p-group, metabelian, nilpotent (class 4), monomial
Aliases: Q32⋊4C4, C42.133D4, C16.8(C2×C4), C4.29(C4×D4), C2.17(C4×D8), (C2×C8).211D4, (C2×C4).110D8, Q16.3(C2×C4), (C2×Q32).6C2, C16⋊5C4.1C2, C16⋊4C4.2C2, C4.15(C4○D8), C8.42(C4○D4), C8.39(C22×C4), (C4×Q16).19C2, C22.65(C2×D8), (C4×C8).219C22, (C2×C8).506C23, (C2×C16).20C22, C2.Q32.6C2, C2.5(Q32⋊C2), C2.D8.154C22, (C2×Q16).105C22, (C2×C4).772(C2×D4), SmallGroup(128,908)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q32⋊4C4
G = < a,b,c | a16=c4=1, b2=a8, bab-1=a-1, cac-1=a9, bc=cb >
Subgroups: 148 in 74 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, C16, C16, C42, C42, C4⋊C4, C2×C8, Q16, Q16, C2×Q8, C4×C8, Q8⋊C4, C2.D8, C2×C16, Q32, C4×Q8, C2×Q16, C16⋊5C4, C2.Q32, C16⋊4C4, C4×Q16, C2×Q32, Q32⋊4C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, C4×D4, C2×D8, C4○D8, C4×D8, Q32⋊C2, Q32⋊4C4
►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)
(1 115 9 123)(2 114 10 122)(3 113 11 121)(4 128 12 120)(5 127 13 119)(6 126 14 118)(7 125 15 117)(8 124 16 116)(17 62 25 54)(18 61 26 53)(19 60 27 52)(20 59 28 51)(21 58 29 50)(22 57 30 49)(23 56 31 64)(24 55 32 63)(33 91 41 83)(34 90 42 82)(35 89 43 81)(36 88 44 96)(37 87 45 95)(38 86 46 94)(39 85 47 93)(40 84 48 92)(65 111 73 103)(66 110 74 102)(67 109 75 101)(68 108 76 100)(69 107 77 99)(70 106 78 98)(71 105 79 97)(72 104 80 112)
(1 97 90 28)(2 106 91 21)(3 99 92 30)(4 108 93 23)(5 101 94 32)(6 110 95 25)(7 103 96 18)(8 112 81 27)(9 105 82 20)(10 98 83 29)(11 107 84 22)(12 100 85 31)(13 109 86 24)(14 102 87 17)(15 111 88 26)(16 104 89 19)(33 50 122 70)(34 59 123 79)(35 52 124 72)(36 61 125 65)(37 54 126 74)(38 63 127 67)(39 56 128 76)(40 49 113 69)(41 58 114 78)(42 51 115 71)(43 60 116 80)(44 53 117 73)(45 62 118 66)(46 55 119 75)(47 64 120 68)(48 57 121 77)
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), (1,115,9,123)(2,114,10,122)(3,113,11,121)(4,128,12,120)(5,127,13,119)(6,126,14,118)(7,125,15,117)(8,124,16,116)(17,62,25,54)(18,61,26,53)(19,60,27,52)(20,59,28,51)(21,58,29,50)(22,57,30,49)(23,56,31,64)(24,55,32,63)(33,91,41,83)(34,90,42,82)(35,89,43,81)(36,88,44,96)(37,87,45,95)(38,86,46,94)(39,85,47,93)(40,84,48,92)(65,111,73,103)(66,110,74,102)(67,109,75,101)(68,108,76,100)(69,107,77,99)(70,106,78,98)(71,105,79,97)(72,104,80,112), (1,97,90,28)(2,106,91,21)(3,99,92,30)(4,108,93,23)(5,101,94,32)(6,110,95,25)(7,103,96,18)(8,112,81,27)(9,105,82,20)(10,98,83,29)(11,107,84,22)(12,100,85,31)(13,109,86,24)(14,102,87,17)(15,111,88,26)(16,104,89,19)(33,50,122,70)(34,59,123,79)(35,52,124,72)(36,61,125,65)(37,54,126,74)(38,63,127,67)(39,56,128,76)(40,49,113,69)(41,58,114,78)(42,51,115,71)(43,60,116,80)(44,53,117,73)(45,62,118,66)(46,55,119,75)(47,64,120,68)(48,57,121,77)>;
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), (1,115,9,123)(2,114,10,122)(3,113,11,121)(4,128,12,120)(5,127,13,119)(6,126,14,118)(7,125,15,117)(8,124,16,116)(17,62,25,54)(18,61,26,53)(19,60,27,52)(20,59,28,51)(21,58,29,50)(22,57,30,49)(23,56,31,64)(24,55,32,63)(33,91,41,83)(34,90,42,82)(35,89,43,81)(36,88,44,96)(37,87,45,95)(38,86,46,94)(39,85,47,93)(40,84,48,92)(65,111,73,103)(66,110,74,102)(67,109,75,101)(68,108,76,100)(69,107,77,99)(70,106,78,98)(71,105,79,97)(72,104,80,112), (1,97,90,28)(2,106,91,21)(3,99,92,30)(4,108,93,23)(5,101,94,32)(6,110,95,25)(7,103,96,18)(8,112,81,27)(9,105,82,20)(10,98,83,29)(11,107,84,22)(12,100,85,31)(13,109,86,24)(14,102,87,17)(15,111,88,26)(16,104,89,19)(33,50,122,70)(34,59,123,79)(35,52,124,72)(36,61,125,65)(37,54,126,74)(38,63,127,67)(39,56,128,76)(40,49,113,69)(41,58,114,78)(42,51,115,71)(43,60,116,80)(44,53,117,73)(45,62,118,66)(46,55,119,75)(47,64,120,68)(48,57,121,77) );
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)], [(1,115,9,123),(2,114,10,122),(3,113,11,121),(4,128,12,120),(5,127,13,119),(6,126,14,118),(7,125,15,117),(8,124,16,116),(17,62,25,54),(18,61,26,53),(19,60,27,52),(20,59,28,51),(21,58,29,50),(22,57,30,49),(23,56,31,64),(24,55,32,63),(33,91,41,83),(34,90,42,82),(35,89,43,81),(36,88,44,96),(37,87,45,95),(38,86,46,94),(39,85,47,93),(40,84,48,92),(65,111,73,103),(66,110,74,102),(67,109,75,101),(68,108,76,100),(69,107,77,99),(70,106,78,98),(71,105,79,97),(72,104,80,112)], [(1,97,90,28),(2,106,91,21),(3,99,92,30),(4,108,93,23),(5,101,94,32),(6,110,95,25),(7,103,96,18),(8,112,81,27),(9,105,82,20),(10,98,83,29),(11,107,84,22),(12,100,85,31),(13,109,86,24),(14,102,87,17),(15,111,88,26),(16,104,89,19),(33,50,122,70),(34,59,123,79),(35,52,124,72),(36,61,125,65),(37,54,126,74),(38,63,127,67),(39,56,128,76),(40,49,113,69),(41,58,114,78),(42,51,115,71),(43,60,116,80),(44,53,117,73),(45,62,118,66),(46,55,119,75),(47,64,120,68),(48,57,121,77)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | ··· | 4N | 8A | 8B | 8C | 8D | 8E | 8F | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | D8 | C4○D8 | Q32⋊C2 |
| kernel | Q32⋊4C4 | C16⋊5C4 | C2.Q32 | C16⋊4C4 | C4×Q16 | C2×Q32 | Q32 | C42 | C2×C8 | C8 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 1 | 1 | 2 | 4 | 4 | 4 |
Matrix representation of Q32⋊4C4 ►in GL6(𝔽17)
| 14 | 1 | 0 | 0 | 0 | 0 |
| 7 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 15 | 16 | 7 |
| 0 | 0 | 2 | 10 | 10 | 16 |
| 0 | 0 | 16 | 7 | 7 | 2 |
| 0 | 0 | 10 | 16 | 15 | 7 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 1 |
| 0 | 0 | 0 | 0 | 1 | 7 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(6,GF(17))| [14,7,0,0,0,0,1,3,0,0,0,0,0,0,10,2,16,10,0,0,15,10,7,16,0,0,16,10,7,15,0,0,7,16,2,7],[16,11,0,0,0,0,0,1,0,0,0,0,0,0,10,1,0,0,0,0,1,7,0,0,0,0,0,0,10,1,0,0,0,0,1,7],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
Q32⋊4C4 in GAP, Magma, Sage, TeX
Q_{32}\rtimes_4C_4 % in TeX
G:=Group("Q32:4C4"); // GroupNames label
G:=SmallGroup(128,908);
// by ID
G=gap.SmallGroup(128,908);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,1430,100,1123,570,360,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c|a^16=c^4=1,b^2=a^8,b*a*b^-1=a^-1,c*a*c^-1=a^9,b*c=c*b>;
// generators/relations