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, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C2×C8, C2×C8, C22×C4, C22×C4, C4×C8, C2×C16, C2×C42, C22×C8, C16⋊5C4, C2×C4×C8, C22×C16, C2×C16⋊5C4
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, C4×C8, M5(2), C2×C42, C22×C8, C16⋊5C4, C2×C4×C8, C2×M5(2), C2×C16⋊5C4
►Regular action on 128 points
Generators in S128
(1 44)(2 45)(3 46)(4 47)(5 48)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 63)(18 64)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(65 118)(66 119)(67 120)(68 121)(69 122)(70 123)(71 124)(72 125)(73 126)(74 127)(75 128)(76 113)(77 114)(78 115)(79 116)(80 117)(81 110)(82 111)(83 112)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)
(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 91 74 23)(2 84 75 32)(3 93 76 25)(4 86 77 18)(5 95 78 27)(6 88 79 20)(7 81 80 29)(8 90 65 22)(9 83 66 31)(10 92 67 24)(11 85 68 17)(12 94 69 26)(13 87 70 19)(14 96 71 28)(15 89 72 21)(16 82 73 30)(33 101 116 50)(34 110 117 59)(35 103 118 52)(36 112 119 61)(37 105 120 54)(38 98 121 63)(39 107 122 56)(40 100 123 49)(41 109 124 58)(42 102 125 51)(43 111 126 60)(44 104 127 53)(45 97 128 62)(46 106 113 55)(47 99 114 64)(48 108 115 57)
G:=sub<Sym(128)| (1,44)(2,45)(3,46)(4,47)(5,48)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,63)(18,64)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(65,118)(66,119)(67,120)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(75,128)(76,113)(77,114)(78,115)(79,116)(80,117)(81,110)(82,111)(83,112)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104)(92,105)(93,106)(94,107)(95,108)(96,109), (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), (1,91,74,23)(2,84,75,32)(3,93,76,25)(4,86,77,18)(5,95,78,27)(6,88,79,20)(7,81,80,29)(8,90,65,22)(9,83,66,31)(10,92,67,24)(11,85,68,17)(12,94,69,26)(13,87,70,19)(14,96,71,28)(15,89,72,21)(16,82,73,30)(33,101,116,50)(34,110,117,59)(35,103,118,52)(36,112,119,61)(37,105,120,54)(38,98,121,63)(39,107,122,56)(40,100,123,49)(41,109,124,58)(42,102,125,51)(43,111,126,60)(44,104,127,53)(45,97,128,62)(46,106,113,55)(47,99,114,64)(48,108,115,57)>;
G:=Group( (1,44)(2,45)(3,46)(4,47)(5,48)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,63)(18,64)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(65,118)(66,119)(67,120)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(75,128)(76,113)(77,114)(78,115)(79,116)(80,117)(81,110)(82,111)(83,112)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104)(92,105)(93,106)(94,107)(95,108)(96,109), (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), (1,91,74,23)(2,84,75,32)(3,93,76,25)(4,86,77,18)(5,95,78,27)(6,88,79,20)(7,81,80,29)(8,90,65,22)(9,83,66,31)(10,92,67,24)(11,85,68,17)(12,94,69,26)(13,87,70,19)(14,96,71,28)(15,89,72,21)(16,82,73,30)(33,101,116,50)(34,110,117,59)(35,103,118,52)(36,112,119,61)(37,105,120,54)(38,98,121,63)(39,107,122,56)(40,100,123,49)(41,109,124,58)(42,102,125,51)(43,111,126,60)(44,104,127,53)(45,97,128,62)(46,106,113,55)(47,99,114,64)(48,108,115,57) );
G=PermutationGroup([[(1,44),(2,45),(3,46),(4,47),(5,48),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,63),(18,64),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(65,118),(66,119),(67,120),(68,121),(69,122),(70,123),(71,124),(72,125),(73,126),(74,127),(75,128),(76,113),(77,114),(78,115),(79,116),(80,117),(81,110),(82,111),(83,112),(84,97),(85,98),(86,99),(87,100),(88,101),(89,102),(90,103),(91,104),(92,105),(93,106),(94,107),(95,108),(96,109)], [(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)], [(1,91,74,23),(2,84,75,32),(3,93,76,25),(4,86,77,18),(5,95,78,27),(6,88,79,20),(7,81,80,29),(8,90,65,22),(9,83,66,31),(10,92,67,24),(11,85,68,17),(12,94,69,26),(13,87,70,19),(14,96,71,28),(15,89,72,21),(16,82,73,30),(33,101,116,50),(34,110,117,59),(35,103,118,52),(36,112,119,61),(37,105,120,54),(38,98,121,63),(39,107,122,56),(40,100,123,49),(41,109,124,58),(42,102,125,51),(43,111,126,60),(44,104,127,53),(45,97,128,62),(46,106,113,55),(47,99,114,64),(48,108,115,57)]])
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