direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4×Q16, C42.351D4, C42.685C23, C4○(C4×Q16), C4.44(C4×D4), C4.15(C23×C4), C8.43(C22×C4), Q8.1(C22×C4), C2.3(C22×Q16), C4⋊C4.355C23, (C2×C8).557C23, (C2×C4).195C24, (C4×C8).406C22, C22.117(C4×D4), C23.842(C2×D4), (C22×C4).826D4, C22.46(C2×Q16), C22.85(C4○D8), (C4×Q8).271C22, (C2×Q8).338C23, (C22×Q16).17C2, C2.D8.233C22, (C22×C8).512C22, (C2×Q16).167C22, C22.139(C22×D4), (C22×C4).1511C23, (C2×C42).1113C22, Q8⋊C4.214C22, (C22×Q8).460C22, (C2×C4×C8).37C2, (C2×C4)○(C4×Q16), C4○3(C2×C2.D8), C2.55(C2×C4×D4), C2.5(C2×C4○D8), C4.3(C2×C4○D4), (C2×C4×Q8).41C2, C4○3(C2×Q8⋊C4), (C2×C4)○4(C2.D8), (C2×C8).176(C2×C4), (C2×C2.D8).41C2, (C2×C4)○4(Q8⋊C4), (C2×C4).1575(C2×D4), (C2×Q8).158(C2×C4), (C2×C4).687(C4○D4), (C2×C4⋊C4).908C22, (C2×C4).466(C22×C4), (C2×Q8⋊C4).39C2, (C2×C4)○3(C2×C2.D8), (C2×C4)○3(C2×Q8⋊C4), SmallGroup(128,1670)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 348 in 244 conjugacy classes, 156 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×14], C22, C22 [×6], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×12], C2×C4 [×22], Q8 [×8], Q8 [×12], C23, C42 [×4], C42 [×8], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×8], C2×C8 [×2], Q16 [×16], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×12], C2×Q8 [×6], C4×C8 [×4], Q8⋊C4 [×8], C2.D8 [×4], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4×Q8 [×8], C4×Q8 [×4], C22×C8 [×2], C2×Q16 [×12], C22×Q8 [×2], C2×C4×C8, C2×Q8⋊C4 [×2], C2×C2.D8, C4×Q16 [×8], C2×C4×Q8 [×2], C22×Q16, C2×C4×Q16
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], Q16 [×4], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C2×Q16 [×6], C4○D8 [×2], C23×C4, C22×D4, C2×C4○D4, C4×Q16 [×4], C2×C4×D4, C22×Q16, C2×C4○D8, C2×C4×Q16
Generators and relations
G = < a,b,c,d | a2=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
►Regular action on 128 points
Generators in S128
(1 119)(2 120)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 54)(10 55)(11 56)(12 49)(13 50)(14 51)(15 52)(16 53)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)(57 121)(58 122)(59 123)(60 124)(61 125)(62 126)(63 127)(64 128)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 105)(73 100)(74 101)(75 102)(76 103)(77 104)(78 97)(79 98)(80 99)(81 94)(82 95)(83 96)(84 89)(85 90)(86 91)(87 92)(88 93)
(1 56 124 107)(2 49 125 108)(3 50 126 109)(4 51 127 110)(5 52 128 111)(6 53 121 112)(7 54 122 105)(8 55 123 106)(9 58 72 117)(10 59 65 118)(11 60 66 119)(12 61 67 120)(13 62 68 113)(14 63 69 114)(15 64 70 115)(16 57 71 116)(17 96 73 34)(18 89 74 35)(19 90 75 36)(20 91 76 37)(21 92 77 38)(22 93 78 39)(23 94 79 40)(24 95 80 33)(25 48 87 104)(26 41 88 97)(27 42 81 98)(28 43 82 99)(29 44 83 100)(30 45 84 101)(31 46 85 102)(32 47 86 103)
(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)
(1 103 5 99)(2 102 6 98)(3 101 7 97)(4 100 8 104)(9 39 13 35)(10 38 14 34)(11 37 15 33)(12 36 16 40)(17 59 21 63)(18 58 22 62)(19 57 23 61)(20 64 24 60)(25 51 29 55)(26 50 30 54)(27 49 31 53)(28 56 32 52)(41 126 45 122)(42 125 46 121)(43 124 47 128)(44 123 48 127)(65 92 69 96)(66 91 70 95)(67 90 71 94)(68 89 72 93)(73 118 77 114)(74 117 78 113)(75 116 79 120)(76 115 80 119)(81 108 85 112)(82 107 86 111)(83 106 87 110)(84 105 88 109)
G:=sub<Sym(128)| (1,119)(2,120)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,54)(10,55)(11,56)(12,49)(13,50)(14,51)(15,52)(16,53)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(57,121)(58,122)(59,123)(60,124)(61,125)(62,126)(63,127)(64,128)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,105)(73,100)(74,101)(75,102)(76,103)(77,104)(78,97)(79,98)(80,99)(81,94)(82,95)(83,96)(84,89)(85,90)(86,91)(87,92)(88,93), (1,56,124,107)(2,49,125,108)(3,50,126,109)(4,51,127,110)(5,52,128,111)(6,53,121,112)(7,54,122,105)(8,55,123,106)(9,58,72,117)(10,59,65,118)(11,60,66,119)(12,61,67,120)(13,62,68,113)(14,63,69,114)(15,64,70,115)(16,57,71,116)(17,96,73,34)(18,89,74,35)(19,90,75,36)(20,91,76,37)(21,92,77,38)(22,93,78,39)(23,94,79,40)(24,95,80,33)(25,48,87,104)(26,41,88,97)(27,42,81,98)(28,43,82,99)(29,44,83,100)(30,45,84,101)(31,46,85,102)(32,47,86,103), (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), (1,103,5,99)(2,102,6,98)(3,101,7,97)(4,100,8,104)(9,39,13,35)(10,38,14,34)(11,37,15,33)(12,36,16,40)(17,59,21,63)(18,58,22,62)(19,57,23,61)(20,64,24,60)(25,51,29,55)(26,50,30,54)(27,49,31,53)(28,56,32,52)(41,126,45,122)(42,125,46,121)(43,124,47,128)(44,123,48,127)(65,92,69,96)(66,91,70,95)(67,90,71,94)(68,89,72,93)(73,118,77,114)(74,117,78,113)(75,116,79,120)(76,115,80,119)(81,108,85,112)(82,107,86,111)(83,106,87,110)(84,105,88,109)>;
G:=Group( (1,119)(2,120)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,54)(10,55)(11,56)(12,49)(13,50)(14,51)(15,52)(16,53)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(57,121)(58,122)(59,123)(60,124)(61,125)(62,126)(63,127)(64,128)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,105)(73,100)(74,101)(75,102)(76,103)(77,104)(78,97)(79,98)(80,99)(81,94)(82,95)(83,96)(84,89)(85,90)(86,91)(87,92)(88,93), (1,56,124,107)(2,49,125,108)(3,50,126,109)(4,51,127,110)(5,52,128,111)(6,53,121,112)(7,54,122,105)(8,55,123,106)(9,58,72,117)(10,59,65,118)(11,60,66,119)(12,61,67,120)(13,62,68,113)(14,63,69,114)(15,64,70,115)(16,57,71,116)(17,96,73,34)(18,89,74,35)(19,90,75,36)(20,91,76,37)(21,92,77,38)(22,93,78,39)(23,94,79,40)(24,95,80,33)(25,48,87,104)(26,41,88,97)(27,42,81,98)(28,43,82,99)(29,44,83,100)(30,45,84,101)(31,46,85,102)(32,47,86,103), (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), (1,103,5,99)(2,102,6,98)(3,101,7,97)(4,100,8,104)(9,39,13,35)(10,38,14,34)(11,37,15,33)(12,36,16,40)(17,59,21,63)(18,58,22,62)(19,57,23,61)(20,64,24,60)(25,51,29,55)(26,50,30,54)(27,49,31,53)(28,56,32,52)(41,126,45,122)(42,125,46,121)(43,124,47,128)(44,123,48,127)(65,92,69,96)(66,91,70,95)(67,90,71,94)(68,89,72,93)(73,118,77,114)(74,117,78,113)(75,116,79,120)(76,115,80,119)(81,108,85,112)(82,107,86,111)(83,106,87,110)(84,105,88,109) );
G=PermutationGroup([(1,119),(2,120),(3,113),(4,114),(5,115),(6,116),(7,117),(8,118),(9,54),(10,55),(11,56),(12,49),(13,50),(14,51),(15,52),(16,53),(17,44),(18,45),(19,46),(20,47),(21,48),(22,41),(23,42),(24,43),(25,38),(26,39),(27,40),(28,33),(29,34),(30,35),(31,36),(32,37),(57,121),(58,122),(59,123),(60,124),(61,125),(62,126),(63,127),(64,128),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,105),(73,100),(74,101),(75,102),(76,103),(77,104),(78,97),(79,98),(80,99),(81,94),(82,95),(83,96),(84,89),(85,90),(86,91),(87,92),(88,93)], [(1,56,124,107),(2,49,125,108),(3,50,126,109),(4,51,127,110),(5,52,128,111),(6,53,121,112),(7,54,122,105),(8,55,123,106),(9,58,72,117),(10,59,65,118),(11,60,66,119),(12,61,67,120),(13,62,68,113),(14,63,69,114),(15,64,70,115),(16,57,71,116),(17,96,73,34),(18,89,74,35),(19,90,75,36),(20,91,76,37),(21,92,77,38),(22,93,78,39),(23,94,79,40),(24,95,80,33),(25,48,87,104),(26,41,88,97),(27,42,81,98),(28,43,82,99),(29,44,83,100),(30,45,84,101),(31,46,85,102),(32,47,86,103)], [(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)], [(1,103,5,99),(2,102,6,98),(3,101,7,97),(4,100,8,104),(9,39,13,35),(10,38,14,34),(11,37,15,33),(12,36,16,40),(17,59,21,63),(18,58,22,62),(19,57,23,61),(20,64,24,60),(25,51,29,55),(26,50,30,54),(27,49,31,53),(28,56,32,52),(41,126,45,122),(42,125,46,121),(43,124,47,128),(44,123,48,127),(65,92,69,96),(66,91,70,95),(67,90,71,94),(68,89,72,93),(73,118,77,114),(74,117,78,113),(75,116,79,120),(76,115,80,119),(81,108,85,112),(82,107,86,111),(83,106,87,110),(84,105,88,109)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 3 | 11 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 0 | 0 | 13 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,0,3,0,0,11,11],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,9,13] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AF | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q16 | C4○D4 | C4○D8 |
| kernel | C2×C4×Q16 | C2×C4×C8 | C2×Q8⋊C4 | C2×C2.D8 | C4×Q16 | C2×C4×Q8 | C22×Q16 | C2×Q16 | C42 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 16 | 2 | 2 | 8 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_4\times Q_{16} % in TeX
G:=Group("C2xC4xQ16"); // GroupNames label
G:=SmallGroup(128,1670);
// by ID
G=gap.SmallGroup(128,1670);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,520,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀