p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊5Q16, C42.504C23, C4.252- (1+4), (Q82).3C2, (C8×Q8).8C2, C4⋊C4.277D4, (C4×Q16).8C2, C4.30(C2×Q16), Q8○3(Q8⋊C4), (C4×C8).93C22, Q8⋊3Q8.4C2, C4.Q16.9C2, (C2×Q8).269D4, C4⋊C4.431C23, C4⋊C8.303C22, (C2×C4).555C24, (C2×C8).206C23, Q8.33(C4○D4), C4⋊2Q16.10C2, C4⋊Q8.184C22, C4.SD16.8C2, C2.21(C22×Q16), C2.63(Q8⋊5D4), C2.98(D4○SD16), (C2×Q8).253C23, (C4×Q8).309C22, C2.D8.201C22, Q8⋊C4.18C22, (C2×Q16).141C22, C22.815(C22×D4), C4.256(C2×C4○D4), (C2×C4).1101(C2×D4), SmallGroup(128,2095)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 272 in 168 conjugacy classes, 96 normal (18 characteristic)
C1, C2 [×3], C4 [×2], C4 [×6], C4 [×12], C22, C8 [×4], C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×6], Q8 [×10], C42 [×3], C42 [×6], C4⋊C4, C4⋊C4 [×6], C4⋊C4 [×15], C2×C8, C2×C8 [×3], Q16 [×6], C2×Q8 [×2], C2×Q8 [×3], C2×Q8 [×3], C4×C8 [×3], Q8⋊C4, Q8⋊C4 [×9], C4⋊C8 [×3], C2.D8 [×3], C4×Q8, C4×Q8 [×6], C4×Q8 [×2], C42.C2 [×3], C4⋊Q8 [×6], C4⋊Q8 [×3], C2×Q16 [×3], C4×Q16 [×3], C8×Q8, C4⋊2Q16 [×3], C4.Q16 [×3], C4.SD16 [×3], Q8⋊3Q8, Q82, Q8⋊5Q16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C4○D4 [×2], C24, C2×Q16 [×6], C22×D4, C2×C4○D4, 2- (1+4), Q8⋊5D4, C22×Q16, D4○SD16, Q8⋊5Q16
Generators and relations
G = < a,b,c,d | a4=c8=1, b2=a2, d2=c4, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=c-1 >
►Regular action on 128 points
Generators in S128
(1 39 107 91)(2 40 108 92)(3 33 109 93)(4 34 110 94)(5 35 111 95)(6 36 112 96)(7 37 105 89)(8 38 106 90)(9 117 41 56)(10 118 42 49)(11 119 43 50)(12 120 44 51)(13 113 45 52)(14 114 46 53)(15 115 47 54)(16 116 48 55)(17 25 57 102)(18 26 58 103)(19 27 59 104)(20 28 60 97)(21 29 61 98)(22 30 62 99)(23 31 63 100)(24 32 64 101)(65 74 122 81)(66 75 123 82)(67 76 124 83)(68 77 125 84)(69 78 126 85)(70 79 127 86)(71 80 128 87)(72 73 121 88)
(1 127 107 70)(2 71 108 128)(3 121 109 72)(4 65 110 122)(5 123 111 66)(6 67 112 124)(7 125 105 68)(8 69 106 126)(9 20 41 60)(10 61 42 21)(11 22 43 62)(12 63 44 23)(13 24 45 64)(14 57 46 17)(15 18 47 58)(16 59 48 19)(25 53 102 114)(26 115 103 54)(27 55 104 116)(28 117 97 56)(29 49 98 118)(30 119 99 50)(31 51 100 120)(32 113 101 52)(33 73 93 88)(34 81 94 74)(35 75 95 82)(36 83 96 76)(37 77 89 84)(38 85 90 78)(39 79 91 86)(40 87 92 80)
(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 26 5 30)(2 25 6 29)(3 32 7 28)(4 31 8 27)(9 73 13 77)(10 80 14 76)(11 79 15 75)(12 78 16 74)(17 96 21 92)(18 95 22 91)(19 94 23 90)(20 93 24 89)(33 64 37 60)(34 63 38 59)(35 62 39 58)(36 61 40 57)(41 88 45 84)(42 87 46 83)(43 86 47 82)(44 85 48 81)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)(97 109 101 105)(98 108 102 112)(99 107 103 111)(100 106 104 110)(113 125 117 121)(114 124 118 128)(115 123 119 127)(116 122 120 126)
G:=sub<Sym(128)| (1,39,107,91)(2,40,108,92)(3,33,109,93)(4,34,110,94)(5,35,111,95)(6,36,112,96)(7,37,105,89)(8,38,106,90)(9,117,41,56)(10,118,42,49)(11,119,43,50)(12,120,44,51)(13,113,45,52)(14,114,46,53)(15,115,47,54)(16,116,48,55)(17,25,57,102)(18,26,58,103)(19,27,59,104)(20,28,60,97)(21,29,61,98)(22,30,62,99)(23,31,63,100)(24,32,64,101)(65,74,122,81)(66,75,123,82)(67,76,124,83)(68,77,125,84)(69,78,126,85)(70,79,127,86)(71,80,128,87)(72,73,121,88), (1,127,107,70)(2,71,108,128)(3,121,109,72)(4,65,110,122)(5,123,111,66)(6,67,112,124)(7,125,105,68)(8,69,106,126)(9,20,41,60)(10,61,42,21)(11,22,43,62)(12,63,44,23)(13,24,45,64)(14,57,46,17)(15,18,47,58)(16,59,48,19)(25,53,102,114)(26,115,103,54)(27,55,104,116)(28,117,97,56)(29,49,98,118)(30,119,99,50)(31,51,100,120)(32,113,101,52)(33,73,93,88)(34,81,94,74)(35,75,95,82)(36,83,96,76)(37,77,89,84)(38,85,90,78)(39,79,91,86)(40,87,92,80), (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), (1,26,5,30)(2,25,6,29)(3,32,7,28)(4,31,8,27)(9,73,13,77)(10,80,14,76)(11,79,15,75)(12,78,16,74)(17,96,21,92)(18,95,22,91)(19,94,23,90)(20,93,24,89)(33,64,37,60)(34,63,38,59)(35,62,39,58)(36,61,40,57)(41,88,45,84)(42,87,46,83)(43,86,47,82)(44,85,48,81)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72)(97,109,101,105)(98,108,102,112)(99,107,103,111)(100,106,104,110)(113,125,117,121)(114,124,118,128)(115,123,119,127)(116,122,120,126)>;
G:=Group( (1,39,107,91)(2,40,108,92)(3,33,109,93)(4,34,110,94)(5,35,111,95)(6,36,112,96)(7,37,105,89)(8,38,106,90)(9,117,41,56)(10,118,42,49)(11,119,43,50)(12,120,44,51)(13,113,45,52)(14,114,46,53)(15,115,47,54)(16,116,48,55)(17,25,57,102)(18,26,58,103)(19,27,59,104)(20,28,60,97)(21,29,61,98)(22,30,62,99)(23,31,63,100)(24,32,64,101)(65,74,122,81)(66,75,123,82)(67,76,124,83)(68,77,125,84)(69,78,126,85)(70,79,127,86)(71,80,128,87)(72,73,121,88), (1,127,107,70)(2,71,108,128)(3,121,109,72)(4,65,110,122)(5,123,111,66)(6,67,112,124)(7,125,105,68)(8,69,106,126)(9,20,41,60)(10,61,42,21)(11,22,43,62)(12,63,44,23)(13,24,45,64)(14,57,46,17)(15,18,47,58)(16,59,48,19)(25,53,102,114)(26,115,103,54)(27,55,104,116)(28,117,97,56)(29,49,98,118)(30,119,99,50)(31,51,100,120)(32,113,101,52)(33,73,93,88)(34,81,94,74)(35,75,95,82)(36,83,96,76)(37,77,89,84)(38,85,90,78)(39,79,91,86)(40,87,92,80), (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), (1,26,5,30)(2,25,6,29)(3,32,7,28)(4,31,8,27)(9,73,13,77)(10,80,14,76)(11,79,15,75)(12,78,16,74)(17,96,21,92)(18,95,22,91)(19,94,23,90)(20,93,24,89)(33,64,37,60)(34,63,38,59)(35,62,39,58)(36,61,40,57)(41,88,45,84)(42,87,46,83)(43,86,47,82)(44,85,48,81)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72)(97,109,101,105)(98,108,102,112)(99,107,103,111)(100,106,104,110)(113,125,117,121)(114,124,118,128)(115,123,119,127)(116,122,120,126) );
G=PermutationGroup([(1,39,107,91),(2,40,108,92),(3,33,109,93),(4,34,110,94),(5,35,111,95),(6,36,112,96),(7,37,105,89),(8,38,106,90),(9,117,41,56),(10,118,42,49),(11,119,43,50),(12,120,44,51),(13,113,45,52),(14,114,46,53),(15,115,47,54),(16,116,48,55),(17,25,57,102),(18,26,58,103),(19,27,59,104),(20,28,60,97),(21,29,61,98),(22,30,62,99),(23,31,63,100),(24,32,64,101),(65,74,122,81),(66,75,123,82),(67,76,124,83),(68,77,125,84),(69,78,126,85),(70,79,127,86),(71,80,128,87),(72,73,121,88)], [(1,127,107,70),(2,71,108,128),(3,121,109,72),(4,65,110,122),(5,123,111,66),(6,67,112,124),(7,125,105,68),(8,69,106,126),(9,20,41,60),(10,61,42,21),(11,22,43,62),(12,63,44,23),(13,24,45,64),(14,57,46,17),(15,18,47,58),(16,59,48,19),(25,53,102,114),(26,115,103,54),(27,55,104,116),(28,117,97,56),(29,49,98,118),(30,119,99,50),(31,51,100,120),(32,113,101,52),(33,73,93,88),(34,81,94,74),(35,75,95,82),(36,83,96,76),(37,77,89,84),(38,85,90,78),(39,79,91,86),(40,87,92,80)], [(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)], [(1,26,5,30),(2,25,6,29),(3,32,7,28),(4,31,8,27),(9,73,13,77),(10,80,14,76),(11,79,15,75),(12,78,16,74),(17,96,21,92),(18,95,22,91),(19,94,23,90),(20,93,24,89),(33,64,37,60),(34,63,38,59),(35,62,39,58),(36,61,40,57),(41,88,45,84),(42,87,46,83),(43,86,47,82),(44,85,48,81),(49,71,53,67),(50,70,54,66),(51,69,55,65),(52,68,56,72),(97,109,101,105),(98,108,102,112),(99,107,103,111),(100,106,104,110),(113,125,117,121),(114,124,118,128),(115,123,119,127),(116,122,120,126)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 7 | 1 |
| 0 | 0 | 1 | 10 |
| 8 | 0 | 0 | 0 |
| 5 | 15 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
| 1 | 15 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,7,1,0,0,1,10],[8,5,0,0,0,15,0,0,0,0,0,4,0,0,13,0],[1,1,0,0,15,16,0,0,0,0,16,0,0,0,0,16] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4H | 4I | ··· | 4O | 4P | ··· | 4U | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q16 | C4○D4 | 2- (1+4) | D4○SD16 |
| kernel | Q8⋊5Q16 | C4×Q16 | C8×Q8 | C4⋊2Q16 | C4.Q16 | C4.SD16 | Q8⋊3Q8 | Q82 | C4⋊C4 | C2×Q8 | Q8 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 1 | 3 | 3 | 3 | 1 | 1 | 3 | 1 | 8 | 4 | 1 | 2 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_5Q_{16} % in TeX
G:=Group("Q8:5Q16"); // GroupNames label
G:=SmallGroup(128,2095);
// by ID
G=gap.SmallGroup(128,2095);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,352,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=1,b^2=a^2,d^2=c^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀