p-group, metabelian, nilpotent (class 5), monomial
Aliases: Q32⋊2C4, C2.1Q64, C16.16D4, C2.2SD64, C4.2SD32, C8.16SD16, C22.9D16, (C2×C32).1C2, (C2×C4).66D8, C16.11(C2×C4), (C2×C8).238D4, (C2×Q32).1C2, C16⋊3C4.1C2, C2.7(C2.D16), C8.14(C22⋊C4), (C2×C16).76C22, C4.14(D4⋊C4), SmallGroup(128,148)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q32⋊2C4
G = < a,b,c | a16=c4=1, b2=a8, bab-1=cac-1=a-1, cbc-1=a-1b >
Smallest permutation representation of Q32⋊2C4
►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 72 9 80)(2 71 10 79)(3 70 11 78)(4 69 12 77)(5 68 13 76)(6 67 14 75)(7 66 15 74)(8 65 16 73)(17 49 25 57)(18 64 26 56)(19 63 27 55)(20 62 28 54)(21 61 29 53)(22 60 30 52)(23 59 31 51)(24 58 32 50)(33 108 41 100)(34 107 42 99)(35 106 43 98)(36 105 44 97)(37 104 45 112)(38 103 46 111)(39 102 47 110)(40 101 48 109)(81 116 89 124)(82 115 90 123)(83 114 91 122)(84 113 92 121)(85 128 93 120)(86 127 94 119)(87 126 95 118)(88 125 96 117)
(1 83 39 22)(2 82 40 21)(3 81 41 20)(4 96 42 19)(5 95 43 18)(6 94 44 17)(7 93 45 32)(8 92 46 31)(9 91 47 30)(10 90 48 29)(11 89 33 28)(12 88 34 27)(13 87 35 26)(14 86 36 25)(15 85 37 24)(16 84 38 23)(49 68 119 98)(50 67 120 97)(51 66 121 112)(52 65 122 111)(53 80 123 110)(54 79 124 109)(55 78 125 108)(56 77 126 107)(57 76 127 106)(58 75 128 105)(59 74 113 104)(60 73 114 103)(61 72 115 102)(62 71 116 101)(63 70 117 100)(64 69 118 99)
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), (1,72,9,80)(2,71,10,79)(3,70,11,78)(4,69,12,77)(5,68,13,76)(6,67,14,75)(7,66,15,74)(8,65,16,73)(17,49,25,57)(18,64,26,56)(19,63,27,55)(20,62,28,54)(21,61,29,53)(22,60,30,52)(23,59,31,51)(24,58,32,50)(33,108,41,100)(34,107,42,99)(35,106,43,98)(36,105,44,97)(37,104,45,112)(38,103,46,111)(39,102,47,110)(40,101,48,109)(81,116,89,124)(82,115,90,123)(83,114,91,122)(84,113,92,121)(85,128,93,120)(86,127,94,119)(87,126,95,118)(88,125,96,117), (1,83,39,22)(2,82,40,21)(3,81,41,20)(4,96,42,19)(5,95,43,18)(6,94,44,17)(7,93,45,32)(8,92,46,31)(9,91,47,30)(10,90,48,29)(11,89,33,28)(12,88,34,27)(13,87,35,26)(14,86,36,25)(15,85,37,24)(16,84,38,23)(49,68,119,98)(50,67,120,97)(51,66,121,112)(52,65,122,111)(53,80,123,110)(54,79,124,109)(55,78,125,108)(56,77,126,107)(57,76,127,106)(58,75,128,105)(59,74,113,104)(60,73,114,103)(61,72,115,102)(62,71,116,101)(63,70,117,100)(64,69,118,99)>;
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), (1,72,9,80)(2,71,10,79)(3,70,11,78)(4,69,12,77)(5,68,13,76)(6,67,14,75)(7,66,15,74)(8,65,16,73)(17,49,25,57)(18,64,26,56)(19,63,27,55)(20,62,28,54)(21,61,29,53)(22,60,30,52)(23,59,31,51)(24,58,32,50)(33,108,41,100)(34,107,42,99)(35,106,43,98)(36,105,44,97)(37,104,45,112)(38,103,46,111)(39,102,47,110)(40,101,48,109)(81,116,89,124)(82,115,90,123)(83,114,91,122)(84,113,92,121)(85,128,93,120)(86,127,94,119)(87,126,95,118)(88,125,96,117), (1,83,39,22)(2,82,40,21)(3,81,41,20)(4,96,42,19)(5,95,43,18)(6,94,44,17)(7,93,45,32)(8,92,46,31)(9,91,47,30)(10,90,48,29)(11,89,33,28)(12,88,34,27)(13,87,35,26)(14,86,36,25)(15,85,37,24)(16,84,38,23)(49,68,119,98)(50,67,120,97)(51,66,121,112)(52,65,122,111)(53,80,123,110)(54,79,124,109)(55,78,125,108)(56,77,126,107)(57,76,127,106)(58,75,128,105)(59,74,113,104)(60,73,114,103)(61,72,115,102)(62,71,116,101)(63,70,117,100)(64,69,118,99) );
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)], [(1,72,9,80),(2,71,10,79),(3,70,11,78),(4,69,12,77),(5,68,13,76),(6,67,14,75),(7,66,15,74),(8,65,16,73),(17,49,25,57),(18,64,26,56),(19,63,27,55),(20,62,28,54),(21,61,29,53),(22,60,30,52),(23,59,31,51),(24,58,32,50),(33,108,41,100),(34,107,42,99),(35,106,43,98),(36,105,44,97),(37,104,45,112),(38,103,46,111),(39,102,47,110),(40,101,48,109),(81,116,89,124),(82,115,90,123),(83,114,91,122),(84,113,92,121),(85,128,93,120),(86,127,94,119),(87,126,95,118),(88,125,96,117)], [(1,83,39,22),(2,82,40,21),(3,81,41,20),(4,96,42,19),(5,95,43,18),(6,94,44,17),(7,93,45,32),(8,92,46,31),(9,91,47,30),(10,90,48,29),(11,89,33,28),(12,88,34,27),(13,87,35,26),(14,86,36,25),(15,85,37,24),(16,84,38,23),(49,68,119,98),(50,67,120,97),(51,66,121,112),(52,65,122,111),(53,80,123,110),(54,79,124,109),(55,78,125,108),(56,77,126,107),(57,76,127,106),(58,75,128,105),(59,74,113,104),(60,73,114,103),(61,72,115,102),(62,71,116,101),(63,70,117,100),(64,69,118,99)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 16A | ··· | 16H | 32A | ··· | 32P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 32 | ··· | 32 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 16 | 16 | 16 | 16 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | SD16 | D8 | SD32 | D16 | SD64 | Q64 |
| kernel | Q32⋊2C4 | C16⋊3C4 | C2×C32 | C2×Q32 | Q32 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of Q32⋊2C4 ►in GL3(𝔽97) generated by
| 1 | 0 | 0 |
| 0 | 79 | 0 |
| 0 | 0 | 70 |
| 1 | 0 | 0 |
| 0 | 0 | 96 |
| 0 | 1 | 0 |
| 75 | 0 | 0 |
| 0 | 0 | 51 |
| 0 | 19 | 0 |
G:=sub<GL(3,GF(97))| [1,0,0,0,79,0,0,0,70],[1,0,0,0,0,1,0,96,0],[75,0,0,0,0,19,0,51,0] >;
Q32⋊2C4 in GAP, Magma, Sage, TeX
Q_{32}\rtimes_2C_4 % in TeX
G:=Group("Q32:2C4"); // GroupNames label
G:=SmallGroup(128,148);
// by ID
G=gap.SmallGroup(128,148);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,456,422,219,268,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c|a^16=c^4=1,b^2=a^8,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^-1*b>;
// generators/relations
Export