direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4.SD16, C42.353D4, C42.704C23, C4.13(C2×Q16), (C2×C4).43Q16, C4⋊C4.82C23, C4.21(C2×SD16), (C2×C4).80SD16, (C2×C8).489C23, (C4×C8).382C22, (C2×C4).327C24, (C22×C4).609D4, C23.871(C2×D4), C4⋊Q8.271C22, (C2×Q8).84C23, C2.10(C22×Q16), C22.49(C2×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
Subgroups: 372 in 212 conjugacy classes, 116 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C4 [×8], C22, C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×16], C2×C4 [×16], Q8 [×20], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×18], C4×C8 [×4], Q8⋊C4 [×16], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×3], C4⋊Q8 [×8], C4⋊Q8 [×4], C22×C8 [×2], C22×Q8 [×2], C22×Q8, C2×C4×C8, C2×Q8⋊C4 [×4], C4.SD16 [×8], C2×C4⋊Q8 [×2], C2×C4.SD16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], Q16 [×4], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C2×SD16 [×6], C2×Q16 [×6], C22×D4, C2×C4○D4 [×2], C4.SD16 [×4], C2×C4.4D4, C22×SD16, C22×Q16, C2×C4.SD16
Generators and relations
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 >
►Regular action on 128 points
Generators in S128
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 73)(24 74)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)(41 70)(42 71)(43 72)(44 65)(45 66)(46 67)(47 68)(48 69)(81 89)(82 90)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(97 126)(98 127)(99 128)(100 121)(101 122)(102 123)(103 124)(104 125)(105 113)(106 114)(107 115)(108 116)(109 117)(110 118)(111 119)(112 120)
(1 62 13 20)(2 63 14 21)(3 64 15 22)(4 57 16 23)(5 58 9 24)(6 59 10 17)(7 60 11 18)(8 61 12 19)(25 33 49 76)(26 34 50 77)(27 35 51 78)(28 36 52 79)(29 37 53 80)(30 38 54 73)(31 39 55 74)(32 40 56 75)(41 89 101 113)(42 90 102 114)(43 91 103 115)(44 92 104 116)(45 93 97 117)(46 94 98 118)(47 95 99 119)(48 96 100 120)(65 84 125 108)(66 85 126 109)(67 86 127 110)(68 87 128 111)(69 88 121 112)(70 81 122 105)(71 82 123 106)(72 83 124 107)
(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 90 13 114)(2 117 14 93)(3 96 15 120)(4 115 16 91)(5 94 9 118)(6 113 10 89)(7 92 11 116)(8 119 12 95)(17 41 59 101)(18 104 60 44)(19 47 61 99)(20 102 62 42)(21 45 63 97)(22 100 64 48)(23 43 57 103)(24 98 58 46)(25 84 49 108)(26 111 50 87)(27 82 51 106)(28 109 52 85)(29 88 53 112)(30 107 54 83)(31 86 55 110)(32 105 56 81)(33 65 76 125)(34 128 77 68)(35 71 78 123)(36 126 79 66)(37 69 80 121)(38 124 73 72)(39 67 74 127)(40 122 75 70)
G:=sub<Sym(128)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,73)(24,74)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59)(41,70)(42,71)(43,72)(44,65)(45,66)(46,67)(47,68)(48,69)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(97,126)(98,127)(99,128)(100,121)(101,122)(102,123)(103,124)(104,125)(105,113)(106,114)(107,115)(108,116)(109,117)(110,118)(111,119)(112,120), (1,62,13,20)(2,63,14,21)(3,64,15,22)(4,57,16,23)(5,58,9,24)(6,59,10,17)(7,60,11,18)(8,61,12,19)(25,33,49,76)(26,34,50,77)(27,35,51,78)(28,36,52,79)(29,37,53,80)(30,38,54,73)(31,39,55,74)(32,40,56,75)(41,89,101,113)(42,90,102,114)(43,91,103,115)(44,92,104,116)(45,93,97,117)(46,94,98,118)(47,95,99,119)(48,96,100,120)(65,84,125,108)(66,85,126,109)(67,86,127,110)(68,87,128,111)(69,88,121,112)(70,81,122,105)(71,82,123,106)(72,83,124,107), (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), (1,90,13,114)(2,117,14,93)(3,96,15,120)(4,115,16,91)(5,94,9,118)(6,113,10,89)(7,92,11,116)(8,119,12,95)(17,41,59,101)(18,104,60,44)(19,47,61,99)(20,102,62,42)(21,45,63,97)(22,100,64,48)(23,43,57,103)(24,98,58,46)(25,84,49,108)(26,111,50,87)(27,82,51,106)(28,109,52,85)(29,88,53,112)(30,107,54,83)(31,86,55,110)(32,105,56,81)(33,65,76,125)(34,128,77,68)(35,71,78,123)(36,126,79,66)(37,69,80,121)(38,124,73,72)(39,67,74,127)(40,122,75,70)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,73)(24,74)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59)(41,70)(42,71)(43,72)(44,65)(45,66)(46,67)(47,68)(48,69)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(97,126)(98,127)(99,128)(100,121)(101,122)(102,123)(103,124)(104,125)(105,113)(106,114)(107,115)(108,116)(109,117)(110,118)(111,119)(112,120), (1,62,13,20)(2,63,14,21)(3,64,15,22)(4,57,16,23)(5,58,9,24)(6,59,10,17)(7,60,11,18)(8,61,12,19)(25,33,49,76)(26,34,50,77)(27,35,51,78)(28,36,52,79)(29,37,53,80)(30,38,54,73)(31,39,55,74)(32,40,56,75)(41,89,101,113)(42,90,102,114)(43,91,103,115)(44,92,104,116)(45,93,97,117)(46,94,98,118)(47,95,99,119)(48,96,100,120)(65,84,125,108)(66,85,126,109)(67,86,127,110)(68,87,128,111)(69,88,121,112)(70,81,122,105)(71,82,123,106)(72,83,124,107), (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), (1,90,13,114)(2,117,14,93)(3,96,15,120)(4,115,16,91)(5,94,9,118)(6,113,10,89)(7,92,11,116)(8,119,12,95)(17,41,59,101)(18,104,60,44)(19,47,61,99)(20,102,62,42)(21,45,63,97)(22,100,64,48)(23,43,57,103)(24,98,58,46)(25,84,49,108)(26,111,50,87)(27,82,51,106)(28,109,52,85)(29,88,53,112)(30,107,54,83)(31,86,55,110)(32,105,56,81)(33,65,76,125)(34,128,77,68)(35,71,78,123)(36,126,79,66)(37,69,80,121)(38,124,73,72)(39,67,74,127)(40,122,75,70) );
G=PermutationGroup([(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,55),(10,56),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,75),(18,76),(19,77),(20,78),(21,79),(22,80),(23,73),(24,74),(33,60),(34,61),(35,62),(36,63),(37,64),(38,57),(39,58),(40,59),(41,70),(42,71),(43,72),(44,65),(45,66),(46,67),(47,68),(48,69),(81,89),(82,90),(83,91),(84,92),(85,93),(86,94),(87,95),(88,96),(97,126),(98,127),(99,128),(100,121),(101,122),(102,123),(103,124),(104,125),(105,113),(106,114),(107,115),(108,116),(109,117),(110,118),(111,119),(112,120)], [(1,62,13,20),(2,63,14,21),(3,64,15,22),(4,57,16,23),(5,58,9,24),(6,59,10,17),(7,60,11,18),(8,61,12,19),(25,33,49,76),(26,34,50,77),(27,35,51,78),(28,36,52,79),(29,37,53,80),(30,38,54,73),(31,39,55,74),(32,40,56,75),(41,89,101,113),(42,90,102,114),(43,91,103,115),(44,92,104,116),(45,93,97,117),(46,94,98,118),(47,95,99,119),(48,96,100,120),(65,84,125,108),(66,85,126,109),(67,86,127,110),(68,87,128,111),(69,88,121,112),(70,81,122,105),(71,82,123,106),(72,83,124,107)], [(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)], [(1,90,13,114),(2,117,14,93),(3,96,15,120),(4,115,16,91),(5,94,9,118),(6,113,10,89),(7,92,11,116),(8,119,12,95),(17,41,59,101),(18,104,60,44),(19,47,61,99),(20,102,62,42),(21,45,63,97),(22,100,64,48),(23,43,57,103),(24,98,58,46),(25,84,49,108),(26,111,50,87),(27,82,51,106),(28,109,52,85),(29,88,53,112),(30,107,54,83),(31,86,55,110),(32,105,56,81),(33,65,76,125),(34,128,77,68),(35,71,78,123),(36,126,79,66),(37,69,80,121),(38,124,73,72),(39,67,74,127),(40,122,75,70)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
C_2\times C_4.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
×
⋊
ℤ
𝔽
○
≀