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⋊C4.374C23, C4⋊C8.280C22, (C2×C4).237C24, (C2×C8).133C23, (C22×C4).792D4, C23.856(C2×D4), C4⋊Q8.254C22, (C22×Q16).8C2, (C2×Q8).33C23, 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
Subgroups: 396 in 240 conjugacy classes, 116 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×12], C22, C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×12], C2×C4 [×20], Q8 [×4], Q8 [×18], C23, C42 [×4], C42 [×4], C4⋊C4 [×2], C4⋊C4 [×13], C2×C8 [×4], C2×C8 [×4], Q16 [×16], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×10], C2×Q8 [×13], Q8⋊C4 [×8], C4⋊C8 [×4], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×3], C4×Q8 [×4], C4×Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C2×Q16 [×8], C2×Q16 [×8], C22×Q8, C22×Q8 [×2], C2×Q8⋊C4 [×2], C2×C4⋊C8, C4⋊2Q16 [×8], C2×C4×Q8, C2×C4⋊Q8, C22×Q16 [×2], C2×C4⋊2Q16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], Q16 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×Q16 [×6], C8.C22 [×2], C22×D4 [×2], C2×C4○D4, C4⋊2Q16 [×4], C2×C4⋊D4, C22×Q16, C2×C8.C22, C2×C4⋊2Q16
Generators and relations
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 >
►Regular action on 128 points
Generators in S128
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 117)(10 118)(11 119)(12 120)(13 113)(14 114)(15 115)(16 116)(17 102)(18 103)(19 104)(20 97)(21 98)(22 99)(23 100)(24 101)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(41 56)(42 49)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(71 87)(72 88)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(89 105)(90 106)(91 107)(92 108)(93 109)(94 110)(95 111)(96 112)
(1 82 107 75)(2 76 108 83)(3 84 109 77)(4 78 110 85)(5 86 111 79)(6 80 112 87)(7 88 105 73)(8 74 106 81)(9 32 41 101)(10 102 42 25)(11 26 43 103)(12 104 44 27)(13 28 45 97)(14 98 46 29)(15 30 47 99)(16 100 48 31)(17 49 57 118)(18 119 58 50)(19 51 59 120)(20 113 60 52)(21 53 61 114)(22 115 62 54)(23 55 63 116)(24 117 64 56)(33 68 93 125)(34 126 94 69)(35 70 95 127)(36 128 96 71)(37 72 89 121)(38 122 90 65)(39 66 91 123)(40 124 92 67)
(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 22 5 18)(2 21 6 17)(3 20 7 24)(4 19 8 23)(9 68 13 72)(10 67 14 71)(11 66 15 70)(12 65 16 69)(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)(49 76 53 80)(50 75 54 79)(51 74 55 78)(52 73 56 77)(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)(9,117)(10,118)(11,119)(12,120)(13,113)(14,114)(15,115)(16,116)(17,102)(18,103)(19,104)(20,97)(21,98)(22,99)(23,100)(24,101)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(41,56)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(89,105)(90,106)(91,107)(92,108)(93,109)(94,110)(95,111)(96,112), (1,82,107,75)(2,76,108,83)(3,84,109,77)(4,78,110,85)(5,86,111,79)(6,80,112,87)(7,88,105,73)(8,74,106,81)(9,32,41,101)(10,102,42,25)(11,26,43,103)(12,104,44,27)(13,28,45,97)(14,98,46,29)(15,30,47,99)(16,100,48,31)(17,49,57,118)(18,119,58,50)(19,51,59,120)(20,113,60,52)(21,53,61,114)(22,115,62,54)(23,55,63,116)(24,117,64,56)(33,68,93,125)(34,126,94,69)(35,70,95,127)(36,128,96,71)(37,72,89,121)(38,122,90,65)(39,66,91,123)(40,124,92,67), (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), (1,22,5,18)(2,21,6,17)(3,20,7,24)(4,19,8,23)(9,68,13,72)(10,67,14,71)(11,66,15,70)(12,65,16,69)(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)(49,76,53,80)(50,75,54,79)(51,74,55,78)(52,73,56,77)(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)(9,117)(10,118)(11,119)(12,120)(13,113)(14,114)(15,115)(16,116)(17,102)(18,103)(19,104)(20,97)(21,98)(22,99)(23,100)(24,101)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(41,56)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(89,105)(90,106)(91,107)(92,108)(93,109)(94,110)(95,111)(96,112), (1,82,107,75)(2,76,108,83)(3,84,109,77)(4,78,110,85)(5,86,111,79)(6,80,112,87)(7,88,105,73)(8,74,106,81)(9,32,41,101)(10,102,42,25)(11,26,43,103)(12,104,44,27)(13,28,45,97)(14,98,46,29)(15,30,47,99)(16,100,48,31)(17,49,57,118)(18,119,58,50)(19,51,59,120)(20,113,60,52)(21,53,61,114)(22,115,62,54)(23,55,63,116)(24,117,64,56)(33,68,93,125)(34,126,94,69)(35,70,95,127)(36,128,96,71)(37,72,89,121)(38,122,90,65)(39,66,91,123)(40,124,92,67), (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), (1,22,5,18)(2,21,6,17)(3,20,7,24)(4,19,8,23)(9,68,13,72)(10,67,14,71)(11,66,15,70)(12,65,16,69)(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)(49,76,53,80)(50,75,54,79)(51,74,55,78)(52,73,56,77)(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),(9,117),(10,118),(11,119),(12,120),(13,113),(14,114),(15,115),(16,116),(17,102),(18,103),(19,104),(20,97),(21,98),(22,99),(23,100),(24,101),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64),(41,56),(42,49),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(65,81),(66,82),(67,83),(68,84),(69,85),(70,86),(71,87),(72,88),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,127),(80,128),(89,105),(90,106),(91,107),(92,108),(93,109),(94,110),(95,111),(96,112)], [(1,82,107,75),(2,76,108,83),(3,84,109,77),(4,78,110,85),(5,86,111,79),(6,80,112,87),(7,88,105,73),(8,74,106,81),(9,32,41,101),(10,102,42,25),(11,26,43,103),(12,104,44,27),(13,28,45,97),(14,98,46,29),(15,30,47,99),(16,100,48,31),(17,49,57,118),(18,119,58,50),(19,51,59,120),(20,113,60,52),(21,53,61,114),(22,115,62,54),(23,55,63,116),(24,117,64,56),(33,68,93,125),(34,126,94,69),(35,70,95,127),(36,128,96,71),(37,72,89,121),(38,122,90,65),(39,66,91,123),(40,124,92,67)], [(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)], [(1,22,5,18),(2,21,6,17),(3,20,7,24),(4,19,8,23),(9,68,13,72),(10,67,14,71),(11,66,15,70),(12,65,16,69),(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),(49,76,53,80),(50,75,54,79),(51,74,55,78),(52,73,56,77),(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)])
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 | 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀