direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4.6Q16, C42.77D4, C42.159C23, C4⋊Q8.21C4, C4.34(C2×Q16), (C2×C4).57Q16, C4.52(C2×SD16), (C22×Q8).9C4, C4⋊C8.255C22, C42.100(C2×C4), (C2×C4).101SD16, (C22×C4).742D4, C4⋊Q8.233C22, C4.11(Q8⋊C4), (C2×C42).203C22, C23.223(C22⋊C4), C22.32(Q8⋊C4), C22.29(C4.D4), (C2×C4⋊Q8).2C2, (C2×C4⋊C8).12C2, (C2×Q8).27(C2×C4), (C2×C4).1230(C2×D4), C2.12(C2×Q8⋊C4), C2.16(C2×C4.D4), (C22×C4).225(C2×C4), (C2×C4).153(C22×C4), (C2×C4).104(C22⋊C4), C22.217(C2×C22⋊C4), SmallGroup(128,273)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4.6Q16
G = < a,b,c,d | a2=b4=c8=1, d2=b2c4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c-1 >
Subgroups: 244 in 128 conjugacy classes, 68 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C4⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C4.6Q16, C2×C4⋊C8, C2×C4⋊Q8, C2×C4.6Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, C4.D4, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C4.6Q16, C2×C4.D4, C2×Q8⋊C4, C2×C4.6Q16
►Regular action on 128 points
Generators in S128
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 97)(8 98)(9 53)(10 54)(11 55)(12 56)(13 49)(14 50)(15 51)(16 52)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)(41 85)(42 86)(43 87)(44 88)(45 81)(46 82)(47 83)(48 84)(65 94)(66 95)(67 96)(68 89)(69 90)(70 91)(71 92)(72 93)(73 109)(74 110)(75 111)(76 112)(77 105)(78 106)(79 107)(80 108)(113 125)(114 126)(115 127)(116 128)(117 121)(118 122)(119 123)(120 124)
(1 33 19 93)(2 94 20 34)(3 35 21 95)(4 96 22 36)(5 37 23 89)(6 90 24 38)(7 39 17 91)(8 92 18 40)(9 83 125 111)(10 112 126 84)(11 85 127 105)(12 106 128 86)(13 87 121 107)(14 108 122 88)(15 81 123 109)(16 110 124 82)(25 66 101 63)(26 64 102 67)(27 68 103 57)(28 58 104 69)(29 70 97 59)(30 60 98 71)(31 72 99 61)(32 62 100 65)(41 115 77 55)(42 56 78 116)(43 117 79 49)(44 50 80 118)(45 119 73 51)(46 52 74 120)(47 113 75 53)(48 54 76 114)
(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 51 23 115)(2 80 24 48)(3 49 17 113)(4 78 18 46)(5 55 19 119)(6 76 20 44)(7 53 21 117)(8 74 22 42)(9 25 121 97)(10 62 122 69)(11 31 123 103)(12 60 124 67)(13 29 125 101)(14 58 126 65)(15 27 127 99)(16 64 128 71)(26 86 98 110)(28 84 100 108)(30 82 102 106)(32 88 104 112)(33 73 89 41)(34 118 90 54)(35 79 91 47)(36 116 92 52)(37 77 93 45)(38 114 94 50)(39 75 95 43)(40 120 96 56)(57 105 72 81)(59 111 66 87)(61 109 68 85)(63 107 70 83)
G:=sub<Sym(128)| (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,97)(8,98)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60)(41,85)(42,86)(43,87)(44,88)(45,81)(46,82)(47,83)(48,84)(65,94)(66,95)(67,96)(68,89)(69,90)(70,91)(71,92)(72,93)(73,109)(74,110)(75,111)(76,112)(77,105)(78,106)(79,107)(80,108)(113,125)(114,126)(115,127)(116,128)(117,121)(118,122)(119,123)(120,124), (1,33,19,93)(2,94,20,34)(3,35,21,95)(4,96,22,36)(5,37,23,89)(6,90,24,38)(7,39,17,91)(8,92,18,40)(9,83,125,111)(10,112,126,84)(11,85,127,105)(12,106,128,86)(13,87,121,107)(14,108,122,88)(15,81,123,109)(16,110,124,82)(25,66,101,63)(26,64,102,67)(27,68,103,57)(28,58,104,69)(29,70,97,59)(30,60,98,71)(31,72,99,61)(32,62,100,65)(41,115,77,55)(42,56,78,116)(43,117,79,49)(44,50,80,118)(45,119,73,51)(46,52,74,120)(47,113,75,53)(48,54,76,114), (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), (1,51,23,115)(2,80,24,48)(3,49,17,113)(4,78,18,46)(5,55,19,119)(6,76,20,44)(7,53,21,117)(8,74,22,42)(9,25,121,97)(10,62,122,69)(11,31,123,103)(12,60,124,67)(13,29,125,101)(14,58,126,65)(15,27,127,99)(16,64,128,71)(26,86,98,110)(28,84,100,108)(30,82,102,106)(32,88,104,112)(33,73,89,41)(34,118,90,54)(35,79,91,47)(36,116,92,52)(37,77,93,45)(38,114,94,50)(39,75,95,43)(40,120,96,56)(57,105,72,81)(59,111,66,87)(61,109,68,85)(63,107,70,83)>;
G:=Group( (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,97)(8,98)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60)(41,85)(42,86)(43,87)(44,88)(45,81)(46,82)(47,83)(48,84)(65,94)(66,95)(67,96)(68,89)(69,90)(70,91)(71,92)(72,93)(73,109)(74,110)(75,111)(76,112)(77,105)(78,106)(79,107)(80,108)(113,125)(114,126)(115,127)(116,128)(117,121)(118,122)(119,123)(120,124), (1,33,19,93)(2,94,20,34)(3,35,21,95)(4,96,22,36)(5,37,23,89)(6,90,24,38)(7,39,17,91)(8,92,18,40)(9,83,125,111)(10,112,126,84)(11,85,127,105)(12,106,128,86)(13,87,121,107)(14,108,122,88)(15,81,123,109)(16,110,124,82)(25,66,101,63)(26,64,102,67)(27,68,103,57)(28,58,104,69)(29,70,97,59)(30,60,98,71)(31,72,99,61)(32,62,100,65)(41,115,77,55)(42,56,78,116)(43,117,79,49)(44,50,80,118)(45,119,73,51)(46,52,74,120)(47,113,75,53)(48,54,76,114), (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), (1,51,23,115)(2,80,24,48)(3,49,17,113)(4,78,18,46)(5,55,19,119)(6,76,20,44)(7,53,21,117)(8,74,22,42)(9,25,121,97)(10,62,122,69)(11,31,123,103)(12,60,124,67)(13,29,125,101)(14,58,126,65)(15,27,127,99)(16,64,128,71)(26,86,98,110)(28,84,100,108)(30,82,102,106)(32,88,104,112)(33,73,89,41)(34,118,90,54)(35,79,91,47)(36,116,92,52)(37,77,93,45)(38,114,94,50)(39,75,95,43)(40,120,96,56)(57,105,72,81)(59,111,66,87)(61,109,68,85)(63,107,70,83) );
G=PermutationGroup([[(1,99),(2,100),(3,101),(4,102),(5,103),(6,104),(7,97),(8,98),(9,53),(10,54),(11,55),(12,56),(13,49),(14,50),(15,51),(16,52),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28),(33,61),(34,62),(35,63),(36,64),(37,57),(38,58),(39,59),(40,60),(41,85),(42,86),(43,87),(44,88),(45,81),(46,82),(47,83),(48,84),(65,94),(66,95),(67,96),(68,89),(69,90),(70,91),(71,92),(72,93),(73,109),(74,110),(75,111),(76,112),(77,105),(78,106),(79,107),(80,108),(113,125),(114,126),(115,127),(116,128),(117,121),(118,122),(119,123),(120,124)], [(1,33,19,93),(2,94,20,34),(3,35,21,95),(4,96,22,36),(5,37,23,89),(6,90,24,38),(7,39,17,91),(8,92,18,40),(9,83,125,111),(10,112,126,84),(11,85,127,105),(12,106,128,86),(13,87,121,107),(14,108,122,88),(15,81,123,109),(16,110,124,82),(25,66,101,63),(26,64,102,67),(27,68,103,57),(28,58,104,69),(29,70,97,59),(30,60,98,71),(31,72,99,61),(32,62,100,65),(41,115,77,55),(42,56,78,116),(43,117,79,49),(44,50,80,118),(45,119,73,51),(46,52,74,120),(47,113,75,53),(48,54,76,114)], [(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)], [(1,51,23,115),(2,80,24,48),(3,49,17,113),(4,78,18,46),(5,55,19,119),(6,76,20,44),(7,53,21,117),(8,74,22,42),(9,25,121,97),(10,62,122,69),(11,31,123,103),(12,60,124,67),(13,29,125,101),(14,58,126,65),(15,27,127,99),(16,64,128,71),(26,86,98,110),(28,84,100,108),(30,82,102,106),(32,88,104,112),(33,73,89,41),(34,118,90,54),(35,79,91,47),(36,116,92,52),(37,77,93,45),(38,114,94,50),(39,75,95,43),(40,120,96,56),(57,105,72,81),(59,111,66,87),(61,109,68,85),(63,107,70,83)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | SD16 | Q16 | C4.D4 |
| kernel | C2×C4.6Q16 | C4.6Q16 | C2×C4⋊C8 | C2×C4⋊Q8 | C4⋊Q8 | C22×Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 2 |
Matrix representation of C2×C4.6Q16 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 2 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 14 | 0 | 0 |
| 0 | 0 | 3 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 9 |
| 0 | 0 | 0 | 0 | 3 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 2 |
| 0 | 0 | 0 | 0 | 7 | 11 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,2,4],[8,0,0,0,0,0,0,2,0,0,0,0,0,0,5,3,0,0,0,0,14,12,0,0,0,0,0,0,5,3,0,0,0,0,9,12],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,6,7,0,0,0,0,2,11] >;
C2×C4.6Q16 in GAP, Magma, Sage, TeX
C_2\times C_4._6Q_{16} % in TeX
G:=Group("C2xC4.6Q16"); // GroupNames label
G:=SmallGroup(128,273);
// by ID
G=gap.SmallGroup(128,273);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,456,1123,1018,248,1971,242]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=b^2*c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c^-1>;
// generators/relations