direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C16⋊5C4, C42.5C8, C8.22C42, C22.8M5(2), (C2×C16)⋊16C4, C16⋊12(C2×C4), C8.24(C2×C8), C4.15(C4×C8), (C4×C8).36C4, (C2×C8).17C8, C8○(C16⋊5C4), C4○(C16⋊5C4), (C22×C4).9C8, C4.35(C2×C42), C22.17(C4×C8), C8.65(C22×C4), (C22×C8).45C4, (C2×C4).90C42, C4.34(C22×C8), (C2×C42).39C4, C23.44(C2×C8), C2.1(C2×M5(2)), C42.296(C2×C4), (C22×C16).16C2, (C2×C8).620C23, (C4×C8).433C22, (C2×C16).103C22, C22.23(C22×C8), (C22×C8).591C22, C2.10(C2×C4×C8), (C2×C4×C8).64C2, (C2×C4).63(C2×C8), (C2×C8)○(C16⋊5C4), (C2×C4)○(C16⋊5C4), (C2×C8).245(C2×C4), (C22×C4).505(C2×C4), (C2×C4).605(C22×C4), SmallGroup(128,838)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C16⋊5C4
G = < a,b,c | a2=b16=c4=1, ab=ba, ac=ca, cbc-1=b9 >
Subgroups: 108 in 100 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], C23, C16 [×8], C42 [×4], C2×C8 [×2], C2×C8 [×10], C22×C4, C22×C4 [×2], C4×C8 [×4], C2×C16 [×12], C2×C42, C22×C8 [×2], C16⋊5C4 [×4], C2×C4×C8, C22×C16 [×2], C2×C16⋊5C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], C22×C4 [×3], C4×C8 [×4], M5(2) [×4], C2×C42, C22×C8 [×2], C16⋊5C4 [×4], C2×C4×C8, C2×M5(2) [×2], C2×C16⋊5C4
►Regular action on 128 points
Generators in S128
(1 62)(2 63)(3 64)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 102)(18 103)(19 104)(20 105)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 97)(29 98)(30 99)(31 100)(32 101)(33 125)(34 126)(35 127)(36 128)(37 113)(38 114)(39 115)(40 116)(41 117)(42 118)(43 119)(44 120)(45 121)(46 122)(47 123)(48 124)(65 95)(66 96)(67 81)(68 82)(69 83)(70 84)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)
(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)
(1 119 91 107)(2 128 92 100)(3 121 93 109)(4 114 94 102)(5 123 95 111)(6 116 96 104)(7 125 81 97)(8 118 82 106)(9 127 83 99)(10 120 84 108)(11 113 85 101)(12 122 86 110)(13 115 87 103)(14 124 88 112)(15 117 89 105)(16 126 90 98)(17 49 38 80)(18 58 39 73)(19 51 40 66)(20 60 41 75)(21 53 42 68)(22 62 43 77)(23 55 44 70)(24 64 45 79)(25 57 46 72)(26 50 47 65)(27 59 48 74)(28 52 33 67)(29 61 34 76)(30 54 35 69)(31 63 36 78)(32 56 37 71)
G:=sub<Sym(128)| (1,62)(2,63)(3,64)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,102)(18,103)(19,104)(20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,97)(29,98)(30,99)(31,100)(32,101)(33,125)(34,126)(35,127)(36,128)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,121)(46,122)(47,123)(48,124)(65,95)(66,96)(67,81)(68,82)(69,83)(70,84)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94), (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), (1,119,91,107)(2,128,92,100)(3,121,93,109)(4,114,94,102)(5,123,95,111)(6,116,96,104)(7,125,81,97)(8,118,82,106)(9,127,83,99)(10,120,84,108)(11,113,85,101)(12,122,86,110)(13,115,87,103)(14,124,88,112)(15,117,89,105)(16,126,90,98)(17,49,38,80)(18,58,39,73)(19,51,40,66)(20,60,41,75)(21,53,42,68)(22,62,43,77)(23,55,44,70)(24,64,45,79)(25,57,46,72)(26,50,47,65)(27,59,48,74)(28,52,33,67)(29,61,34,76)(30,54,35,69)(31,63,36,78)(32,56,37,71)>;
G:=Group( (1,62)(2,63)(3,64)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,102)(18,103)(19,104)(20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,97)(29,98)(30,99)(31,100)(32,101)(33,125)(34,126)(35,127)(36,128)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,121)(46,122)(47,123)(48,124)(65,95)(66,96)(67,81)(68,82)(69,83)(70,84)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94), (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), (1,119,91,107)(2,128,92,100)(3,121,93,109)(4,114,94,102)(5,123,95,111)(6,116,96,104)(7,125,81,97)(8,118,82,106)(9,127,83,99)(10,120,84,108)(11,113,85,101)(12,122,86,110)(13,115,87,103)(14,124,88,112)(15,117,89,105)(16,126,90,98)(17,49,38,80)(18,58,39,73)(19,51,40,66)(20,60,41,75)(21,53,42,68)(22,62,43,77)(23,55,44,70)(24,64,45,79)(25,57,46,72)(26,50,47,65)(27,59,48,74)(28,52,33,67)(29,61,34,76)(30,54,35,69)(31,63,36,78)(32,56,37,71) );
G=PermutationGroup([(1,62),(2,63),(3,64),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,61),(17,102),(18,103),(19,104),(20,105),(21,106),(22,107),(23,108),(24,109),(25,110),(26,111),(27,112),(28,97),(29,98),(30,99),(31,100),(32,101),(33,125),(34,126),(35,127),(36,128),(37,113),(38,114),(39,115),(40,116),(41,117),(42,118),(43,119),(44,120),(45,121),(46,122),(47,123),(48,124),(65,95),(66,96),(67,81),(68,82),(69,83),(70,84),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94)], [(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)], [(1,119,91,107),(2,128,92,100),(3,121,93,109),(4,114,94,102),(5,123,95,111),(6,116,96,104),(7,125,81,97),(8,118,82,106),(9,127,83,99),(10,120,84,108),(11,113,85,101),(12,122,86,110),(13,115,87,103),(14,124,88,112),(15,117,89,105),(16,126,90,98),(17,49,38,80),(18,58,39,73),(19,51,40,66),(20,60,41,75),(21,53,42,68),(22,62,43,77),(23,55,44,70),(24,64,45,79),(25,57,46,72),(26,50,47,65),(27,59,48,74),(28,52,33,67),(29,61,34,76),(30,54,35,69),(31,63,36,78),(32,56,37,71)])
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 | 1 | 2 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | C8 | C8 | M5(2) |
| kernel | C2×C16⋊5C4 | C16⋊5C4 | C2×C4×C8 | C22×C16 | C4×C8 | C2×C16 | C2×C42 | C22×C8 | C42 | C2×C8 | C22×C4 | C22 |
| # reps | 1 | 4 | 1 | 2 | 4 | 16 | 2 | 2 | 8 | 16 | 8 | 16 |
Matrix representation of C2×C16⋊5C4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 9 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 2 | 15 |
| 0 | 0 | 15 | 15 |
| 4 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[9,0,0,0,0,16,0,0,0,0,2,15,0,0,15,15],[4,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0] >;
C2×C16⋊5C4 in GAP, Magma, Sage, TeX
C_2\times C_{16}\rtimes_5C_4 % in TeX
G:=Group("C2xC16:5C4"); // GroupNames label
G:=SmallGroup(128,838);
// by ID
G=gap.SmallGroup(128,838);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,925,120,136,124]);
// 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^9>;
// generators/relations