direct product, p-group, abelian, monomial
Aliases: C23×C16, SmallGroup(128,2136)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C23×C16 |
| C1 — C23×C16 |
| C1 — C23×C16 |
Generators and relations for C23×C16
G = < a,b,c,d | a2=b2=c2=d16=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 220, all normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, C23, C16, C2×C8, C22×C4, C24, C2×C16, C22×C8, C23×C4, C22×C16, C23×C8, C23×C16
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C16, C2×C8, C22×C4, C24, C2×C16, C22×C8, C23×C4, C22×C16, C23×C8, C23×C16
►Regular action on 128 points
Generators in S128
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 92)(18 93)(19 94)(20 95)(21 96)(22 81)(23 82)(24 83)(25 84)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 121)(72 122)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 113)(80 114)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(33 127)(34 128)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 65)(62 66)(63 67)(64 68)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(91 109)(92 110)(93 111)(94 112)(95 97)(96 98)
(1 64)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 73)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 121)(90 122)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)
(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)| (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,92)(18,93)(19,94)(20,95)(21,96)(22,81)(23,82)(24,83)(25,84)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,121)(72,122)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,113)(80,114), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(33,127)(34,128)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,65)(62,66)(63,67)(64,68)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,97)(96,98), (1,64)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,73)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128), (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( (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,92)(18,93)(19,94)(20,95)(21,96)(22,81)(23,82)(24,83)(25,84)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,121)(72,122)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,113)(80,114), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(33,127)(34,128)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,65)(62,66)(63,67)(64,68)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,97)(96,98), (1,64)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,73)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128), (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([[(1,104),(2,105),(3,106),(4,107),(5,108),(6,109),(7,110),(8,111),(9,112),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,92),(18,93),(19,94),(20,95),(21,96),(22,81),(23,82),(24,83),(25,84),(26,85),(27,86),(28,87),(29,88),(30,89),(31,90),(32,91),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56),(65,115),(66,116),(67,117),(68,118),(69,119),(70,120),(71,121),(72,122),(73,123),(74,124),(75,125),(76,126),(77,127),(78,128),(79,113),(80,114)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(33,127),(34,128),(35,113),(36,114),(37,115),(38,116),(39,117),(40,118),(41,119),(42,120),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,65),(62,66),(63,67),(64,68),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(91,109),(92,110),(93,111),(94,112),(95,97),(96,98)], [(1,64),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,74),(18,75),(19,76),(20,77),(21,78),(22,79),(23,80),(24,65),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,73),(33,97),(34,98),(35,99),(36,100),(37,101),(38,102),(39,103),(40,104),(41,105),(42,106),(43,107),(44,108),(45,109),(46,110),(47,111),(48,112),(81,113),(82,114),(83,115),(84,116),(85,117),(86,118),(87,119),(88,120),(89,121),(90,122),(91,123),(92,124),(93,125),(94,126),(95,127),(96,128)], [(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)]])
128 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4P | 8A | ··· | 8AF | 16A | ··· | 16BL |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 |
| kernel | C23×C16 | C22×C16 | C23×C8 | C22×C8 | C23×C4 | C22×C4 | C24 | C23 |
| # reps | 1 | 14 | 1 | 14 | 2 | 28 | 4 | 64 |
Matrix representation of C23×C16 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[1,0,0,0,0,11,0,0,0,0,6,0,0,0,0,16] >;
C23×C16 in GAP, Magma, Sage, TeX
C_2^3\times C_{16} % in TeX
G:=Group("C2^3xC16"); // GroupNames label
G:=SmallGroup(128,2136);
// by ID
G=gap.SmallGroup(128,2136);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,112,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^2=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations