direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C4⋊C16, C42.11C8, C22.10M5(2), C8○(C4⋊C16), C4○(C4⋊C16), (C2×C4)⋊3C16, C4⋊2(C2×C16), (C4×C8).22C4, C4.28(C4⋊C8), C8.45(C4⋊C4), (C2×C8).66Q8, C8.40(C2×Q8), (C2×C8).403D4, C8.133(C2×D4), C23.45(C2×C8), (C2×C42).47C4, C2.2(C22×C16), (C22×C16).8C2, (C22×C8).30C4, (C22×C4).14C8, C2.4(C2×M5(2)), C22.23(C4⋊C8), C22.10(C2×C16), (C2×C8).626C23, (C4×C8).372C22, C42.327(C2×C4), (C2×C16).64C22, (C2×C4).93M4(2), C4.64(C2×M4(2)), C22.27(C22×C8), (C22×C8).593C22, C2.3(C2×C4⋊C8), (C2×C4×C8).27C2, (C2×C4)○(C4⋊C16), (C2×C8)○(C4⋊C16), C4.78(C2×C4⋊C4), (C2×C4).87(C2×C8), (C2×C8).215(C2×C4), (C2×C4).167(C4⋊C4), (C2×C4).611(C22×C4), (C22×C4).506(C2×C4), SmallGroup(128,881)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4⋊C16
G = < a,b,c | a2=b4=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 108 in 92 conjugacy classes, 76 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C2×C8, C2×C8, C2×C8, C22×C4, C4×C8, C2×C16, C2×C16, C2×C42, C22×C8, C4⋊C16, C2×C4×C8, C22×C16, C2×C4⋊C16
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C16, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4⋊C8, C2×C16, M5(2), C2×C4⋊C4, C22×C8, C2×M4(2), C4⋊C16, C2×C4⋊C8, C22×C16, C2×M5(2), C2×C4⋊C16
►Regular action on 128 points
Generators in S128
(1 107)(2 108)(3 109)(4 110)(5 111)(6 112)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 76)(34 77)(35 78)(36 79)(37 80)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 126)(50 127)(51 128)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 121)(61 122)(62 123)(63 124)(64 125)
(1 87 65 118)(2 119 66 88)(3 89 67 120)(4 121 68 90)(5 91 69 122)(6 123 70 92)(7 93 71 124)(8 125 72 94)(9 95 73 126)(10 127 74 96)(11 81 75 128)(12 113 76 82)(13 83 77 114)(14 115 78 84)(15 85 79 116)(16 117 80 86)(17 42 61 111)(18 112 62 43)(19 44 63 97)(20 98 64 45)(21 46 49 99)(22 100 50 47)(23 48 51 101)(24 102 52 33)(25 34 53 103)(26 104 54 35)(27 36 55 105)(28 106 56 37)(29 38 57 107)(30 108 58 39)(31 40 59 109)(32 110 60 41)
(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)
G:=sub<Sym(128)| (1,107)(2,108)(3,109)(4,110)(5,111)(6,112)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,76)(34,77)(35,78)(36,79)(37,80)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,126)(50,127)(51,128)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122)(62,123)(63,124)(64,125), (1,87,65,118)(2,119,66,88)(3,89,67,120)(4,121,68,90)(5,91,69,122)(6,123,70,92)(7,93,71,124)(8,125,72,94)(9,95,73,126)(10,127,74,96)(11,81,75,128)(12,113,76,82)(13,83,77,114)(14,115,78,84)(15,85,79,116)(16,117,80,86)(17,42,61,111)(18,112,62,43)(19,44,63,97)(20,98,64,45)(21,46,49,99)(22,100,50,47)(23,48,51,101)(24,102,52,33)(25,34,53,103)(26,104,54,35)(27,36,55,105)(28,106,56,37)(29,38,57,107)(30,108,58,39)(31,40,59,109)(32,110,60,41), (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)>;
G:=Group( (1,107)(2,108)(3,109)(4,110)(5,111)(6,112)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,76)(34,77)(35,78)(36,79)(37,80)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,126)(50,127)(51,128)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122)(62,123)(63,124)(64,125), (1,87,65,118)(2,119,66,88)(3,89,67,120)(4,121,68,90)(5,91,69,122)(6,123,70,92)(7,93,71,124)(8,125,72,94)(9,95,73,126)(10,127,74,96)(11,81,75,128)(12,113,76,82)(13,83,77,114)(14,115,78,84)(15,85,79,116)(16,117,80,86)(17,42,61,111)(18,112,62,43)(19,44,63,97)(20,98,64,45)(21,46,49,99)(22,100,50,47)(23,48,51,101)(24,102,52,33)(25,34,53,103)(26,104,54,35)(27,36,55,105)(28,106,56,37)(29,38,57,107)(30,108,58,39)(31,40,59,109)(32,110,60,41), (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) );
G=PermutationGroup([[(1,107),(2,108),(3,109),(4,110),(5,111),(6,112),(7,97),(8,98),(9,99),(10,100),(11,101),(12,102),(13,103),(14,104),(15,105),(16,106),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,81),(24,82),(25,83),(26,84),(27,85),(28,86),(29,87),(30,88),(31,89),(32,90),(33,76),(34,77),(35,78),(36,79),(37,80),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,73),(47,74),(48,75),(49,126),(50,127),(51,128),(52,113),(53,114),(54,115),(55,116),(56,117),(57,118),(58,119),(59,120),(60,121),(61,122),(62,123),(63,124),(64,125)], [(1,87,65,118),(2,119,66,88),(3,89,67,120),(4,121,68,90),(5,91,69,122),(6,123,70,92),(7,93,71,124),(8,125,72,94),(9,95,73,126),(10,127,74,96),(11,81,75,128),(12,113,76,82),(13,83,77,114),(14,115,78,84),(15,85,79,116),(16,117,80,86),(17,42,61,111),(18,112,62,43),(19,44,63,97),(20,98,64,45),(21,46,49,99),(22,100,50,47),(23,48,51,101),(24,102,52,33),(25,34,53,103),(26,104,54,35),(27,36,55,105),(28,106,56,37),(29,38,57,107),(30,108,58,39),(31,40,59,109),(32,110,60,41)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 8A | ··· | 8P | 8Q | ··· | 8X | 16A | ··· | 16AF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C16 | D4 | Q8 | M4(2) | M5(2) |
| kernel | C2×C4⋊C16 | C4⋊C16 | C2×C4×C8 | C22×C16 | C4×C8 | C2×C42 | C22×C8 | C42 | C22×C4 | C2×C4 | C2×C8 | C2×C8 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 2 | 8 | 8 | 32 | 2 | 2 | 4 | 8 |
Matrix representation of C2×C4⋊C16 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
| 14 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,13],[14,0,0,0,0,13,0,0,0,0,0,9,0,0,1,0] >;
C2×C4⋊C16 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes C_{16} % in TeX
G:=Group("C2xC4:C16"); // GroupNames label
G:=SmallGroup(128,881);
// by ID
G=gap.SmallGroup(128,881);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,102,124]);
// Polycyclic
G:=Group<a,b,c|a^2=b^4=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations