direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C2×C16⋊4C4, C23.60D8, C22.12SD32, (C2×C16)⋊15C4, C16⋊11(C2×C4), (C2×C4).80D8, C8.24(C4⋊C4), (C2×C8).46Q8, C8.15(C2×Q8), C4.6(C2×Q16), (C2×C8).249D4, (C2×C4).40Q16, C2.3(C2×SD32), C8.52(C22×C4), C4.17(C2.D8), C22.58(C2×D8), (C22×C16).15C2, (C2×C16).97C22, (C2×C8).499C23, (C22×C4).587D4, C2.D8.147C22, C22.23(C2.D8), (C22×C8).529C22, C4.51(C2×C4⋊C4), C2.12(C2×C2.D8), (C2×C8).223(C2×C4), (C2×C4).763(C2×D4), (C2×C2.D8).24C2, (C2×C4).143(C4⋊C4), SmallGroup(128,889)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C16⋊4C4
G = < a,b,c | a2=b16=c4=1, ab=ba, ac=ca, cbc-1=b7 >
Subgroups: 172 in 84 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.D8, C2.D8, C2×C16, C2×C4⋊C4, C22×C8, C16⋊4C4, C2×C2.D8, C22×C16, C2×C16⋊4C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C2.D8, SD32, C2×C4⋊C4, C2×D8, C2×Q16, C16⋊4C4, C2×C2.D8, C2×SD32, C2×C16⋊4C4
►Regular action on 128 points
Generators in S128
(1 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 81)(13 82)(14 83)(15 84)(16 85)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 121)(25 122)(26 123)(27 124)(28 125)(29 126)(30 127)(31 128)(32 113)(33 80)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 97)(64 98)
(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)
(1 112 26 75)(2 103 27 66)(3 110 28 73)(4 101 29 80)(5 108 30 71)(6 99 31 78)(7 106 32 69)(8 97 17 76)(9 104 18 67)(10 111 19 74)(11 102 20 65)(12 109 21 72)(13 100 22 79)(14 107 23 70)(15 98 24 77)(16 105 25 68)(33 89 51 126)(34 96 52 117)(35 87 53 124)(36 94 54 115)(37 85 55 122)(38 92 56 113)(39 83 57 120)(40 90 58 127)(41 81 59 118)(42 88 60 125)(43 95 61 116)(44 86 62 123)(45 93 63 114)(46 84 64 121)(47 91 49 128)(48 82 50 119)
G:=sub<Sym(128)| (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,81)(13,82)(14,83)(15,84)(16,85)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,121)(25,122)(26,123)(27,124)(28,125)(29,126)(30,127)(31,128)(32,113)(33,80)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,97)(64,98), (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), (1,112,26,75)(2,103,27,66)(3,110,28,73)(4,101,29,80)(5,108,30,71)(6,99,31,78)(7,106,32,69)(8,97,17,76)(9,104,18,67)(10,111,19,74)(11,102,20,65)(12,109,21,72)(13,100,22,79)(14,107,23,70)(15,98,24,77)(16,105,25,68)(33,89,51,126)(34,96,52,117)(35,87,53,124)(36,94,54,115)(37,85,55,122)(38,92,56,113)(39,83,57,120)(40,90,58,127)(41,81,59,118)(42,88,60,125)(43,95,61,116)(44,86,62,123)(45,93,63,114)(46,84,64,121)(47,91,49,128)(48,82,50,119)>;
G:=Group( (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,81)(13,82)(14,83)(15,84)(16,85)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,121)(25,122)(26,123)(27,124)(28,125)(29,126)(30,127)(31,128)(32,113)(33,80)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,97)(64,98), (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), (1,112,26,75)(2,103,27,66)(3,110,28,73)(4,101,29,80)(5,108,30,71)(6,99,31,78)(7,106,32,69)(8,97,17,76)(9,104,18,67)(10,111,19,74)(11,102,20,65)(12,109,21,72)(13,100,22,79)(14,107,23,70)(15,98,24,77)(16,105,25,68)(33,89,51,126)(34,96,52,117)(35,87,53,124)(36,94,54,115)(37,85,55,122)(38,92,56,113)(39,83,57,120)(40,90,58,127)(41,81,59,118)(42,88,60,125)(43,95,61,116)(44,86,62,123)(45,93,63,114)(46,84,64,121)(47,91,49,128)(48,82,50,119) );
G=PermutationGroup([[(1,86),(2,87),(3,88),(4,89),(5,90),(6,91),(7,92),(8,93),(9,94),(10,95),(11,96),(12,81),(13,82),(14,83),(15,84),(16,85),(17,114),(18,115),(19,116),(20,117),(21,118),(22,119),(23,120),(24,121),(25,122),(26,123),(27,124),(28,125),(29,126),(30,127),(31,128),(32,113),(33,80),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71),(41,72),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(49,99),(50,100),(51,101),(52,102),(53,103),(54,104),(55,105),(56,106),(57,107),(58,108),(59,109),(60,110),(61,111),(62,112),(63,97),(64,98)], [(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)], [(1,112,26,75),(2,103,27,66),(3,110,28,73),(4,101,29,80),(5,108,30,71),(6,99,31,78),(7,106,32,69),(8,97,17,76),(9,104,18,67),(10,111,19,74),(11,102,20,65),(12,109,21,72),(13,100,22,79),(14,107,23,70),(15,98,24,77),(16,105,25,68),(33,89,51,126),(34,96,52,117),(35,87,53,124),(36,94,54,115),(37,85,55,122),(38,92,56,113),(39,83,57,120),(40,90,58,127),(41,81,59,118),(42,88,60,125),(43,95,61,116),(44,86,62,123),(45,93,63,114),(46,84,64,121),(47,91,49,128),(48,82,50,119)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D8 | Q16 | D8 | SD32 |
| kernel | C2×C16⋊4C4 | C16⋊4C4 | C2×C2.D8 | C22×C16 | C2×C16 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 1 | 2 | 1 | 2 | 4 | 2 | 16 |
Matrix representation of C2×C16⋊4C4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 14 | 0 |
| 0 | 0 | 0 | 6 |
| 4 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,14,0,0,0,0,6],[4,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0] >;
C2×C16⋊4C4 in GAP, Magma, Sage, TeX
C_2\times C_{16}\rtimes_4C_4 % in TeX
G:=Group("C2xC16:4C4"); // GroupNames label
G:=SmallGroup(128,889);
// by ID
G=gap.SmallGroup(128,889);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,736,1123,360,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^2=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^7>;
// generators/relations