p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.61Q8, C42.429D4, C4⋊C4⋊4C8, C4⋊2(C4⋊C8), C2.4(C8×Q8), C2.14(C8×D4), (C2×C8).39Q8, (C2×C8).227D4, C4.48(C4⋊Q8), C2.4(C8⋊6D4), C2.4(C8⋊4Q8), C22.26(C4×Q8), C22.102(C4×D4), (C2×C4).46M4(2), C4.197(C4⋊D4), C22.33(C8○D4), C4.117(C22⋊Q8), C22.43(C22×C8), (C22×C8).52C22, C4.38(C42.C2), C2.C42.24C4, (C2×C42).305C22, C23.272(C22×C4), C22.54(C2×M4(2)), (C22×C4).1635C23, C22.7C42.8C2, C2.4(C42.6C22), C2.3(C23.65C23), (C2×C4×C8).22C2, C2.12(C2×C4⋊C8), (C2×C4⋊C8).29C2, (C2×C4⋊C4).58C4, (C4×C4⋊C4).20C2, (C2×C4).22(C2×C8), C22.67(C2×C4⋊C4), (C2×C4).345(C2×Q8), (C2×C4).134(C4⋊C4), (C2×C4).1538(C2×D4), (C2×C4).941(C4○D4), (C22×C4).125(C2×C4), SmallGroup(128,671)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.61Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=b-1c3 >
Subgroups: 188 in 130 conjugacy classes, 80 normal (38 characteristic)
C1, C2, C4, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4×C8, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C22×C8, C22×C8, C22.7C42, C4×C4⋊C4, C2×C4×C8, C2×C4⋊C8, C2×C4⋊C8, C42.61Q8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4⋊C8, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C22×C8, C2×M4(2), C8○D4, C23.65C23, C2×C4⋊C8, C42.6C22, C8×D4, C8⋊6D4, C8×Q8, C8⋊4Q8, C42.61Q8
►Regular action on 128 points
Generators in S128
(1 63 55 43)(2 64 56 44)(3 57 49 45)(4 58 50 46)(5 59 51 47)(6 60 52 48)(7 61 53 41)(8 62 54 42)(9 18 74 26)(10 19 75 27)(11 20 76 28)(12 21 77 29)(13 22 78 30)(14 23 79 31)(15 24 80 32)(16 17 73 25)(33 109 104 85)(34 110 97 86)(35 111 98 87)(36 112 99 88)(37 105 100 81)(38 106 101 82)(39 107 102 83)(40 108 103 84)(65 120 128 96)(66 113 121 89)(67 114 122 90)(68 115 123 91)(69 116 124 92)(70 117 125 93)(71 118 126 94)(72 119 127 95)
(1 17 5 21)(2 18 6 22)(3 19 7 23)(4 20 8 24)(9 48 13 44)(10 41 14 45)(11 42 15 46)(12 43 16 47)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(33 127 37 123)(34 128 38 124)(35 121 39 125)(36 122 40 126)(57 75 61 79)(58 76 62 80)(59 77 63 73)(60 78 64 74)(65 101 69 97)(66 102 70 98)(67 103 71 99)(68 104 72 100)(81 115 85 119)(82 116 86 120)(83 117 87 113)(84 118 88 114)(89 107 93 111)(90 108 94 112)(91 109 95 105)(92 110 96 106)
(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)
(1 105 3 107 5 109 7 111)(2 90 4 92 6 94 8 96)(9 103 11 97 13 99 15 101)(10 70 12 72 14 66 16 68)(17 91 19 93 21 95 23 89)(18 108 20 110 22 112 24 106)(25 115 27 117 29 119 31 113)(26 84 28 86 30 88 32 82)(33 61 35 63 37 57 39 59)(34 78 36 80 38 74 40 76)(41 98 43 100 45 102 47 104)(42 65 44 67 46 69 48 71)(49 83 51 85 53 87 55 81)(50 116 52 118 54 120 56 114)(58 124 60 126 62 128 64 122)(73 123 75 125 77 127 79 121)
G:=sub<Sym(128)| (1,63,55,43)(2,64,56,44)(3,57,49,45)(4,58,50,46)(5,59,51,47)(6,60,52,48)(7,61,53,41)(8,62,54,42)(9,18,74,26)(10,19,75,27)(11,20,76,28)(12,21,77,29)(13,22,78,30)(14,23,79,31)(15,24,80,32)(16,17,73,25)(33,109,104,85)(34,110,97,86)(35,111,98,87)(36,112,99,88)(37,105,100,81)(38,106,101,82)(39,107,102,83)(40,108,103,84)(65,120,128,96)(66,113,121,89)(67,114,122,90)(68,115,123,91)(69,116,124,92)(70,117,125,93)(71,118,126,94)(72,119,127,95), (1,17,5,21)(2,18,6,22)(3,19,7,23)(4,20,8,24)(9,48,13,44)(10,41,14,45)(11,42,15,46)(12,43,16,47)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,127,37,123)(34,128,38,124)(35,121,39,125)(36,122,40,126)(57,75,61,79)(58,76,62,80)(59,77,63,73)(60,78,64,74)(65,101,69,97)(66,102,70,98)(67,103,71,99)(68,104,72,100)(81,115,85,119)(82,116,86,120)(83,117,87,113)(84,118,88,114)(89,107,93,111)(90,108,94,112)(91,109,95,105)(92,110,96,106), (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), (1,105,3,107,5,109,7,111)(2,90,4,92,6,94,8,96)(9,103,11,97,13,99,15,101)(10,70,12,72,14,66,16,68)(17,91,19,93,21,95,23,89)(18,108,20,110,22,112,24,106)(25,115,27,117,29,119,31,113)(26,84,28,86,30,88,32,82)(33,61,35,63,37,57,39,59)(34,78,36,80,38,74,40,76)(41,98,43,100,45,102,47,104)(42,65,44,67,46,69,48,71)(49,83,51,85,53,87,55,81)(50,116,52,118,54,120,56,114)(58,124,60,126,62,128,64,122)(73,123,75,125,77,127,79,121)>;
G:=Group( (1,63,55,43)(2,64,56,44)(3,57,49,45)(4,58,50,46)(5,59,51,47)(6,60,52,48)(7,61,53,41)(8,62,54,42)(9,18,74,26)(10,19,75,27)(11,20,76,28)(12,21,77,29)(13,22,78,30)(14,23,79,31)(15,24,80,32)(16,17,73,25)(33,109,104,85)(34,110,97,86)(35,111,98,87)(36,112,99,88)(37,105,100,81)(38,106,101,82)(39,107,102,83)(40,108,103,84)(65,120,128,96)(66,113,121,89)(67,114,122,90)(68,115,123,91)(69,116,124,92)(70,117,125,93)(71,118,126,94)(72,119,127,95), (1,17,5,21)(2,18,6,22)(3,19,7,23)(4,20,8,24)(9,48,13,44)(10,41,14,45)(11,42,15,46)(12,43,16,47)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,127,37,123)(34,128,38,124)(35,121,39,125)(36,122,40,126)(57,75,61,79)(58,76,62,80)(59,77,63,73)(60,78,64,74)(65,101,69,97)(66,102,70,98)(67,103,71,99)(68,104,72,100)(81,115,85,119)(82,116,86,120)(83,117,87,113)(84,118,88,114)(89,107,93,111)(90,108,94,112)(91,109,95,105)(92,110,96,106), (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), (1,105,3,107,5,109,7,111)(2,90,4,92,6,94,8,96)(9,103,11,97,13,99,15,101)(10,70,12,72,14,66,16,68)(17,91,19,93,21,95,23,89)(18,108,20,110,22,112,24,106)(25,115,27,117,29,119,31,113)(26,84,28,86,30,88,32,82)(33,61,35,63,37,57,39,59)(34,78,36,80,38,74,40,76)(41,98,43,100,45,102,47,104)(42,65,44,67,46,69,48,71)(49,83,51,85,53,87,55,81)(50,116,52,118,54,120,56,114)(58,124,60,126,62,128,64,122)(73,123,75,125,77,127,79,121) );
G=PermutationGroup([[(1,63,55,43),(2,64,56,44),(3,57,49,45),(4,58,50,46),(5,59,51,47),(6,60,52,48),(7,61,53,41),(8,62,54,42),(9,18,74,26),(10,19,75,27),(11,20,76,28),(12,21,77,29),(13,22,78,30),(14,23,79,31),(15,24,80,32),(16,17,73,25),(33,109,104,85),(34,110,97,86),(35,111,98,87),(36,112,99,88),(37,105,100,81),(38,106,101,82),(39,107,102,83),(40,108,103,84),(65,120,128,96),(66,113,121,89),(67,114,122,90),(68,115,123,91),(69,116,124,92),(70,117,125,93),(71,118,126,94),(72,119,127,95)], [(1,17,5,21),(2,18,6,22),(3,19,7,23),(4,20,8,24),(9,48,13,44),(10,41,14,45),(11,42,15,46),(12,43,16,47),(25,51,29,55),(26,52,30,56),(27,53,31,49),(28,54,32,50),(33,127,37,123),(34,128,38,124),(35,121,39,125),(36,122,40,126),(57,75,61,79),(58,76,62,80),(59,77,63,73),(60,78,64,74),(65,101,69,97),(66,102,70,98),(67,103,71,99),(68,104,72,100),(81,115,85,119),(82,116,86,120),(83,117,87,113),(84,118,88,114),(89,107,93,111),(90,108,94,112),(91,109,95,105),(92,110,96,106)], [(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)], [(1,105,3,107,5,109,7,111),(2,90,4,92,6,94,8,96),(9,103,11,97,13,99,15,101),(10,70,12,72,14,66,16,68),(17,91,19,93,21,95,23,89),(18,108,20,110,22,112,24,106),(25,115,27,117,29,119,31,113),(26,84,28,86,30,88,32,82),(33,61,35,63,37,57,39,59),(34,78,36,80,38,74,40,76),(41,98,43,100,45,102,47,104),(42,65,44,67,46,69,48,71),(49,83,51,85,53,87,55,81),(50,116,52,118,54,120,56,114),(58,124,60,126,62,128,64,122),(73,123,75,125,77,127,79,121)]])
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 |
| type | + | + | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D4 | Q8 | M4(2) | C4○D4 | C8○D4 |
| kernel | C42.61Q8 | C22.7C42 | C4×C4⋊C4 | C2×C4×C8 | C2×C4⋊C8 | C2.C42 | C2×C4⋊C4 | C4⋊C4 | C42 | C42 | C2×C8 | C2×C8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 1 | 3 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C42.61Q8 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 1 | 1 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 1 | 1 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 12 | 9 | 0 | 0 |
| 0 | 9 | 5 | 0 | 0 |
| 0 | 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,15,1],[13,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,16,1,0,0,0,15,1],[9,0,0,0,0,0,12,9,0,0,0,9,5,0,0,0,0,0,13,0,0,0,0,9,4] >;
C42.61Q8 in GAP, Magma, Sage, TeX
C_4^2._{61}Q_8 % in TeX
G:=Group("C4^2.61Q8"); // GroupNames label
G:=SmallGroup(128,671);
// by ID
G=gap.SmallGroup(128,671);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,100,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations