direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C2×C16⋊3C4, C23.59D8, C22.6Q32, C22.13D16, (C2×C16)⋊13C4, C16⋊10(C2×C4), (C2×C4).79D8, C2.2(C2×D16), C8.23(C4⋊C4), C8.14(C2×Q8), C4.5(C2×Q16), (C2×C8).45Q8, C2.2(C2×Q32), (C2×C8).248D4, (C2×C4).39Q16, C8.51(C22×C4), C4.16(C2.D8), C22.57(C2×D8), (C22×C16).10C2, (C2×C8).498C23, (C2×C16).79C22, (C22×C4).586D4, C2.D8.146C22, C22.22(C2.D8), (C22×C8).528C22, C4.50(C2×C4⋊C4), C2.11(C2×C2.D8), (C2×C8).222(C2×C4), (C2×C4).762(C2×D4), (C2×C2.D8).23C2, (C2×C4).142(C4⋊C4), SmallGroup(128,888)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C16⋊3C4
G = < a,b,c | a2=b16=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 172 in 84 conjugacy classes, 60 normal (18 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⋊3C4, C2×C2.D8, C22×C16, C2×C16⋊3C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C2.D8, D16, Q32, C2×C4⋊C4, C2×D8, C2×Q16, C16⋊3C4, C2×C2.D8, C2×D16, C2×Q32, C2×C16⋊3C4
►Regular action on 128 points
Generators in S128
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 49)(17 110)(18 111)(19 112)(20 97)(21 98)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 106)(30 107)(31 108)(32 109)(33 79)(34 80)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(81 127)(82 128)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)
(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 45 120 23)(2 44 121 22)(3 43 122 21)(4 42 123 20)(5 41 124 19)(6 40 125 18)(7 39 126 17)(8 38 127 32)(9 37 128 31)(10 36 113 30)(11 35 114 29)(12 34 115 28)(13 33 116 27)(14 48 117 26)(15 47 118 25)(16 46 119 24)(49 76 89 101)(50 75 90 100)(51 74 91 99)(52 73 92 98)(53 72 93 97)(54 71 94 112)(55 70 95 111)(56 69 96 110)(57 68 81 109)(58 67 82 108)(59 66 83 107)(60 65 84 106)(61 80 85 105)(62 79 86 104)(63 78 87 103)(64 77 88 102)
G:=sub<Sym(128)| (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,49)(17,110)(18,111)(19,112)(20,97)(21,98)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,106)(30,107)(31,108)(32,109)(33,79)(34,80)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(81,127)(82,128)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126), (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), (1,45,120,23)(2,44,121,22)(3,43,122,21)(4,42,123,20)(5,41,124,19)(6,40,125,18)(7,39,126,17)(8,38,127,32)(9,37,128,31)(10,36,113,30)(11,35,114,29)(12,34,115,28)(13,33,116,27)(14,48,117,26)(15,47,118,25)(16,46,119,24)(49,76,89,101)(50,75,90,100)(51,74,91,99)(52,73,92,98)(53,72,93,97)(54,71,94,112)(55,70,95,111)(56,69,96,110)(57,68,81,109)(58,67,82,108)(59,66,83,107)(60,65,84,106)(61,80,85,105)(62,79,86,104)(63,78,87,103)(64,77,88,102)>;
G:=Group( (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,49)(17,110)(18,111)(19,112)(20,97)(21,98)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,106)(30,107)(31,108)(32,109)(33,79)(34,80)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(81,127)(82,128)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126), (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), (1,45,120,23)(2,44,121,22)(3,43,122,21)(4,42,123,20)(5,41,124,19)(6,40,125,18)(7,39,126,17)(8,38,127,32)(9,37,128,31)(10,36,113,30)(11,35,114,29)(12,34,115,28)(13,33,116,27)(14,48,117,26)(15,47,118,25)(16,46,119,24)(49,76,89,101)(50,75,90,100)(51,74,91,99)(52,73,92,98)(53,72,93,97)(54,71,94,112)(55,70,95,111)(56,69,96,110)(57,68,81,109)(58,67,82,108)(59,66,83,107)(60,65,84,106)(61,80,85,105)(62,79,86,104)(63,78,87,103)(64,77,88,102) );
G=PermutationGroup([[(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,49),(17,110),(18,111),(19,112),(20,97),(21,98),(22,99),(23,100),(24,101),(25,102),(26,103),(27,104),(28,105),(29,106),(30,107),(31,108),(32,109),(33,79),(34,80),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(81,127),(82,128),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120),(91,121),(92,122),(93,123),(94,124),(95,125),(96,126)], [(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)], [(1,45,120,23),(2,44,121,22),(3,43,122,21),(4,42,123,20),(5,41,124,19),(6,40,125,18),(7,39,126,17),(8,38,127,32),(9,37,128,31),(10,36,113,30),(11,35,114,29),(12,34,115,28),(13,33,116,27),(14,48,117,26),(15,47,118,25),(16,46,119,24),(49,76,89,101),(50,75,90,100),(51,74,91,99),(52,73,92,98),(53,72,93,97),(54,71,94,112),(55,70,95,111),(56,69,96,110),(57,68,81,109),(58,67,82,108),(59,66,83,107),(60,65,84,106),(61,80,85,105),(62,79,86,104),(63,78,87,103),(64,77,88,102)]])
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 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D8 | Q16 | D8 | D16 | Q32 |
| kernel | C2×C16⋊3C4 | C16⋊3C4 | C2×C2.D8 | C22×C16 | C2×C16 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 1 | 2 | 1 | 2 | 4 | 2 | 8 | 8 |
Matrix representation of C2×C16⋊3C4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 12 |
| 16 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,10,0,0,0,0,12],[16,0,0,0,0,4,0,0,0,0,0,16,0,0,16,0] >;
C2×C16⋊3C4 in GAP, Magma, Sage, TeX
C_2\times C_{16}\rtimes_3C_4 % in TeX
G:=Group("C2xC16:3C4"); // GroupNames label
G:=SmallGroup(128,888);
// by ID
G=gap.SmallGroup(128,888);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,288,1123,360,4037,124]);
// 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^-1>;
// generators/relations