p-group, metabelian, nilpotent (class 2), monomial
Aliases: C16⋊4Q8, C4.3M5(2), C4⋊C4.11C8, C4⋊C8.24C4, C2.6(C8×Q8), C4⋊C16.13C2, (C2×Q8).9C8, C4.48(C4×Q8), C8.44(C2×Q8), (C4×C16).18C2, (C4×Q8).19C4, (C8×Q8).16C2, C16⋊5C4.9C2, C2.9(D4○C16), C4.61(C8○D4), C8.105(C4○D4), (C2×C8).636C23, C42.173(C2×C4), (C2×C16).57C22, (C4×C8).378C22, C2.11(C2×M5(2)), C22.54(C22×C8), (C2×C4).31(C2×C8), (C2×C8).152(C2×C4), (C2×C4).621(C22×C4), SmallGroup(128,915)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16⋊4Q8
G = < a,b,c | a16=b4=1, c2=b2, ab=ba, cac-1=a9, cbc-1=b-1 >
Subgroups: 76 in 63 conjugacy classes, 50 normal (26 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×5], C22, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], Q8 [×2], C16 [×2], C16 [×3], C42, C42 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C2×Q8, C4×C8, C4×C8 [×2], C4⋊C8, C4⋊C8 [×2], C2×C16 [×2], C2×C16 [×2], C4×Q8, C4×C16, C16⋊5C4 [×2], C4⋊C16, C4⋊C16 [×2], C8×Q8, C16⋊4Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], Q8 [×2], C23, C2×C8 [×6], C22×C4, C2×Q8, C4○D4, M5(2) [×2], C4×Q8, C22×C8, C8○D4, C8×Q8, C2×M5(2), D4○C16, C16⋊4Q8
►Regular action on 128 points
Generators in S128
(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 59 48 76)(2 60 33 77)(3 61 34 78)(4 62 35 79)(5 63 36 80)(6 64 37 65)(7 49 38 66)(8 50 39 67)(9 51 40 68)(10 52 41 69)(11 53 42 70)(12 54 43 71)(13 55 44 72)(14 56 45 73)(15 57 46 74)(16 58 47 75)(17 99 84 124)(18 100 85 125)(19 101 86 126)(20 102 87 127)(21 103 88 128)(22 104 89 113)(23 105 90 114)(24 106 91 115)(25 107 92 116)(26 108 93 117)(27 109 94 118)(28 110 95 119)(29 111 96 120)(30 112 81 121)(31 97 82 122)(32 98 83 123)
(1 28 48 95)(2 21 33 88)(3 30 34 81)(4 23 35 90)(5 32 36 83)(6 25 37 92)(7 18 38 85)(8 27 39 94)(9 20 40 87)(10 29 41 96)(11 22 42 89)(12 31 43 82)(13 24 44 91)(14 17 45 84)(15 26 46 93)(16 19 47 86)(49 125 66 100)(50 118 67 109)(51 127 68 102)(52 120 69 111)(53 113 70 104)(54 122 71 97)(55 115 72 106)(56 124 73 99)(57 117 74 108)(58 126 75 101)(59 119 76 110)(60 128 77 103)(61 121 78 112)(62 114 79 105)(63 123 80 98)(64 116 65 107)
G:=sub<Sym(128)| (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), (1,59,48,76)(2,60,33,77)(3,61,34,78)(4,62,35,79)(5,63,36,80)(6,64,37,65)(7,49,38,66)(8,50,39,67)(9,51,40,68)(10,52,41,69)(11,53,42,70)(12,54,43,71)(13,55,44,72)(14,56,45,73)(15,57,46,74)(16,58,47,75)(17,99,84,124)(18,100,85,125)(19,101,86,126)(20,102,87,127)(21,103,88,128)(22,104,89,113)(23,105,90,114)(24,106,91,115)(25,107,92,116)(26,108,93,117)(27,109,94,118)(28,110,95,119)(29,111,96,120)(30,112,81,121)(31,97,82,122)(32,98,83,123), (1,28,48,95)(2,21,33,88)(3,30,34,81)(4,23,35,90)(5,32,36,83)(6,25,37,92)(7,18,38,85)(8,27,39,94)(9,20,40,87)(10,29,41,96)(11,22,42,89)(12,31,43,82)(13,24,44,91)(14,17,45,84)(15,26,46,93)(16,19,47,86)(49,125,66,100)(50,118,67,109)(51,127,68,102)(52,120,69,111)(53,113,70,104)(54,122,71,97)(55,115,72,106)(56,124,73,99)(57,117,74,108)(58,126,75,101)(59,119,76,110)(60,128,77,103)(61,121,78,112)(62,114,79,105)(63,123,80,98)(64,116,65,107)>;
G:=Group( (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), (1,59,48,76)(2,60,33,77)(3,61,34,78)(4,62,35,79)(5,63,36,80)(6,64,37,65)(7,49,38,66)(8,50,39,67)(9,51,40,68)(10,52,41,69)(11,53,42,70)(12,54,43,71)(13,55,44,72)(14,56,45,73)(15,57,46,74)(16,58,47,75)(17,99,84,124)(18,100,85,125)(19,101,86,126)(20,102,87,127)(21,103,88,128)(22,104,89,113)(23,105,90,114)(24,106,91,115)(25,107,92,116)(26,108,93,117)(27,109,94,118)(28,110,95,119)(29,111,96,120)(30,112,81,121)(31,97,82,122)(32,98,83,123), (1,28,48,95)(2,21,33,88)(3,30,34,81)(4,23,35,90)(5,32,36,83)(6,25,37,92)(7,18,38,85)(8,27,39,94)(9,20,40,87)(10,29,41,96)(11,22,42,89)(12,31,43,82)(13,24,44,91)(14,17,45,84)(15,26,46,93)(16,19,47,86)(49,125,66,100)(50,118,67,109)(51,127,68,102)(52,120,69,111)(53,113,70,104)(54,122,71,97)(55,115,72,106)(56,124,73,99)(57,117,74,108)(58,126,75,101)(59,119,76,110)(60,128,77,103)(61,121,78,112)(62,114,79,105)(63,123,80,98)(64,116,65,107) );
G=PermutationGroup([(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)], [(1,59,48,76),(2,60,33,77),(3,61,34,78),(4,62,35,79),(5,63,36,80),(6,64,37,65),(7,49,38,66),(8,50,39,67),(9,51,40,68),(10,52,41,69),(11,53,42,70),(12,54,43,71),(13,55,44,72),(14,56,45,73),(15,57,46,74),(16,58,47,75),(17,99,84,124),(18,100,85,125),(19,101,86,126),(20,102,87,127),(21,103,88,128),(22,104,89,113),(23,105,90,114),(24,106,91,115),(25,107,92,116),(26,108,93,117),(27,109,94,118),(28,110,95,119),(29,111,96,120),(30,112,81,121),(31,97,82,122),(32,98,83,123)], [(1,28,48,95),(2,21,33,88),(3,30,34,81),(4,23,35,90),(5,32,36,83),(6,25,37,92),(7,18,38,85),(8,27,39,94),(9,20,40,87),(10,29,41,96),(11,22,42,89),(12,31,43,82),(13,24,44,91),(14,17,45,84),(15,26,46,93),(16,19,47,86),(49,125,66,100),(50,118,67,109),(51,127,68,102),(52,120,69,111),(53,113,70,104),(54,122,71,97),(55,115,72,106),(56,124,73,99),(57,117,74,108),(58,126,75,101),(59,119,76,110),(60,128,77,103),(61,121,78,112),(62,114,79,105),(63,123,80,98),(64,116,65,107)])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 8O | 8P | 16A | ··· | 16P | 16Q | ··· | 16X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | Q8 | C4○D4 | M5(2) | C8○D4 | D4○C16 |
| kernel | C16⋊4Q8 | C4×C16 | C16⋊5C4 | C4⋊C16 | C8×Q8 | C4⋊C8 | C4×Q8 | C4⋊C4 | C2×Q8 | C16 | C8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 3 | 1 | 6 | 2 | 12 | 4 | 2 | 2 | 8 | 4 | 8 |
Matrix representation of C16⋊4Q8 ►in GL4(𝔽17) generated by
| 0 | 9 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 4 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 3 | 0 |
| 11 | 11 | 0 | 0 |
| 3 | 6 | 0 | 0 |
| 0 | 0 | 16 | 9 |
| 0 | 0 | 13 | 1 |
G:=sub<GL(4,GF(17))| [0,13,0,0,9,0,0,0,0,0,0,4,0,0,9,0],[16,0,0,0,0,16,0,0,0,0,0,3,0,0,11,0],[11,3,0,0,11,6,0,0,0,0,16,13,0,0,9,1] >;
C16⋊4Q8 in GAP, Magma, Sage, TeX
C_{16}\rtimes_4Q_8 % in TeX
G:=Group("C16:4Q8"); // GroupNames label
G:=SmallGroup(128,915);
// by ID
G=gap.SmallGroup(128,915);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,1430,142,102,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^9,c*b*c^-1=b^-1>;
// generators/relations