direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×Q16⋊C4, C42.203D4, C42.270C23, C4.46(C4×D4), Q16⋊10(C2×C4), (C2×Q16)⋊17C4, C8.18(C22×C4), C4.18(C23×C4), Q8.3(C22×C4), C4⋊C4.358C23, (C2×C8).409C23, (C2×C4).198C24, C23.844(C2×D4), C22.119(C4×D4), (C22×C4).708D4, (C22×Q16).15C2, (C4×Q8).273C22, (C2×Q8).341C23, C8⋊C4.110C22, C4.Q8.124C22, (C22×C8).437C22, (C2×C42).763C22, (C2×Q16).150C22, C22.142(C22×D4), (C22×C4).1514C23, Q8⋊C4.194C22, C22.97(C8.C22), (C22×Q8).462C22, C2.58(C2×C4×D4), C4.6(C2×C4○D4), (C2×C4×Q8).42C2, (C2×C8).96(C2×C4), (C2×C8⋊C4).10C2, (C2×C4.Q8).10C2, C2.5(C2×C8.C22), (C2×C4).1210(C2×D4), (C2×Q8).159(C2×C4), (C2×C4).690(C4○D4), (C2×C4⋊C4).910C22, (C2×C4).469(C22×C4), (C2×Q8⋊C4).36C2, SmallGroup(128,1673)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 348 in 240 conjugacy classes, 148 normal (16 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×2], C4 [×16], C22, C22 [×6], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×24], 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, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×12], C2×Q8 [×6], C8⋊C4 [×4], Q8⋊C4 [×8], C4.Q8 [×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×C8⋊C4, C2×Q8⋊C4 [×2], C2×C4.Q8, Q16⋊C4 [×8], C2×C4×Q8 [×2], C22×Q16, C2×Q16⋊C4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C8.C22 [×4], C23×C4, C22×D4, C2×C4○D4, Q16⋊C4 [×4], C2×C4×D4, C2×C8.C22 [×2], C2×Q16⋊C4
Generators and relations
G = < a,b,c,d | a2=b8=d4=1, c2=b4, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b5, cd=dc >
►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 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 47)(18 48)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(57 124)(58 125)(59 126)(60 127)(61 128)(62 121)(63 122)(64 123)(65 110)(66 111)(67 112)(68 105)(69 106)(70 107)(71 108)(72 109)(73 104)(74 97)(75 98)(76 99)(77 100)(78 101)(79 102)(80 103)(81 90)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 89)
(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 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(17 57 21 61)(18 64 22 60)(19 63 23 59)(20 62 24 58)(25 55 29 51)(26 54 30 50)(27 53 31 49)(28 52 32 56)(41 122 45 126)(42 121 46 125)(43 128 47 124)(44 127 48 123)(65 96 69 92)(66 95 70 91)(67 94 71 90)(68 93 72 89)(73 114 77 118)(74 113 78 117)(75 120 79 116)(76 119 80 115)(81 112 85 108)(82 111 86 107)(83 110 87 106)(84 109 88 105)
(1 55 128 107)(2 52 121 112)(3 49 122 109)(4 54 123 106)(5 51 124 111)(6 56 125 108)(7 53 126 105)(8 50 127 110)(9 63 72 113)(10 60 65 118)(11 57 66 115)(12 62 67 120)(13 59 68 117)(14 64 69 114)(15 61 70 119)(16 58 71 116)(17 91 80 37)(18 96 73 34)(19 93 74 39)(20 90 75 36)(21 95 76 33)(22 92 77 38)(23 89 78 35)(24 94 79 40)(25 43 86 99)(26 48 87 104)(27 45 88 101)(28 42 81 98)(29 47 82 103)(30 44 83 100)(31 41 84 97)(32 46 85 102)
G:=sub<Sym(128)| (1,119)(2,120)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,47)(18,48)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(57,124)(58,125)(59,126)(60,127)(61,128)(62,121)(63,122)(64,123)(65,110)(66,111)(67,112)(68,105)(69,106)(70,107)(71,108)(72,109)(73,104)(74,97)(75,98)(76,99)(77,100)(78,101)(79,102)(80,103)(81,90)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,89), (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)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(17,57,21,61)(18,64,22,60)(19,63,23,59)(20,62,24,58)(25,55,29,51)(26,54,30,50)(27,53,31,49)(28,52,32,56)(41,122,45,126)(42,121,46,125)(43,128,47,124)(44,127,48,123)(65,96,69,92)(66,95,70,91)(67,94,71,90)(68,93,72,89)(73,114,77,118)(74,113,78,117)(75,120,79,116)(76,119,80,115)(81,112,85,108)(82,111,86,107)(83,110,87,106)(84,109,88,105), (1,55,128,107)(2,52,121,112)(3,49,122,109)(4,54,123,106)(5,51,124,111)(6,56,125,108)(7,53,126,105)(8,50,127,110)(9,63,72,113)(10,60,65,118)(11,57,66,115)(12,62,67,120)(13,59,68,117)(14,64,69,114)(15,61,70,119)(16,58,71,116)(17,91,80,37)(18,96,73,34)(19,93,74,39)(20,90,75,36)(21,95,76,33)(22,92,77,38)(23,89,78,35)(24,94,79,40)(25,43,86,99)(26,48,87,104)(27,45,88,101)(28,42,81,98)(29,47,82,103)(30,44,83,100)(31,41,84,97)(32,46,85,102)>;
G:=Group( (1,119)(2,120)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,47)(18,48)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(57,124)(58,125)(59,126)(60,127)(61,128)(62,121)(63,122)(64,123)(65,110)(66,111)(67,112)(68,105)(69,106)(70,107)(71,108)(72,109)(73,104)(74,97)(75,98)(76,99)(77,100)(78,101)(79,102)(80,103)(81,90)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,89), (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)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(17,57,21,61)(18,64,22,60)(19,63,23,59)(20,62,24,58)(25,55,29,51)(26,54,30,50)(27,53,31,49)(28,52,32,56)(41,122,45,126)(42,121,46,125)(43,128,47,124)(44,127,48,123)(65,96,69,92)(66,95,70,91)(67,94,71,90)(68,93,72,89)(73,114,77,118)(74,113,78,117)(75,120,79,116)(76,119,80,115)(81,112,85,108)(82,111,86,107)(83,110,87,106)(84,109,88,105), (1,55,128,107)(2,52,121,112)(3,49,122,109)(4,54,123,106)(5,51,124,111)(6,56,125,108)(7,53,126,105)(8,50,127,110)(9,63,72,113)(10,60,65,118)(11,57,66,115)(12,62,67,120)(13,59,68,117)(14,64,69,114)(15,61,70,119)(16,58,71,116)(17,91,80,37)(18,96,73,34)(19,93,74,39)(20,90,75,36)(21,95,76,33)(22,92,77,38)(23,89,78,35)(24,94,79,40)(25,43,86,99)(26,48,87,104)(27,45,88,101)(28,42,81,98)(29,47,82,103)(30,44,83,100)(31,41,84,97)(32,46,85,102) );
G=PermutationGroup([(1,119),(2,120),(3,113),(4,114),(5,115),(6,116),(7,117),(8,118),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,47),(18,48),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(31,39),(32,40),(57,124),(58,125),(59,126),(60,127),(61,128),(62,121),(63,122),(64,123),(65,110),(66,111),(67,112),(68,105),(69,106),(70,107),(71,108),(72,109),(73,104),(74,97),(75,98),(76,99),(77,100),(78,101),(79,102),(80,103),(81,90),(82,91),(83,92),(84,93),(85,94),(86,95),(87,96),(88,89)], [(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),(9,35,13,39),(10,34,14,38),(11,33,15,37),(12,40,16,36),(17,57,21,61),(18,64,22,60),(19,63,23,59),(20,62,24,58),(25,55,29,51),(26,54,30,50),(27,53,31,49),(28,52,32,56),(41,122,45,126),(42,121,46,125),(43,128,47,124),(44,127,48,123),(65,96,69,92),(66,95,70,91),(67,94,71,90),(68,93,72,89),(73,114,77,118),(74,113,78,117),(75,120,79,116),(76,119,80,115),(81,112,85,108),(82,111,86,107),(83,110,87,106),(84,109,88,105)], [(1,55,128,107),(2,52,121,112),(3,49,122,109),(4,54,123,106),(5,51,124,111),(6,56,125,108),(7,53,126,105),(8,50,127,110),(9,63,72,113),(10,60,65,118),(11,57,66,115),(12,62,67,120),(13,59,68,117),(14,64,69,114),(15,61,70,119),(16,58,71,116),(17,91,80,37),(18,96,73,34),(19,93,74,39),(20,90,75,36),(21,95,76,33),(22,92,77,38),(23,89,78,35),(24,94,79,40),(25,43,86,99),(26,48,87,104),(27,45,88,101),(28,42,81,98),(29,47,82,103),(30,44,83,100),(31,41,84,97),(32,46,85,102)])
Matrix representation ►G ⊆ GL8(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 10 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 13 | 16 |
| 0 | 0 | 0 | 0 | 2 | 12 | 12 | 4 |
| 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 14 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 3 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 6 | 6 | 16 | 16 |
G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[10,7,0,0,0,0,0,0,5,7,0,0,0,0,0,0,0,0,15,13,0,0,0,0,0,0,14,2,0,0,0,0,0,0,0,0,16,3,3,2,0,0,0,0,4,1,0,12,0,0,0,0,0,0,13,12,0,0,0,0,0,0,16,4],[1,0,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,14,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,11,6,0,0,0,0,2,16,0,6,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4AB | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | C8.C22 |
| kernel | C2×Q16⋊C4 | C2×C8⋊C4 | C2×Q8⋊C4 | C2×C4.Q8 | Q16⋊C4 | C2×C4×Q8 | C22×Q16 | C2×Q16 | C42 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 16 | 2 | 2 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_2\times Q_{16}\rtimes C_4 % in TeX
G:=Group("C2xQ16:C4"); // GroupNames label
G:=SmallGroup(128,1673);
// by ID
G=gap.SmallGroup(128,1673);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,1430,184,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=d^4=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^5,c*d=d*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀