(12,39,75,49)(13,40,76,50)(14,33,77,51)(15,34,78,52)(16,35,79,53)(41,69,99,114)(42,70,100,115)(43,71,101,116)(44,72,102,117)(45,65,103,118)(46,66,104,119)(47,67,97,120)(48,68,98,113)(81,127,95,109)(82,128,96,110)(83,121,89,111)(84,122,90,112)(85,123,91,105)(86,124,92,106)(87,125,93,107)(88,126,94,108), (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), (1,72,5,68)(2,71,6,67)(3,70,7,66)(4,69,8,65)(9,110,13,106)(10,109,14,105)(11,108,15,112)(12,107,16,111)(17,115,21,119)(18,114,22,118)(19,113,23,117)(20,120,24,116)(25,103,29,99)(26,102,30,98)(27,101,31,97)(28,100,32,104)(33,91,37,95)(34,90,38,94)(35,89,39,93)(36,96,40,92)(41,60,45,64)(42,59,46,63)(43,58,47,62)(44,57,48,61)(49,87,53,83)(50,86,54,82)(51,85,55,81)(52,84,56,88)(73,127,77,123)(74,126,78,122)(75,125,79,121)(76,124,80,128)>;
G:=Group( (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,80)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,105)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,81)(89,119)(90,120)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,122)(98,123)(99,124)(100,125)(101,126)(102,127)(103,128)(104,121), (1,61,23,26)(2,62,24,27)(3,63,17,28)(4,64,18,29)(5,57,19,30)(6,58,20,31)(7,59,21,32)(8,60,22,25)(9,36,80,54)(10,37,73,55)(11,38,74,56)(12,39,75,49)(13,40,76,50)(14,33,77,51)(15,34,78,52)(16,35,79,53)(41,69,99,114)(42,70,100,115)(43,71,101,116)(44,72,102,117)(45,65,103,118)(46,66,104,119)(47,67,97,120)(48,68,98,113)(81,127,95,109)(82,128,96,110)(83,121,89,111)(84,122,90,112)(85,123,91,105)(86,124,92,106)(87,125,93,107)(88,126,94,108), (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), (1,72,5,68)(2,71,6,67)(3,70,7,66)(4,69,8,65)(9,110,13,106)(10,109,14,105)(11,108,15,112)(12,107,16,111)(17,115,21,119)(18,114,22,118)(19,113,23,117)(20,120,24,116)(25,103,29,99)(26,102,30,98)(27,101,31,97)(28,100,32,104)(33,91,37,95)(34,90,38,94)(35,89,39,93)(36,96,40,92)(41,60,45,64)(42,59,46,63)(43,58,47,62)(44,57,48,61)(49,87,53,83)(50,86,54,82)(51,85,55,81)(52,84,56,88)(73,127,77,123)(74,126,78,122)(75,125,79,121)(76,124,80,128) );
G=PermutationGroup([[(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,60),(10,61),(11,62),(12,63),(13,64),(14,57),(15,58),(16,59),(17,39),(18,40),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,80),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(41,106),(42,107),(43,108),(44,109),(45,110),(46,111),(47,112),(48,105),(65,82),(66,83),(67,84),(68,85),(69,86),(70,87),(71,88),(72,81),(89,119),(90,120),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118),(97,122),(98,123),(99,124),(100,125),(101,126),(102,127),(103,128),(104,121)], [(1,61,23,26),(2,62,24,27),(3,63,17,28),(4,64,18,29),(5,57,19,30),(6,58,20,31),(7,59,21,32),(8,60,22,25),(9,36,80,54),(10,37,73,55),(11,38,74,56),(12,39,75,49),(13,40,76,50),(14,33,77,51),(15,34,78,52),(16,35,79,53),(41,69,99,114),(42,70,100,115),(43,71,101,116),(44,72,102,117),(45,65,103,118),(46,66,104,119),(47,67,97,120),(48,68,98,113),(81,127,95,109),(82,128,96,110),(83,121,89,111),(84,122,90,112),(85,123,91,105),(86,124,92,106),(87,125,93,107),(88,126,94,108)], [(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)], [(1,72,5,68),(2,71,6,67),(3,70,7,66),(4,69,8,65),(9,110,13,106),(10,109,14,105),(11,108,15,112),(12,107,16,111),(17,115,21,119),(18,114,22,118),(19,113,23,117),(20,120,24,116),(25,103,29,99),(26,102,30,98),(27,101,31,97),(28,100,32,104),(33,91,37,95),(34,90,38,94),(35,89,39,93),(36,96,40,92),(41,60,45,64),(42,59,46,63),(43,58,47,62),(44,57,48,61),(49,87,53,83),(50,86,54,82),(51,85,55,81),(52,84,56,88),(73,127,77,123),(74,126,78,122),(75,125,79,121),(76,124,80,128)]])
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 |
| type | + | + | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | Q16 |
| kernel | C2×C4⋊Q16 | C2×C4×C8 | C4⋊Q16 | C2×C4⋊Q8 | C22×Q16 | C42 | C2×C8 | C22×C4 | C2×C4 |
| # reps | 1 | 1 | 8 | 2 | 4 | 2 | 8 | 2 | 16 |
Matrix representation of C2×C4⋊Q16 ►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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 8 | 4 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 12 | 15 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 2 | 0 | 0 |
| 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 16 |
| 0 | 0 | 0 | 0 | 1 |
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],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,13,8,0,0,0,0,4],[16,0,0,0,0,0,8,12,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,16,0,0,0,2,16,0,0,0,0,0,16,0,0,0,0,16,1] >;
C2×C4⋊Q16 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes Q_{16} % in TeX
G:=Group("C2xC4:Q16"); // GroupNames label
G:=SmallGroup(128,1877);
// by ID
G=gap.SmallGroup(128,1877);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,184,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=c^4,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