direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4.SD16, C42.353D4, C42.704C23, (C2×C4).43Q16, C4.13(C2×Q16), C4⋊C4.82C23, (C2×C4).80SD16, C4.21(C2×SD16), (C2×C4).327C24, (C4×C8).382C22, (C2×C8).489C23, (C22×C4).609D4, C23.871(C2×D4), C4⋊Q8.271C22, (C2×Q8).84C23, C22.49(C2×Q16), C2.10(C22×Q16), C4.19(C4.4D4), C22.87(C2×SD16), C2.17(C22×SD16), (C22×C8).518C22, C22.587(C22×D4), (C2×C42).1122C22, (C22×C4).1549C23, Q8⋊C4.135C22, C22.81(C4.4D4), (C22×Q8).297C22, (C2×C4×C8).30C2, C4.36(C2×C4○D4), (C2×C4⋊Q8).45C2, (C2×C4).850(C2×D4), C2.38(C2×C4.4D4), (C2×C4).706(C4○D4), (C2×C4⋊C4).619C22, (C2×Q8⋊C4).18C2, SmallGroup(128,1861)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4.SD16
G = < a,b,c,d | a2=b4=c8=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c3 >
Subgroups: 372 in 212 conjugacy classes, 116 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, Q8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C22×Q8, C2×C4×C8, C2×Q8⋊C4, C4.SD16, C2×C4⋊Q8, C2×C4.SD16
Quotients: C1, C2, C22, D4, C23, SD16, Q16, C2×D4, C4○D4, C24, C4.4D4, C2×SD16, C2×Q16, C22×D4, C2×C4○D4, C4.SD16, C2×C4.4D4, C22×SD16, C22×Q16, C2×C4.SD16
►Regular action on 128 points
Generators in S128
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 80)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(81 95)(82 96)(83 89)(84 90)(85 91)(86 92)(87 93)(88 94)(97 128)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(111 113)(112 114)
(1 62 11 31)(2 63 12 32)(3 64 13 25)(4 57 14 26)(5 58 15 27)(6 59 16 28)(7 60 9 29)(8 61 10 30)(17 54 73 38)(18 55 74 39)(19 56 75 40)(20 49 76 33)(21 50 77 34)(22 51 78 35)(23 52 79 36)(24 53 80 37)(41 95 99 113)(42 96 100 114)(43 89 101 115)(44 90 102 116)(45 91 103 117)(46 92 104 118)(47 93 97 119)(48 94 98 120)(65 81 122 111)(66 82 123 112)(67 83 124 105)(68 84 125 106)(69 85 126 107)(70 86 127 108)(71 87 128 109)(72 88 121 110)
(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 96 11 114)(2 117 12 91)(3 94 13 120)(4 115 14 89)(5 92 15 118)(6 113 16 95)(7 90 9 116)(8 119 10 93)(17 124 73 67)(18 70 74 127)(19 122 75 65)(20 68 76 125)(21 128 77 71)(22 66 78 123)(23 126 79 69)(24 72 80 121)(25 98 64 48)(26 43 57 101)(27 104 58 46)(28 41 59 99)(29 102 60 44)(30 47 61 97)(31 100 62 42)(32 45 63 103)(33 84 49 106)(34 109 50 87)(35 82 51 112)(36 107 52 85)(37 88 53 110)(38 105 54 83)(39 86 55 108)(40 111 56 81)
G:=sub<Sym(128)| (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,80)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(81,95)(82,96)(83,89)(84,90)(85,91)(86,92)(87,93)(88,94)(97,128)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(111,113)(112,114), (1,62,11,31)(2,63,12,32)(3,64,13,25)(4,57,14,26)(5,58,15,27)(6,59,16,28)(7,60,9,29)(8,61,10,30)(17,54,73,38)(18,55,74,39)(19,56,75,40)(20,49,76,33)(21,50,77,34)(22,51,78,35)(23,52,79,36)(24,53,80,37)(41,95,99,113)(42,96,100,114)(43,89,101,115)(44,90,102,116)(45,91,103,117)(46,92,104,118)(47,93,97,119)(48,94,98,120)(65,81,122,111)(66,82,123,112)(67,83,124,105)(68,84,125,106)(69,85,126,107)(70,86,127,108)(71,87,128,109)(72,88,121,110), (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), (1,96,11,114)(2,117,12,91)(3,94,13,120)(4,115,14,89)(5,92,15,118)(6,113,16,95)(7,90,9,116)(8,119,10,93)(17,124,73,67)(18,70,74,127)(19,122,75,65)(20,68,76,125)(21,128,77,71)(22,66,78,123)(23,126,79,69)(24,72,80,121)(25,98,64,48)(26,43,57,101)(27,104,58,46)(28,41,59,99)(29,102,60,44)(30,47,61,97)(31,100,62,42)(32,45,63,103)(33,84,49,106)(34,109,50,87)(35,82,51,112)(36,107,52,85)(37,88,53,110)(38,105,54,83)(39,86,55,108)(40,111,56,81)>;
G:=Group( (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,80)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(81,95)(82,96)(83,89)(84,90)(85,91)(86,92)(87,93)(88,94)(97,128)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(111,113)(112,114), (1,62,11,31)(2,63,12,32)(3,64,13,25)(4,57,14,26)(5,58,15,27)(6,59,16,28)(7,60,9,29)(8,61,10,30)(17,54,73,38)(18,55,74,39)(19,56,75,40)(20,49,76,33)(21,50,77,34)(22,51,78,35)(23,52,79,36)(24,53,80,37)(41,95,99,113)(42,96,100,114)(43,89,101,115)(44,90,102,116)(45,91,103,117)(46,92,104,118)(47,93,97,119)(48,94,98,120)(65,81,122,111)(66,82,123,112)(67,83,124,105)(68,84,125,106)(69,85,126,107)(70,86,127,108)(71,87,128,109)(72,88,121,110), (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), (1,96,11,114)(2,117,12,91)(3,94,13,120)(4,115,14,89)(5,92,15,118)(6,113,16,95)(7,90,9,116)(8,119,10,93)(17,124,73,67)(18,70,74,127)(19,122,75,65)(20,68,76,125)(21,128,77,71)(22,66,78,123)(23,126,79,69)(24,72,80,121)(25,98,64,48)(26,43,57,101)(27,104,58,46)(28,41,59,99)(29,102,60,44)(30,47,61,97)(31,100,62,42)(32,45,63,103)(33,84,49,106)(34,109,50,87)(35,82,51,112)(36,107,52,85)(37,88,53,110)(38,105,54,83)(39,86,55,108)(40,111,56,81) );
G=PermutationGroup([[(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,80),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(81,95),(82,96),(83,89),(84,90),(85,91),(86,92),(87,93),(88,94),(97,128),(98,121),(99,122),(100,123),(101,124),(102,125),(103,126),(104,127),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(111,113),(112,114)], [(1,62,11,31),(2,63,12,32),(3,64,13,25),(4,57,14,26),(5,58,15,27),(6,59,16,28),(7,60,9,29),(8,61,10,30),(17,54,73,38),(18,55,74,39),(19,56,75,40),(20,49,76,33),(21,50,77,34),(22,51,78,35),(23,52,79,36),(24,53,80,37),(41,95,99,113),(42,96,100,114),(43,89,101,115),(44,90,102,116),(45,91,103,117),(46,92,104,118),(47,93,97,119),(48,94,98,120),(65,81,122,111),(66,82,123,112),(67,83,124,105),(68,84,125,106),(69,85,126,107),(70,86,127,108),(71,87,128,109),(72,88,121,110)], [(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)], [(1,96,11,114),(2,117,12,91),(3,94,13,120),(4,115,14,89),(5,92,15,118),(6,113,16,95),(7,90,9,116),(8,119,10,93),(17,124,73,67),(18,70,74,127),(19,122,75,65),(20,68,76,125),(21,128,77,71),(22,66,78,123),(23,126,79,69),(24,72,80,121),(25,98,64,48),(26,43,57,101),(27,104,58,46),(28,41,59,99),(29,102,60,44),(30,47,61,97),(31,100,62,42),(32,45,63,103),(33,84,49,106),(34,109,50,87),(35,82,51,112),(36,107,52,85),(37,88,53,110),(38,105,54,83),(39,86,55,108),(40,111,56,81)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | SD16 | Q16 | C4○D4 |
| kernel | C2×C4.SD16 | C2×C4×C8 | C2×Q8⋊C4 | C4.SD16 | C2×C4⋊Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 |
| # reps | 1 | 1 | 4 | 8 | 2 | 2 | 2 | 8 | 8 | 8 |
Matrix representation of C2×C4.SD16 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 9 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 14 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 4 |
| 0 | 0 | 0 | 0 | 4 | 11 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,13,0,0,0,0,0,9,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,5,12,0,0,0,0,5,5],[5,14,0,0,0,0,3,12,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,6,4,0,0,0,0,4,11] >;
C2×C4.SD16 in GAP, Magma, Sage, TeX
C_2\times C_4.{\rm SD}_{16} % in TeX
G:=Group("C2xC4.SD16"); // GroupNames label
G:=SmallGroup(128,1861);
// by ID
G=gap.SmallGroup(128,1861);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,232,758,100,2804,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^3>;
// generators/relations