direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4.Q16, C42.217D4, C42.330C23, Q8⋊2(C2×Q8), (C2×Q8)⋊14Q8, (C2×C4).64Q16, C4.45(C2×Q16), C4⋊C4.37C23, C2.7(C22×Q16), C4.25(C22×Q8), C4⋊C8.282C22, (C2×C4).272C24, (C2×C8).137C23, (C22×C4).798D4, C23.868(C2×D4), C4⋊Q8.259C22, C4.65(C22⋊Q8), C22.48(C2×Q16), (C2×Q8).362C23, (C4×Q8).294C22, C2.D8.164C22, (C22×C8).143C22, (C2×C42).818C22, C22.532(C22×D4), C22.119(C8⋊C22), (C22×C4).1542C23, Q8⋊C4.147C22, C22.100(C22⋊Q8), (C22×Q8).472C22, (C2×C4⋊C8).42C2, (C2×C4×Q8).50C2, C4.82(C2×C4○D4), (C2×C4⋊Q8).44C2, C2.22(C2×C8⋊C22), (C2×C4).320(C2×Q8), (C2×C2.D8).27C2, C2.53(C2×C22⋊Q8), (C2×C4).1434(C2×D4), (C2×C4).838(C4○D4), (C2×C4⋊C4).601C22, (C2×Q8⋊C4).24C2, SmallGroup(128,1806)
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, cbc-1=b-1, bd=db, dcd-1=b2c-1 >
Subgroups: 348 in 208 conjugacy classes, 116 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, Q8⋊C4, C4⋊C8, C2.D8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4×Q8, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C22×Q8, C2×Q8⋊C4, C2×C4⋊C8, C2×C2.D8, C4.Q16, C2×C4×Q8, C2×C4⋊Q8, C2×C4.Q16
Quotients: C1, C2, C22, D4, Q8, C23, Q16, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C2×Q16, C8⋊C22, C22×D4, C22×Q8, C2×C4○D4, C4.Q16, C2×C22⋊Q8, C22×Q16, C2×C8⋊C22, C2×C4.Q16
►Regular action on 128 points
Generators in S128
(1 56)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 123)(18 124)(19 125)(20 126)(21 127)(22 128)(23 121)(24 122)(25 100)(26 101)(27 102)(28 103)(29 104)(30 97)(31 98)(32 99)(33 93)(34 94)(35 95)(36 96)(37 89)(38 90)(39 91)(40 92)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 65)(64 66)(81 110)(82 111)(83 112)(84 105)(85 106)(86 107)(87 108)(88 109)
(1 21 95 86)(2 87 96 22)(3 23 89 88)(4 81 90 24)(5 17 91 82)(6 83 92 18)(7 19 93 84)(8 85 94 20)(9 70 41 101)(10 102 42 71)(11 72 43 103)(12 104 44 65)(13 66 45 97)(14 98 46 67)(15 68 47 99)(16 100 48 69)(25 80 59 120)(26 113 60 73)(27 74 61 114)(28 115 62 75)(29 76 63 116)(30 117 64 77)(31 78 57 118)(32 119 58 79)(33 105 54 125)(34 126 55 106)(35 107 56 127)(36 128 49 108)(37 109 50 121)(38 122 51 110)(39 111 52 123)(40 124 53 112)
(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 62 5 58)(2 27 6 31)(3 60 7 64)(4 25 8 29)(9 105 13 109)(10 124 14 128)(11 111 15 107)(12 122 16 126)(17 79 21 75)(18 118 22 114)(19 77 23 73)(20 116 24 120)(26 93 30 89)(28 91 32 95)(33 97 37 101)(34 65 38 69)(35 103 39 99)(36 71 40 67)(41 125 45 121)(42 112 46 108)(43 123 47 127)(44 110 48 106)(49 102 53 98)(50 70 54 66)(51 100 55 104)(52 68 56 72)(57 96 61 92)(59 94 63 90)(74 83 78 87)(76 81 80 85)(82 119 86 115)(84 117 88 113)
G:=sub<Sym(128)| (1,56)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,123)(18,124)(19,125)(20,126)(21,127)(22,128)(23,121)(24,122)(25,100)(26,101)(27,102)(28,103)(29,104)(30,97)(31,98)(32,99)(33,93)(34,94)(35,95)(36,96)(37,89)(38,90)(39,91)(40,92)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,65)(64,66)(81,110)(82,111)(83,112)(84,105)(85,106)(86,107)(87,108)(88,109), (1,21,95,86)(2,87,96,22)(3,23,89,88)(4,81,90,24)(5,17,91,82)(6,83,92,18)(7,19,93,84)(8,85,94,20)(9,70,41,101)(10,102,42,71)(11,72,43,103)(12,104,44,65)(13,66,45,97)(14,98,46,67)(15,68,47,99)(16,100,48,69)(25,80,59,120)(26,113,60,73)(27,74,61,114)(28,115,62,75)(29,76,63,116)(30,117,64,77)(31,78,57,118)(32,119,58,79)(33,105,54,125)(34,126,55,106)(35,107,56,127)(36,128,49,108)(37,109,50,121)(38,122,51,110)(39,111,52,123)(40,124,53,112), (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), (1,62,5,58)(2,27,6,31)(3,60,7,64)(4,25,8,29)(9,105,13,109)(10,124,14,128)(11,111,15,107)(12,122,16,126)(17,79,21,75)(18,118,22,114)(19,77,23,73)(20,116,24,120)(26,93,30,89)(28,91,32,95)(33,97,37,101)(34,65,38,69)(35,103,39,99)(36,71,40,67)(41,125,45,121)(42,112,46,108)(43,123,47,127)(44,110,48,106)(49,102,53,98)(50,70,54,66)(51,100,55,104)(52,68,56,72)(57,96,61,92)(59,94,63,90)(74,83,78,87)(76,81,80,85)(82,119,86,115)(84,117,88,113)>;
G:=Group( (1,56)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,123)(18,124)(19,125)(20,126)(21,127)(22,128)(23,121)(24,122)(25,100)(26,101)(27,102)(28,103)(29,104)(30,97)(31,98)(32,99)(33,93)(34,94)(35,95)(36,96)(37,89)(38,90)(39,91)(40,92)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,65)(64,66)(81,110)(82,111)(83,112)(84,105)(85,106)(86,107)(87,108)(88,109), (1,21,95,86)(2,87,96,22)(3,23,89,88)(4,81,90,24)(5,17,91,82)(6,83,92,18)(7,19,93,84)(8,85,94,20)(9,70,41,101)(10,102,42,71)(11,72,43,103)(12,104,44,65)(13,66,45,97)(14,98,46,67)(15,68,47,99)(16,100,48,69)(25,80,59,120)(26,113,60,73)(27,74,61,114)(28,115,62,75)(29,76,63,116)(30,117,64,77)(31,78,57,118)(32,119,58,79)(33,105,54,125)(34,126,55,106)(35,107,56,127)(36,128,49,108)(37,109,50,121)(38,122,51,110)(39,111,52,123)(40,124,53,112), (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), (1,62,5,58)(2,27,6,31)(3,60,7,64)(4,25,8,29)(9,105,13,109)(10,124,14,128)(11,111,15,107)(12,122,16,126)(17,79,21,75)(18,118,22,114)(19,77,23,73)(20,116,24,120)(26,93,30,89)(28,91,32,95)(33,97,37,101)(34,65,38,69)(35,103,39,99)(36,71,40,67)(41,125,45,121)(42,112,46,108)(43,123,47,127)(44,110,48,106)(49,102,53,98)(50,70,54,66)(51,100,55,104)(52,68,56,72)(57,96,61,92)(59,94,63,90)(74,83,78,87)(76,81,80,85)(82,119,86,115)(84,117,88,113) );
G=PermutationGroup([[(1,56),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,113),(10,114),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,123),(18,124),(19,125),(20,126),(21,127),(22,128),(23,121),(24,122),(25,100),(26,101),(27,102),(28,103),(29,104),(30,97),(31,98),(32,99),(33,93),(34,94),(35,95),(36,96),(37,89),(38,90),(39,91),(40,92),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(57,67),(58,68),(59,69),(60,70),(61,71),(62,72),(63,65),(64,66),(81,110),(82,111),(83,112),(84,105),(85,106),(86,107),(87,108),(88,109)], [(1,21,95,86),(2,87,96,22),(3,23,89,88),(4,81,90,24),(5,17,91,82),(6,83,92,18),(7,19,93,84),(8,85,94,20),(9,70,41,101),(10,102,42,71),(11,72,43,103),(12,104,44,65),(13,66,45,97),(14,98,46,67),(15,68,47,99),(16,100,48,69),(25,80,59,120),(26,113,60,73),(27,74,61,114),(28,115,62,75),(29,76,63,116),(30,117,64,77),(31,78,57,118),(32,119,58,79),(33,105,54,125),(34,126,55,106),(35,107,56,127),(36,128,49,108),(37,109,50,121),(38,122,51,110),(39,111,52,123),(40,124,53,112)], [(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)], [(1,62,5,58),(2,27,6,31),(3,60,7,64),(4,25,8,29),(9,105,13,109),(10,124,14,128),(11,111,15,107),(12,122,16,126),(17,79,21,75),(18,118,22,114),(19,77,23,73),(20,116,24,120),(26,93,30,89),(28,91,32,95),(33,97,37,101),(34,65,38,69),(35,103,39,99),(36,71,40,67),(41,125,45,121),(42,112,46,108),(43,123,47,127),(44,110,48,106),(49,102,53,98),(50,70,54,66),(51,100,55,104),(52,68,56,72),(57,96,61,92),(59,94,63,90),(74,83,78,87),(76,81,80,85),(82,119,86,115),(84,117,88,113)]])
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 | C2 | D4 | D4 | Q8 | Q16 | C4○D4 | C8⋊C22 |
| kernel | C2×C4.Q16 | C2×Q8⋊C4 | C2×C4⋊C8 | C2×C2.D8 | C4.Q16 | C2×C4×Q8 | C2×C4⋊Q8 | C42 | C22×C4 | C2×Q8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | 8 | 4 | 2 |
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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 11 |
| 0 | 0 | 0 | 3 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 6 | 15 |
| 0 | 0 | 0 | 10 | 11 |
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],[1,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,11,3,0,0,0,11,0],[1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,6,10,0,0,0,15,11] >;
C2×C4.Q16 in GAP, Magma, Sage, TeX
C_2\times C_4.Q_{16} % in TeX
G:=Group("C2xC4.Q16"); // GroupNames label
G:=SmallGroup(128,1806);
// by ID
G=gap.SmallGroup(128,1806);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,352,4037,1027,124]);
// 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=b^-1,b*d=d*b,d*c*d^-1=b^2*c^-1>;
// generators/relations