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 |
Subgroups: 220, all normal (8 characteristic)
C1, C2, C2 [×14], C4, C4 [×7], C22 [×35], C8, C8 [×7], C2×C4 [×28], C23 [×15], C16 [×8], C2×C8 [×28], C22×C4 [×14], C24, C2×C16 [×28], C22×C8 [×14], C23×C4, C22×C16 [×14], C23×C8, C23×C16
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C16 [×8], C2×C8 [×28], C22×C4 [×14], C24, C2×C16 [×28], C22×C8 [×14], C23×C4, C22×C16 [×14], C23×C8, C23×C16
Generators and relations
G = < a,b,c,d | a2=b2=c2=d16=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
►Regular action on 128 points
Generators in S128
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 65)(13 66)(14 67)(15 68)(16 69)(17 127)(18 128)(19 113)(20 114)(21 115)(22 116)(23 117)(24 118)(25 119)(26 120)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 96)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 109)(57 110)(58 111)(59 112)(60 97)(61 98)(62 99)(63 100)(64 101)
(1 111)(2 112)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 103)(10 104)(11 105)(12 106)(13 107)(14 108)(15 109)(16 110)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 118)(34 119)(35 120)(36 121)(37 122)(38 123)(39 124)(40 125)(41 126)(42 127)(43 128)(44 113)(45 114)(46 115)(47 116)(48 117)(49 77)(50 78)(51 79)(52 80)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)
(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 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(65 116)(66 117)(67 118)(68 119)(69 120)(70 121)(71 122)(72 123)(73 124)(74 125)(75 126)(76 127)(77 128)(78 113)(79 114)(80 115)(81 109)(82 110)(83 111)(84 112)(85 97)(86 98)(87 99)(88 100)(89 101)(90 102)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)
(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)| (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,65)(13,66)(14,67)(15,68)(16,69)(17,127)(18,128)(19,113)(20,114)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,96)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109)(57,110)(58,111)(59,112)(60,97)(61,98)(62,99)(63,100)(64,101), (1,111)(2,112)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,103)(10,104)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,118)(34,119)(35,120)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,113)(45,114)(46,115)(47,116)(48,117)(49,77)(50,78)(51,79)(52,80)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76), (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)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(65,116)(66,117)(67,118)(68,119)(69,120)(70,121)(71,122)(72,123)(73,124)(74,125)(75,126)(76,127)(77,128)(78,113)(79,114)(80,115)(81,109)(82,110)(83,111)(84,112)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108), (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( (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,65)(13,66)(14,67)(15,68)(16,69)(17,127)(18,128)(19,113)(20,114)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,96)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109)(57,110)(58,111)(59,112)(60,97)(61,98)(62,99)(63,100)(64,101), (1,111)(2,112)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,103)(10,104)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,118)(34,119)(35,120)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,113)(45,114)(46,115)(47,116)(48,117)(49,77)(50,78)(51,79)(52,80)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76), (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)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(65,116)(66,117)(67,118)(68,119)(69,120)(70,121)(71,122)(72,123)(73,124)(74,125)(75,126)(76,127)(77,128)(78,113)(79,114)(80,115)(81,109)(82,110)(83,111)(84,112)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108), (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([(1,70),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,80),(12,65),(13,66),(14,67),(15,68),(16,69),(17,127),(18,128),(19,113),(20,114),(21,115),(22,116),(23,117),(24,118),(25,119),(26,120),(27,121),(28,122),(29,123),(30,124),(31,125),(32,126),(33,96),(34,81),(35,82),(36,83),(37,84),(38,85),(39,86),(40,87),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,102),(50,103),(51,104),(52,105),(53,106),(54,107),(55,108),(56,109),(57,110),(58,111),(59,112),(60,97),(61,98),(62,99),(63,100),(64,101)], [(1,111),(2,112),(3,97),(4,98),(5,99),(6,100),(7,101),(8,102),(9,103),(10,104),(11,105),(12,106),(13,107),(14,108),(15,109),(16,110),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,118),(34,119),(35,120),(36,121),(37,122),(38,123),(39,124),(40,125),(41,126),(42,127),(43,128),(44,113),(45,114),(46,115),(47,116),(48,117),(49,77),(50,78),(51,79),(52,80),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(61,73),(62,74),(63,75),(64,76)], [(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),(33,55),(34,56),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(41,63),(42,64),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54),(65,116),(66,117),(67,118),(68,119),(69,120),(70,121),(71,122),(72,123),(73,124),(74,125),(75,126),(76,127),(77,128),(78,113),(79,114),(80,115),(81,109),(82,110),(83,111),(84,112),(85,97),(86,98),(87,99),(88,100),(89,101),(90,102),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108)], [(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀