p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.46Q8, C42.386D4, C4⋊C8⋊6C4, C4⋊C4⋊2C8, (C4×C8)⋊10C4, C4.7(C4×C8), C4.14(C4⋊C8), C2.1(D4⋊C8), (C2×C4).157D8, C2.1(Q8⋊C8), (C2×C4).66Q16, C4.7(C8⋊C4), C22.31C4≀C2, (C2×C4).52C42, C4.12(C4.Q8), C4.19(C2.D8), C42.303(C2×C4), (C2×C4).124SD16, (C22×C4).807D4, (C2×C4).39M4(2), C4.42(D4⋊C4), C2.3(C42⋊6C4), C4.28(Q8⋊C4), C22.31(C22⋊C8), C2.2(C22.4Q16), C22.37(D4⋊C4), C23.212(C22⋊C4), (C2×C42).1027C22, C22.27(Q8⋊C4), C2.8(C22.7C42), C22.20(C2.C42), (C2×C4×C8).4C2, (C2×C4⋊C8).2C2, (C4×C4⋊C4).1C2, (C2×C4⋊C4).27C4, (C2×C4).44(C2×C8), (C2×C4).93(C4⋊C4), (C22×C4).390(C2×C4), (C2×C4).289(C22⋊C4), SmallGroup(128,11)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.46Q8
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=a2b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=a-1b2c3 >
Subgroups: 176 in 104 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2.C42, C4×C8, C4×C8, C4⋊C8, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C22×C8, C4×C4⋊C4, C2×C4×C8, C2×C4⋊C8, C42.46Q8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), D8, SD16, Q16, C2.C42, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4≀C2, C4⋊C8, C4.Q8, C2.D8, D4⋊C8, Q8⋊C8, C22.7C42, C42⋊6C4, C22.4Q16, C42.46Q8
►Regular action on 128 points
Generators in S128
(1 21 15 61)(2 22 16 62)(3 23 9 63)(4 24 10 64)(5 17 11 57)(6 18 12 58)(7 19 13 59)(8 20 14 60)(25 53 41 85)(26 54 42 86)(27 55 43 87)(28 56 44 88)(29 49 45 81)(30 50 46 82)(31 51 47 83)(32 52 48 84)(33 66 100 79)(34 67 101 80)(35 68 102 73)(36 69 103 74)(37 70 104 75)(38 71 97 76)(39 72 98 77)(40 65 99 78)(89 113 123 108)(90 114 124 109)(91 115 125 110)(92 116 126 111)(93 117 127 112)(94 118 128 105)(95 119 121 106)(96 120 122 107)
(1 41 11 29)(2 42 12 30)(3 43 13 31)(4 44 14 32)(5 45 15 25)(6 46 16 26)(7 47 9 27)(8 48 10 28)(17 81 61 53)(18 82 62 54)(19 83 63 55)(20 84 64 56)(21 85 57 49)(22 86 58 50)(23 87 59 51)(24 88 60 52)(33 123 104 93)(34 124 97 94)(35 125 98 95)(36 126 99 96)(37 127 100 89)(38 128 101 90)(39 121 102 91)(40 122 103 92)(65 107 74 116)(66 108 75 117)(67 109 76 118)(68 110 77 119)(69 111 78 120)(70 112 79 113)(71 105 80 114)(72 106 73 115)
(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 35 47 123 15 102 31 89)(2 67 48 107 16 80 32 120)(3 100 41 95 9 33 25 121)(4 78 42 118 10 65 26 105)(5 39 43 127 11 98 27 93)(6 71 44 111 12 76 28 116)(7 104 45 91 13 37 29 125)(8 74 46 114 14 69 30 109)(17 72 87 112 57 77 55 117)(18 97 88 92 58 38 56 126)(19 75 81 115 59 70 49 110)(20 36 82 124 60 103 50 90)(21 68 83 108 61 73 51 113)(22 101 84 96 62 34 52 122)(23 79 85 119 63 66 53 106)(24 40 86 128 64 99 54 94)
G:=sub<Sym(128)| (1,21,15,61)(2,22,16,62)(3,23,9,63)(4,24,10,64)(5,17,11,57)(6,18,12,58)(7,19,13,59)(8,20,14,60)(25,53,41,85)(26,54,42,86)(27,55,43,87)(28,56,44,88)(29,49,45,81)(30,50,46,82)(31,51,47,83)(32,52,48,84)(33,66,100,79)(34,67,101,80)(35,68,102,73)(36,69,103,74)(37,70,104,75)(38,71,97,76)(39,72,98,77)(40,65,99,78)(89,113,123,108)(90,114,124,109)(91,115,125,110)(92,116,126,111)(93,117,127,112)(94,118,128,105)(95,119,121,106)(96,120,122,107), (1,41,11,29)(2,42,12,30)(3,43,13,31)(4,44,14,32)(5,45,15,25)(6,46,16,26)(7,47,9,27)(8,48,10,28)(17,81,61,53)(18,82,62,54)(19,83,63,55)(20,84,64,56)(21,85,57,49)(22,86,58,50)(23,87,59,51)(24,88,60,52)(33,123,104,93)(34,124,97,94)(35,125,98,95)(36,126,99,96)(37,127,100,89)(38,128,101,90)(39,121,102,91)(40,122,103,92)(65,107,74,116)(66,108,75,117)(67,109,76,118)(68,110,77,119)(69,111,78,120)(70,112,79,113)(71,105,80,114)(72,106,73,115), (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), (1,35,47,123,15,102,31,89)(2,67,48,107,16,80,32,120)(3,100,41,95,9,33,25,121)(4,78,42,118,10,65,26,105)(5,39,43,127,11,98,27,93)(6,71,44,111,12,76,28,116)(7,104,45,91,13,37,29,125)(8,74,46,114,14,69,30,109)(17,72,87,112,57,77,55,117)(18,97,88,92,58,38,56,126)(19,75,81,115,59,70,49,110)(20,36,82,124,60,103,50,90)(21,68,83,108,61,73,51,113)(22,101,84,96,62,34,52,122)(23,79,85,119,63,66,53,106)(24,40,86,128,64,99,54,94)>;
G:=Group( (1,21,15,61)(2,22,16,62)(3,23,9,63)(4,24,10,64)(5,17,11,57)(6,18,12,58)(7,19,13,59)(8,20,14,60)(25,53,41,85)(26,54,42,86)(27,55,43,87)(28,56,44,88)(29,49,45,81)(30,50,46,82)(31,51,47,83)(32,52,48,84)(33,66,100,79)(34,67,101,80)(35,68,102,73)(36,69,103,74)(37,70,104,75)(38,71,97,76)(39,72,98,77)(40,65,99,78)(89,113,123,108)(90,114,124,109)(91,115,125,110)(92,116,126,111)(93,117,127,112)(94,118,128,105)(95,119,121,106)(96,120,122,107), (1,41,11,29)(2,42,12,30)(3,43,13,31)(4,44,14,32)(5,45,15,25)(6,46,16,26)(7,47,9,27)(8,48,10,28)(17,81,61,53)(18,82,62,54)(19,83,63,55)(20,84,64,56)(21,85,57,49)(22,86,58,50)(23,87,59,51)(24,88,60,52)(33,123,104,93)(34,124,97,94)(35,125,98,95)(36,126,99,96)(37,127,100,89)(38,128,101,90)(39,121,102,91)(40,122,103,92)(65,107,74,116)(66,108,75,117)(67,109,76,118)(68,110,77,119)(69,111,78,120)(70,112,79,113)(71,105,80,114)(72,106,73,115), (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), (1,35,47,123,15,102,31,89)(2,67,48,107,16,80,32,120)(3,100,41,95,9,33,25,121)(4,78,42,118,10,65,26,105)(5,39,43,127,11,98,27,93)(6,71,44,111,12,76,28,116)(7,104,45,91,13,37,29,125)(8,74,46,114,14,69,30,109)(17,72,87,112,57,77,55,117)(18,97,88,92,58,38,56,126)(19,75,81,115,59,70,49,110)(20,36,82,124,60,103,50,90)(21,68,83,108,61,73,51,113)(22,101,84,96,62,34,52,122)(23,79,85,119,63,66,53,106)(24,40,86,128,64,99,54,94) );
G=PermutationGroup([[(1,21,15,61),(2,22,16,62),(3,23,9,63),(4,24,10,64),(5,17,11,57),(6,18,12,58),(7,19,13,59),(8,20,14,60),(25,53,41,85),(26,54,42,86),(27,55,43,87),(28,56,44,88),(29,49,45,81),(30,50,46,82),(31,51,47,83),(32,52,48,84),(33,66,100,79),(34,67,101,80),(35,68,102,73),(36,69,103,74),(37,70,104,75),(38,71,97,76),(39,72,98,77),(40,65,99,78),(89,113,123,108),(90,114,124,109),(91,115,125,110),(92,116,126,111),(93,117,127,112),(94,118,128,105),(95,119,121,106),(96,120,122,107)], [(1,41,11,29),(2,42,12,30),(3,43,13,31),(4,44,14,32),(5,45,15,25),(6,46,16,26),(7,47,9,27),(8,48,10,28),(17,81,61,53),(18,82,62,54),(19,83,63,55),(20,84,64,56),(21,85,57,49),(22,86,58,50),(23,87,59,51),(24,88,60,52),(33,123,104,93),(34,124,97,94),(35,125,98,95),(36,126,99,96),(37,127,100,89),(38,128,101,90),(39,121,102,91),(40,122,103,92),(65,107,74,116),(66,108,75,117),(67,109,76,118),(68,110,77,119),(69,111,78,120),(70,112,79,113),(71,105,80,114),(72,106,73,115)], [(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)], [(1,35,47,123,15,102,31,89),(2,67,48,107,16,80,32,120),(3,100,41,95,9,33,25,121),(4,78,42,118,10,65,26,105),(5,39,43,127,11,98,27,93),(6,71,44,111,12,76,28,116),(7,104,45,91,13,37,29,125),(8,74,46,114,14,69,30,109),(17,72,87,112,57,77,55,117),(18,97,88,92,58,38,56,126),(19,75,81,115,59,70,49,110),(20,36,82,124,60,103,50,90),(21,68,83,108,61,73,51,113),(22,101,84,96,62,34,52,122),(23,79,85,119,63,66,53,106),(24,40,86,128,64,99,54,94)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4X | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | Q8 | D4 | M4(2) | D8 | SD16 | Q16 | C4≀C2 |
| kernel | C42.46Q8 | C4×C4⋊C4 | C2×C4×C8 | C2×C4⋊C8 | C4×C8 | C4⋊C8 | C2×C4⋊C4 | C4⋊C4 | C42 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 16 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 8 |
Matrix representation of C42.46Q8 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
| 9 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 8 |
| 2 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,4,0,0,0,0,13],[9,0,0,0,0,4,0,0,0,0,2,0,0,0,0,8],[2,0,0,0,0,16,0,0,0,0,0,1,0,0,1,0] >;
C42.46Q8 in GAP, Magma, Sage, TeX
C_4^2._{46}Q_8 % in TeX
G:=Group("C4^2.46Q8"); // GroupNames label
G:=SmallGroup(128,11);
// by ID
G=gap.SmallGroup(128,11);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,184,248,1684,242]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=a^2*b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^-1*b^2*c^3>;
// generators/relations