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, C4, C4, C4, C22, C8, C8, C2×C4, C2×C4, Q8, C16, C16, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C4×C8, C4×C8, C4⋊C8, C4⋊C8, C2×C16, C2×C16, C4×Q8, C4×C16, C16⋊5C4, C4⋊C16, C4⋊C16, C8×Q8, C16⋊4Q8
Quotients: C1, C2, C4, C22, C8, C2×C4, Q8, C23, C2×C8, C22×C4, C2×Q8, C4○D4, M5(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 34 80 52)(2 35 65 53)(3 36 66 54)(4 37 67 55)(5 38 68 56)(6 39 69 57)(7 40 70 58)(8 41 71 59)(9 42 72 60)(10 43 73 61)(11 44 74 62)(12 45 75 63)(13 46 76 64)(14 47 77 49)(15 48 78 50)(16 33 79 51)(17 86 109 123)(18 87 110 124)(19 88 111 125)(20 89 112 126)(21 90 97 127)(22 91 98 128)(23 92 99 113)(24 93 100 114)(25 94 101 115)(26 95 102 116)(27 96 103 117)(28 81 104 118)(29 82 105 119)(30 83 106 120)(31 84 107 121)(32 85 108 122)
(1 114 80 93)(2 123 65 86)(3 116 66 95)(4 125 67 88)(5 118 68 81)(6 127 69 90)(7 120 70 83)(8 113 71 92)(9 122 72 85)(10 115 73 94)(11 124 74 87)(12 117 75 96)(13 126 76 89)(14 119 77 82)(15 128 78 91)(16 121 79 84)(17 35 109 53)(18 44 110 62)(19 37 111 55)(20 46 112 64)(21 39 97 57)(22 48 98 50)(23 41 99 59)(24 34 100 52)(25 43 101 61)(26 36 102 54)(27 45 103 63)(28 38 104 56)(29 47 105 49)(30 40 106 58)(31 33 107 51)(32 42 108 60)
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), (1,34,80,52)(2,35,65,53)(3,36,66,54)(4,37,67,55)(5,38,68,56)(6,39,69,57)(7,40,70,58)(8,41,71,59)(9,42,72,60)(10,43,73,61)(11,44,74,62)(12,45,75,63)(13,46,76,64)(14,47,77,49)(15,48,78,50)(16,33,79,51)(17,86,109,123)(18,87,110,124)(19,88,111,125)(20,89,112,126)(21,90,97,127)(22,91,98,128)(23,92,99,113)(24,93,100,114)(25,94,101,115)(26,95,102,116)(27,96,103,117)(28,81,104,118)(29,82,105,119)(30,83,106,120)(31,84,107,121)(32,85,108,122), (1,114,80,93)(2,123,65,86)(3,116,66,95)(4,125,67,88)(5,118,68,81)(6,127,69,90)(7,120,70,83)(8,113,71,92)(9,122,72,85)(10,115,73,94)(11,124,74,87)(12,117,75,96)(13,126,76,89)(14,119,77,82)(15,128,78,91)(16,121,79,84)(17,35,109,53)(18,44,110,62)(19,37,111,55)(20,46,112,64)(21,39,97,57)(22,48,98,50)(23,41,99,59)(24,34,100,52)(25,43,101,61)(26,36,102,54)(27,45,103,63)(28,38,104,56)(29,47,105,49)(30,40,106,58)(31,33,107,51)(32,42,108,60)>;
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), (1,34,80,52)(2,35,65,53)(3,36,66,54)(4,37,67,55)(5,38,68,56)(6,39,69,57)(7,40,70,58)(8,41,71,59)(9,42,72,60)(10,43,73,61)(11,44,74,62)(12,45,75,63)(13,46,76,64)(14,47,77,49)(15,48,78,50)(16,33,79,51)(17,86,109,123)(18,87,110,124)(19,88,111,125)(20,89,112,126)(21,90,97,127)(22,91,98,128)(23,92,99,113)(24,93,100,114)(25,94,101,115)(26,95,102,116)(27,96,103,117)(28,81,104,118)(29,82,105,119)(30,83,106,120)(31,84,107,121)(32,85,108,122), (1,114,80,93)(2,123,65,86)(3,116,66,95)(4,125,67,88)(5,118,68,81)(6,127,69,90)(7,120,70,83)(8,113,71,92)(9,122,72,85)(10,115,73,94)(11,124,74,87)(12,117,75,96)(13,126,76,89)(14,119,77,82)(15,128,78,91)(16,121,79,84)(17,35,109,53)(18,44,110,62)(19,37,111,55)(20,46,112,64)(21,39,97,57)(22,48,98,50)(23,41,99,59)(24,34,100,52)(25,43,101,61)(26,36,102,54)(27,45,103,63)(28,38,104,56)(29,47,105,49)(30,40,106,58)(31,33,107,51)(32,42,108,60) );
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)], [(1,34,80,52),(2,35,65,53),(3,36,66,54),(4,37,67,55),(5,38,68,56),(6,39,69,57),(7,40,70,58),(8,41,71,59),(9,42,72,60),(10,43,73,61),(11,44,74,62),(12,45,75,63),(13,46,76,64),(14,47,77,49),(15,48,78,50),(16,33,79,51),(17,86,109,123),(18,87,110,124),(19,88,111,125),(20,89,112,126),(21,90,97,127),(22,91,98,128),(23,92,99,113),(24,93,100,114),(25,94,101,115),(26,95,102,116),(27,96,103,117),(28,81,104,118),(29,82,105,119),(30,83,106,120),(31,84,107,121),(32,85,108,122)], [(1,114,80,93),(2,123,65,86),(3,116,66,95),(4,125,67,88),(5,118,68,81),(6,127,69,90),(7,120,70,83),(8,113,71,92),(9,122,72,85),(10,115,73,94),(11,124,74,87),(12,117,75,96),(13,126,76,89),(14,119,77,82),(15,128,78,91),(16,121,79,84),(17,35,109,53),(18,44,110,62),(19,37,111,55),(20,46,112,64),(21,39,97,57),(22,48,98,50),(23,41,99,59),(24,34,100,52),(25,43,101,61),(26,36,102,54),(27,45,103,63),(28,38,104,56),(29,47,105,49),(30,40,106,58),(31,33,107,51),(32,42,108,60)]])
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