p-group, metabelian, nilpotent (class 4), monomial
Aliases: C4⋊1Q32, C8.16D8, C16.8D4, C42.337D4, C4.8(C2×D8), (C4×C16).9C2, (C2×C4).84D8, C8.40(C2×D4), (C2×C8).253D4, (C2×Q32).3C2, C2.10(C2×Q32), C4.2(C4⋊1D4), C2.13(C8⋊4D4), (C2×C16).84C22, (C4×C8).402C22, (C2×C8).545C23, C4⋊Q16.10C2, C22.131(C2×D8), (C2×Q16).17C22, (C2×C4).813(C2×D4), SmallGroup(128,979)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊Q32
G = < a,b,c | a4=b16=1, c2=b8, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 200 in 88 conjugacy classes, 40 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C16, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C4×C8, C2×C16, Q32, C4⋊Q8, C2×Q16, C2×Q16, C4×C16, C4⋊Q16, C2×Q32, C4⋊Q32
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, Q32, C4⋊1D4, C2×D8, C8⋊4D4, C2×Q32, C4⋊Q32
►Regular action on 128 points
Generators in S128
(1 70 21 108)(2 71 22 109)(3 72 23 110)(4 73 24 111)(5 74 25 112)(6 75 26 97)(7 76 27 98)(8 77 28 99)(9 78 29 100)(10 79 30 101)(11 80 31 102)(12 65 32 103)(13 66 17 104)(14 67 18 105)(15 68 19 106)(16 69 20 107)(33 119 96 64)(34 120 81 49)(35 121 82 50)(36 122 83 51)(37 123 84 52)(38 124 85 53)(39 125 86 54)(40 126 87 55)(41 127 88 56)(42 128 89 57)(43 113 90 58)(44 114 91 59)(45 115 92 60)(46 116 93 61)(47 117 94 62)(48 118 95 63)
(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)
(1 126 9 118)(2 125 10 117)(3 124 11 116)(4 123 12 115)(5 122 13 114)(6 121 14 113)(7 120 15 128)(8 119 16 127)(17 59 25 51)(18 58 26 50)(19 57 27 49)(20 56 28 64)(21 55 29 63)(22 54 30 62)(23 53 31 61)(24 52 32 60)(33 69 41 77)(34 68 42 76)(35 67 43 75)(36 66 44 74)(37 65 45 73)(38 80 46 72)(39 79 47 71)(40 78 48 70)(81 106 89 98)(82 105 90 97)(83 104 91 112)(84 103 92 111)(85 102 93 110)(86 101 94 109)(87 100 95 108)(88 99 96 107)
G:=sub<Sym(128)| (1,70,21,108)(2,71,22,109)(3,72,23,110)(4,73,24,111)(5,74,25,112)(6,75,26,97)(7,76,27,98)(8,77,28,99)(9,78,29,100)(10,79,30,101)(11,80,31,102)(12,65,32,103)(13,66,17,104)(14,67,18,105)(15,68,19,106)(16,69,20,107)(33,119,96,64)(34,120,81,49)(35,121,82,50)(36,122,83,51)(37,123,84,52)(38,124,85,53)(39,125,86,54)(40,126,87,55)(41,127,88,56)(42,128,89,57)(43,113,90,58)(44,114,91,59)(45,115,92,60)(46,116,93,61)(47,117,94,62)(48,118,95,63), (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), (1,126,9,118)(2,125,10,117)(3,124,11,116)(4,123,12,115)(5,122,13,114)(6,121,14,113)(7,120,15,128)(8,119,16,127)(17,59,25,51)(18,58,26,50)(19,57,27,49)(20,56,28,64)(21,55,29,63)(22,54,30,62)(23,53,31,61)(24,52,32,60)(33,69,41,77)(34,68,42,76)(35,67,43,75)(36,66,44,74)(37,65,45,73)(38,80,46,72)(39,79,47,71)(40,78,48,70)(81,106,89,98)(82,105,90,97)(83,104,91,112)(84,103,92,111)(85,102,93,110)(86,101,94,109)(87,100,95,108)(88,99,96,107)>;
G:=Group( (1,70,21,108)(2,71,22,109)(3,72,23,110)(4,73,24,111)(5,74,25,112)(6,75,26,97)(7,76,27,98)(8,77,28,99)(9,78,29,100)(10,79,30,101)(11,80,31,102)(12,65,32,103)(13,66,17,104)(14,67,18,105)(15,68,19,106)(16,69,20,107)(33,119,96,64)(34,120,81,49)(35,121,82,50)(36,122,83,51)(37,123,84,52)(38,124,85,53)(39,125,86,54)(40,126,87,55)(41,127,88,56)(42,128,89,57)(43,113,90,58)(44,114,91,59)(45,115,92,60)(46,116,93,61)(47,117,94,62)(48,118,95,63), (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), (1,126,9,118)(2,125,10,117)(3,124,11,116)(4,123,12,115)(5,122,13,114)(6,121,14,113)(7,120,15,128)(8,119,16,127)(17,59,25,51)(18,58,26,50)(19,57,27,49)(20,56,28,64)(21,55,29,63)(22,54,30,62)(23,53,31,61)(24,52,32,60)(33,69,41,77)(34,68,42,76)(35,67,43,75)(36,66,44,74)(37,65,45,73)(38,80,46,72)(39,79,47,71)(40,78,48,70)(81,106,89,98)(82,105,90,97)(83,104,91,112)(84,103,92,111)(85,102,93,110)(86,101,94,109)(87,100,95,108)(88,99,96,107) );
G=PermutationGroup([[(1,70,21,108),(2,71,22,109),(3,72,23,110),(4,73,24,111),(5,74,25,112),(6,75,26,97),(7,76,27,98),(8,77,28,99),(9,78,29,100),(10,79,30,101),(11,80,31,102),(12,65,32,103),(13,66,17,104),(14,67,18,105),(15,68,19,106),(16,69,20,107),(33,119,96,64),(34,120,81,49),(35,121,82,50),(36,122,83,51),(37,123,84,52),(38,124,85,53),(39,125,86,54),(40,126,87,55),(41,127,88,56),(42,128,89,57),(43,113,90,58),(44,114,91,59),(45,115,92,60),(46,116,93,61),(47,117,94,62),(48,118,95,63)], [(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)], [(1,126,9,118),(2,125,10,117),(3,124,11,116),(4,123,12,115),(5,122,13,114),(6,121,14,113),(7,120,15,128),(8,119,16,127),(17,59,25,51),(18,58,26,50),(19,57,27,49),(20,56,28,64),(21,55,29,63),(22,54,30,62),(23,53,31,61),(24,52,32,60),(33,69,41,77),(34,68,42,76),(35,67,43,75),(36,66,44,74),(37,65,45,73),(38,80,46,72),(39,79,47,71),(40,78,48,70),(81,106,89,98),(82,105,90,97),(83,104,91,112),(84,103,92,111),(85,102,93,110),(86,101,94,109),(87,100,95,108),(88,99,96,107)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 16 | 16 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | D4 | D4 | D4 | D8 | D8 | Q32 |
| kernel | C4⋊Q32 | C4×C16 | C4⋊Q16 | C2×Q32 | C16 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 4 | 4 | 16 |
Matrix representation of C4⋊Q32 ►in GL4(𝔽17) generated by
| 16 | 2 | 0 | 0 |
| 16 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 11 | 0 | 0 |
| 3 | 11 | 0 | 0 |
| 0 | 0 | 13 | 11 |
| 0 | 0 | 6 | 13 |
| 15 | 9 | 0 | 0 |
| 11 | 2 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 12 | 12 |
G:=sub<GL(4,GF(17))| [16,16,0,0,2,1,0,0,0,0,16,0,0,0,0,16],[0,3,0,0,11,11,0,0,0,0,13,6,0,0,11,13],[15,11,0,0,9,2,0,0,0,0,5,12,0,0,12,12] >;
C4⋊Q32 in GAP, Magma, Sage, TeX
C_4\rtimes Q_{32} % in TeX
G:=Group("C4:Q32"); // GroupNames label
G:=SmallGroup(128,979);
// by ID
G=gap.SmallGroup(128,979);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,288,422,100,1123,360,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^16=1,c^2=b^8,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations