p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4)⋊2Q16, (C2×C8).42D4, (C2×Q8).85D4, C4.12C22≀C2, C4.48(C4⋊D4), C2.8(C4⋊Q16), C23.904(C2×D4), (C22×C4).310D4, (C22×Q16).3C2, C22.53(C2×Q16), C2.12(C4⋊2Q16), C2.15(C8.2D4), C22.206C22≀C2, C22.77(C4⋊1D4), C2.21(C23⋊2D4), (C22×C8).109C22, (C2×C42).352C22, C2.19(C22⋊Q16), (C22×Q8).57C22, C22.225(C4⋊D4), (C22×C4).1438C23, C22.120(C8.C22), C22.7C42.23C2, C23.67C23.12C2, (C2×C4⋊Q8).18C2, (C2×C4).747(C2×D4), (C2×C4).877(C4○D4), (C2×C4⋊C4).111C22, (C2×Q8⋊C4).20C2, SmallGroup(128,748)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4)⋊2Q16
G = < a,b,c,d | a2=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 360 in 185 conjugacy classes, 58 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C22×C8, C2×Q16, C22×Q8, C22×Q8, C22.7C42, C23.67C23, C2×Q8⋊C4, C2×C4⋊Q8, C22×Q16, (C2×C4)⋊2Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C4○D4, C22≀C2, C4⋊D4, C4⋊1D4, C2×Q16, C8.C22, C23⋊2D4, C22⋊Q16, C4⋊2Q16, C4⋊Q16, C8.2D4, (C2×C4)⋊2Q16
►Regular action on 128 points
Generators in S128
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 73)(16 74)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 25)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 89)(57 102)(58 103)(59 104)(60 97)(61 98)(62 99)(63 100)(64 101)(81 126)(82 127)(83 128)(84 121)(85 122)(86 123)(87 124)(88 125)
(1 92 111 117)(2 46 112 52)(3 94 105 119)(4 48 106 54)(5 96 107 113)(6 42 108 56)(7 90 109 115)(8 44 110 50)(9 31 83 102)(10 58 84 23)(11 25 85 104)(12 60 86 17)(13 27 87 98)(14 62 88 19)(15 29 81 100)(16 64 82 21)(18 124 61 79)(20 126 63 73)(22 128 57 75)(24 122 59 77)(26 78 97 123)(28 80 99 125)(30 74 101 127)(32 76 103 121)(33 47 67 53)(34 95 68 120)(35 41 69 55)(36 89 70 114)(37 43 71 49)(38 91 72 116)(39 45 65 51)(40 93 66 118)
(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 28 5 32)(2 27 6 31)(3 26 7 30)(4 25 8 29)(9 46 13 42)(10 45 14 41)(11 44 15 48)(12 43 16 47)(17 71 21 67)(18 70 22 66)(19 69 23 65)(20 68 24 72)(33 60 37 64)(34 59 38 63)(35 58 39 62)(36 57 40 61)(49 82 53 86)(50 81 54 85)(51 88 55 84)(52 87 56 83)(73 120 77 116)(74 119 78 115)(75 118 79 114)(76 117 80 113)(89 128 93 124)(90 127 94 123)(91 126 95 122)(92 125 96 121)(97 109 101 105)(98 108 102 112)(99 107 103 111)(100 106 104 110)
G:=sub<Sym(128)| (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,73)(16,74)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,25)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,89)(57,102)(58,103)(59,104)(60,97)(61,98)(62,99)(63,100)(64,101)(81,126)(82,127)(83,128)(84,121)(85,122)(86,123)(87,124)(88,125), (1,92,111,117)(2,46,112,52)(3,94,105,119)(4,48,106,54)(5,96,107,113)(6,42,108,56)(7,90,109,115)(8,44,110,50)(9,31,83,102)(10,58,84,23)(11,25,85,104)(12,60,86,17)(13,27,87,98)(14,62,88,19)(15,29,81,100)(16,64,82,21)(18,124,61,79)(20,126,63,73)(22,128,57,75)(24,122,59,77)(26,78,97,123)(28,80,99,125)(30,74,101,127)(32,76,103,121)(33,47,67,53)(34,95,68,120)(35,41,69,55)(36,89,70,114)(37,43,71,49)(38,91,72,116)(39,45,65,51)(40,93,66,118), (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), (1,28,5,32)(2,27,6,31)(3,26,7,30)(4,25,8,29)(9,46,13,42)(10,45,14,41)(11,44,15,48)(12,43,16,47)(17,71,21,67)(18,70,22,66)(19,69,23,65)(20,68,24,72)(33,60,37,64)(34,59,38,63)(35,58,39,62)(36,57,40,61)(49,82,53,86)(50,81,54,85)(51,88,55,84)(52,87,56,83)(73,120,77,116)(74,119,78,115)(75,118,79,114)(76,117,80,113)(89,128,93,124)(90,127,94,123)(91,126,95,122)(92,125,96,121)(97,109,101,105)(98,108,102,112)(99,107,103,111)(100,106,104,110)>;
G:=Group( (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,73)(16,74)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,25)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,89)(57,102)(58,103)(59,104)(60,97)(61,98)(62,99)(63,100)(64,101)(81,126)(82,127)(83,128)(84,121)(85,122)(86,123)(87,124)(88,125), (1,92,111,117)(2,46,112,52)(3,94,105,119)(4,48,106,54)(5,96,107,113)(6,42,108,56)(7,90,109,115)(8,44,110,50)(9,31,83,102)(10,58,84,23)(11,25,85,104)(12,60,86,17)(13,27,87,98)(14,62,88,19)(15,29,81,100)(16,64,82,21)(18,124,61,79)(20,126,63,73)(22,128,57,75)(24,122,59,77)(26,78,97,123)(28,80,99,125)(30,74,101,127)(32,76,103,121)(33,47,67,53)(34,95,68,120)(35,41,69,55)(36,89,70,114)(37,43,71,49)(38,91,72,116)(39,45,65,51)(40,93,66,118), (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), (1,28,5,32)(2,27,6,31)(3,26,7,30)(4,25,8,29)(9,46,13,42)(10,45,14,41)(11,44,15,48)(12,43,16,47)(17,71,21,67)(18,70,22,66)(19,69,23,65)(20,68,24,72)(33,60,37,64)(34,59,38,63)(35,58,39,62)(36,57,40,61)(49,82,53,86)(50,81,54,85)(51,88,55,84)(52,87,56,83)(73,120,77,116)(74,119,78,115)(75,118,79,114)(76,117,80,113)(89,128,93,124)(90,127,94,123)(91,126,95,122)(92,125,96,121)(97,109,101,105)(98,108,102,112)(99,107,103,111)(100,106,104,110) );
G=PermutationGroup([[(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,73),(16,74),(17,26),(18,27),(19,28),(20,29),(21,30),(22,31),(23,32),(24,25),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(49,90),(50,91),(51,92),(52,93),(53,94),(54,95),(55,96),(56,89),(57,102),(58,103),(59,104),(60,97),(61,98),(62,99),(63,100),(64,101),(81,126),(82,127),(83,128),(84,121),(85,122),(86,123),(87,124),(88,125)], [(1,92,111,117),(2,46,112,52),(3,94,105,119),(4,48,106,54),(5,96,107,113),(6,42,108,56),(7,90,109,115),(8,44,110,50),(9,31,83,102),(10,58,84,23),(11,25,85,104),(12,60,86,17),(13,27,87,98),(14,62,88,19),(15,29,81,100),(16,64,82,21),(18,124,61,79),(20,126,63,73),(22,128,57,75),(24,122,59,77),(26,78,97,123),(28,80,99,125),(30,74,101,127),(32,76,103,121),(33,47,67,53),(34,95,68,120),(35,41,69,55),(36,89,70,114),(37,43,71,49),(38,91,72,116),(39,45,65,51),(40,93,66,118)], [(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)], [(1,28,5,32),(2,27,6,31),(3,26,7,30),(4,25,8,29),(9,46,13,42),(10,45,14,41),(11,44,15,48),(12,43,16,47),(17,71,21,67),(18,70,22,66),(19,69,23,65),(20,68,24,72),(33,60,37,64),(34,59,38,63),(35,58,39,62),(36,57,40,61),(49,82,53,86),(50,81,54,85),(51,88,55,84),(52,87,56,83),(73,120,77,116),(74,119,78,115),(75,118,79,114),(76,117,80,113),(89,128,93,124),(90,127,94,123),(91,126,95,122),(92,125,96,121),(97,109,101,105),(98,108,102,112),(99,107,103,111),(100,106,104,110)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | Q16 | C4○D4 | C8.C22 |
| kernel | (C2×C4)⋊2Q16 | C22.7C42 | C23.67C23 | C2×Q8⋊C4 | C2×C4⋊Q8 | C22×Q16 | C2×C8 | C22×C4 | C2×Q8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 6 | 8 | 2 | 2 |
Matrix representation of (C2×C4)⋊2Q16 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 10 | 16 | 0 | 0 | 0 | 0 |
| 16 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[10,16,0,0,0,0,16,7,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
(C2×C4)⋊2Q16 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_2Q_{16} % in TeX
G:=Group("(C2xC4):2Q16"); // GroupNames label
G:=SmallGroup(128,748);
// by ID
G=gap.SmallGroup(128,748);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,456,422,387,352,2019,1018,521,248,2804,1411,718,172]);
// 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,c*b*c^-1=a*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations