direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4⋊2Q16, C42.209D4, C42.318C23, C4⋊3(C2×Q16), (C2×C4)⋊10Q16, Q8.1(C2×D4), (C2×Q8).169D4, C4.64(C22×D4), C2.5(C22×Q16), C4.67(C4⋊D4), C4⋊C8.280C22, C4⋊C4.374C23, (C2×C8).133C23, (C2×C4).237C24, C23.856(C2×D4), (C22×C4).792D4, C4⋊Q8.254C22, (C2×Q8).33C23, (C22×Q16).8C2, C22.47(C2×Q16), (C4×Q8).289C22, (C2×C42).806C22, (C22×C8).138C22, (C2×Q16).115C22, C22.497(C22×D4), C22.169(C4⋊D4), (C22×C4).1527C23, Q8⋊C4.144C22, (C22×Q8).269C22, C22.102(C8.C22), (C2×C4⋊C8).41C2, (C2×C4×Q8).48C2, (C2×C4⋊Q8).42C2, C4.147(C2×C4○D4), C2.55(C2×C4⋊D4), (C2×C4).1417(C2×D4), C2.13(C2×C8.C22), (C2×C4).904(C4○D4), (C2×C4⋊C4).918C22, (C2×Q8⋊C4).23C2, SmallGroup(128,1765)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4⋊2Q16
G = < a,b,c,d | a2=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 396 in 240 conjugacy classes, 116 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, Q8⋊C4, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4×Q8, C4⋊Q8, C4⋊Q8, C22×C8, C2×Q16, C2×Q16, C22×Q8, C22×Q8, C2×Q8⋊C4, C2×C4⋊C8, C4⋊2Q16, C2×C4×Q8, C2×C4⋊Q8, C22×Q16, C2×C4⋊2Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C4○D4, C24, C4⋊D4, C2×Q16, C8.C22, C22×D4, C2×C4○D4, C4⋊2Q16, C2×C4⋊D4, C22×Q16, C2×C8.C22, C2×C4⋊2Q16
►Regular action on 128 points
Generators in S128
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 115)(10 116)(11 117)(12 118)(13 119)(14 120)(15 113)(16 114)(17 125)(18 126)(19 127)(20 128)(21 121)(22 122)(23 123)(24 124)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 49)(65 102)(66 103)(67 104)(68 97)(69 98)(70 99)(71 100)(72 101)(73 88)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)(89 107)(90 108)(91 109)(92 110)(93 111)(94 112)(95 105)(96 106)
(1 82 107 23)(2 24 108 83)(3 84 109 17)(4 18 110 85)(5 86 111 19)(6 20 112 87)(7 88 105 21)(8 22 106 81)(9 32 47 99)(10 100 48 25)(11 26 41 101)(12 102 42 27)(13 28 43 103)(14 104 44 29)(15 30 45 97)(16 98 46 31)(33 77 91 125)(34 126 92 78)(35 79 93 127)(36 128 94 80)(37 73 95 121)(38 122 96 74)(39 75 89 123)(40 124 90 76)(49 63 116 71)(50 72 117 64)(51 57 118 65)(52 66 119 58)(53 59 120 67)(54 68 113 60)(55 61 114 69)(56 70 115 62)
(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 70 5 66)(2 69 6 65)(3 68 7 72)(4 67 8 71)(9 79 13 75)(10 78 14 74)(11 77 15 73)(12 76 16 80)(17 54 21 50)(18 53 22 49)(19 52 23 56)(20 51 24 55)(25 92 29 96)(26 91 30 95)(27 90 31 94)(28 89 32 93)(33 97 37 101)(34 104 38 100)(35 103 39 99)(36 102 40 98)(41 125 45 121)(42 124 46 128)(43 123 47 127)(44 122 48 126)(57 108 61 112)(58 107 62 111)(59 106 63 110)(60 105 64 109)(81 116 85 120)(82 115 86 119)(83 114 87 118)(84 113 88 117)
G:=sub<Sym(128)| (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,115)(10,116)(11,117)(12,118)(13,119)(14,120)(15,113)(16,114)(17,125)(18,126)(19,127)(20,128)(21,121)(22,122)(23,123)(24,124)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,49)(65,102)(66,103)(67,104)(68,97)(69,98)(70,99)(71,100)(72,101)(73,88)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,105)(96,106), (1,82,107,23)(2,24,108,83)(3,84,109,17)(4,18,110,85)(5,86,111,19)(6,20,112,87)(7,88,105,21)(8,22,106,81)(9,32,47,99)(10,100,48,25)(11,26,41,101)(12,102,42,27)(13,28,43,103)(14,104,44,29)(15,30,45,97)(16,98,46,31)(33,77,91,125)(34,126,92,78)(35,79,93,127)(36,128,94,80)(37,73,95,121)(38,122,96,74)(39,75,89,123)(40,124,90,76)(49,63,116,71)(50,72,117,64)(51,57,118,65)(52,66,119,58)(53,59,120,67)(54,68,113,60)(55,61,114,69)(56,70,115,62), (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), (1,70,5,66)(2,69,6,65)(3,68,7,72)(4,67,8,71)(9,79,13,75)(10,78,14,74)(11,77,15,73)(12,76,16,80)(17,54,21,50)(18,53,22,49)(19,52,23,56)(20,51,24,55)(25,92,29,96)(26,91,30,95)(27,90,31,94)(28,89,32,93)(33,97,37,101)(34,104,38,100)(35,103,39,99)(36,102,40,98)(41,125,45,121)(42,124,46,128)(43,123,47,127)(44,122,48,126)(57,108,61,112)(58,107,62,111)(59,106,63,110)(60,105,64,109)(81,116,85,120)(82,115,86,119)(83,114,87,118)(84,113,88,117)>;
G:=Group( (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,115)(10,116)(11,117)(12,118)(13,119)(14,120)(15,113)(16,114)(17,125)(18,126)(19,127)(20,128)(21,121)(22,122)(23,123)(24,124)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,49)(65,102)(66,103)(67,104)(68,97)(69,98)(70,99)(71,100)(72,101)(73,88)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,105)(96,106), (1,82,107,23)(2,24,108,83)(3,84,109,17)(4,18,110,85)(5,86,111,19)(6,20,112,87)(7,88,105,21)(8,22,106,81)(9,32,47,99)(10,100,48,25)(11,26,41,101)(12,102,42,27)(13,28,43,103)(14,104,44,29)(15,30,45,97)(16,98,46,31)(33,77,91,125)(34,126,92,78)(35,79,93,127)(36,128,94,80)(37,73,95,121)(38,122,96,74)(39,75,89,123)(40,124,90,76)(49,63,116,71)(50,72,117,64)(51,57,118,65)(52,66,119,58)(53,59,120,67)(54,68,113,60)(55,61,114,69)(56,70,115,62), (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), (1,70,5,66)(2,69,6,65)(3,68,7,72)(4,67,8,71)(9,79,13,75)(10,78,14,74)(11,77,15,73)(12,76,16,80)(17,54,21,50)(18,53,22,49)(19,52,23,56)(20,51,24,55)(25,92,29,96)(26,91,30,95)(27,90,31,94)(28,89,32,93)(33,97,37,101)(34,104,38,100)(35,103,39,99)(36,102,40,98)(41,125,45,121)(42,124,46,128)(43,123,47,127)(44,122,48,126)(57,108,61,112)(58,107,62,111)(59,106,63,110)(60,105,64,109)(81,116,85,120)(82,115,86,119)(83,114,87,118)(84,113,88,117) );
G=PermutationGroup([[(1,39),(2,40),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,115),(10,116),(11,117),(12,118),(13,119),(14,120),(15,113),(16,114),(17,125),(18,126),(19,127),(20,128),(21,121),(22,122),(23,123),(24,124),(25,63),(26,64),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(41,50),(42,51),(43,52),(44,53),(45,54),(46,55),(47,56),(48,49),(65,102),(66,103),(67,104),(68,97),(69,98),(70,99),(71,100),(72,101),(73,88),(74,81),(75,82),(76,83),(77,84),(78,85),(79,86),(80,87),(89,107),(90,108),(91,109),(92,110),(93,111),(94,112),(95,105),(96,106)], [(1,82,107,23),(2,24,108,83),(3,84,109,17),(4,18,110,85),(5,86,111,19),(6,20,112,87),(7,88,105,21),(8,22,106,81),(9,32,47,99),(10,100,48,25),(11,26,41,101),(12,102,42,27),(13,28,43,103),(14,104,44,29),(15,30,45,97),(16,98,46,31),(33,77,91,125),(34,126,92,78),(35,79,93,127),(36,128,94,80),(37,73,95,121),(38,122,96,74),(39,75,89,123),(40,124,90,76),(49,63,116,71),(50,72,117,64),(51,57,118,65),(52,66,119,58),(53,59,120,67),(54,68,113,60),(55,61,114,69),(56,70,115,62)], [(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)], [(1,70,5,66),(2,69,6,65),(3,68,7,72),(4,67,8,71),(9,79,13,75),(10,78,14,74),(11,77,15,73),(12,76,16,80),(17,54,21,50),(18,53,22,49),(19,52,23,56),(20,51,24,55),(25,92,29,96),(26,91,30,95),(27,90,31,94),(28,89,32,93),(33,97,37,101),(34,104,38,100),(35,103,39,99),(36,102,40,98),(41,125,45,121),(42,124,46,128),(43,123,47,127),(44,122,48,126),(57,108,61,112),(58,107,62,111),(59,106,63,110),(60,105,64,109),(81,116,85,120),(82,115,86,119),(83,114,87,118),(84,113,88,117)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 4U | 4V | 8A | ··· | 8H |
| 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 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | Q16 | C4○D4 | C8.C22 |
| kernel | C2×C4⋊2Q16 | C2×Q8⋊C4 | C2×C4⋊C8 | C4⋊2Q16 | C2×C4×Q8 | C2×C4⋊Q8 | C22×Q16 | C42 | C22×C4 | C2×Q8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 4 | 2 |
Matrix representation of C2×C4⋊2Q16 ►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 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 11 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 12 |
| 0 | 0 | 0 | 0 | 2 | 14 |
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],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,15,11,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,3,2,0,0,0,0,12,14] >;
C2×C4⋊2Q16 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes_2Q_{16} % in TeX
G:=Group("C2xC4:2Q16"); // GroupNames label
G:=SmallGroup(128,1765);
// by ID
G=gap.SmallGroup(128,1765);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,352,4037,1027,124]);
// 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,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations