p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.27Q8, C42.108D4, (C2×C8).31Q8, (C2×C8).202D4, C4.49(C4⋊Q8), C2.7(C8⋊4Q8), C22.41(C4×Q8), C2.20(C8⋊9D4), C22.173(C4×D4), (C2×C4).21M4(2), C4.198(C4⋊D4), C22.55(C8○D4), C4.118(C22⋊Q8), (C22×C8).53C22, C4.39(C42.C2), C2.C42.25C4, (C2×C42).306C22, C23.316(C22×C4), C22.69(C2×M4(2)), C2.11(C4⋊M4(2)), (C22×C4).1636C23, C22.7C42.45C2, C2.14(C42.6C22), C2.7(C23.65C23), (C4×C4⋊C4).21C2, (C2×C4⋊C8).56C2, (C2×C4⋊C4).59C4, (C2×C4).54(C4⋊C4), (C2×C4).346(C2×Q8), (C2×C8⋊C4).32C2, (C2×C4).1539(C2×D4), C22.113(C2×C4⋊C4), (C2×C4).942(C4○D4), (C22×C4).126(C2×C4), SmallGroup(128,672)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.27Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=c2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=b-1c3 >
Subgroups: 188 in 124 conjugacy classes, 68 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C8⋊C4, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C22×C8, C22×C8, C22.7C42, C4×C4⋊C4, C2×C8⋊C4, C2×C4⋊C8, C2×C4⋊C8, C42.27Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C2×M4(2), C8○D4, C23.65C23, C4⋊M4(2), C42.6C22, C8⋊9D4, C8⋊4Q8, C42.27Q8
►Regular action on 128 points
Generators in S128
(1 63 55 12)(2 60 56 9)(3 57 49 14)(4 62 50 11)(5 59 51 16)(6 64 52 13)(7 61 53 10)(8 58 54 15)(17 73 25 41)(18 78 26 46)(19 75 27 43)(20 80 28 48)(21 77 29 45)(22 74 30 42)(23 79 31 47)(24 76 32 44)(33 109 104 85)(34 106 97 82)(35 111 98 87)(36 108 99 84)(37 105 100 81)(38 110 101 86)(39 107 102 83)(40 112 103 88)(65 117 125 93)(66 114 126 90)(67 119 127 95)(68 116 128 92)(69 113 121 89)(70 118 122 94)(71 115 123 91)(72 120 124 96)
(1 17 5 21)(2 18 6 22)(3 19 7 23)(4 20 8 24)(9 46 13 42)(10 47 14 43)(11 48 15 44)(12 41 16 45)(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 98 69 102)(66 99 70 103)(67 100 71 104)(68 101 72 97)(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 66 11 68 13 70 15 72)(10 98 12 100 14 102 16 104)(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 71 43 65 45 67 47 69)(42 103 44 97 46 99 48 101)(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)| (1,63,55,12)(2,60,56,9)(3,57,49,14)(4,62,50,11)(5,59,51,16)(6,64,52,13)(7,61,53,10)(8,58,54,15)(17,73,25,41)(18,78,26,46)(19,75,27,43)(20,80,28,48)(21,77,29,45)(22,74,30,42)(23,79,31,47)(24,76,32,44)(33,109,104,85)(34,106,97,82)(35,111,98,87)(36,108,99,84)(37,105,100,81)(38,110,101,86)(39,107,102,83)(40,112,103,88)(65,117,125,93)(66,114,126,90)(67,119,127,95)(68,116,128,92)(69,113,121,89)(70,118,122,94)(71,115,123,91)(72,120,124,96), (1,17,5,21)(2,18,6,22)(3,19,7,23)(4,20,8,24)(9,46,13,42)(10,47,14,43)(11,48,15,44)(12,41,16,45)(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)(65,98,69,102)(66,99,70,103)(67,100,71,104)(68,101,72,97)(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)(9,66,11,68,13,70,15,72)(10,98,12,100,14,102,16,104)(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)(41,71,43,65,45,67,47,69)(42,103,44,97,46,99,48,101)(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( (1,63,55,12)(2,60,56,9)(3,57,49,14)(4,62,50,11)(5,59,51,16)(6,64,52,13)(7,61,53,10)(8,58,54,15)(17,73,25,41)(18,78,26,46)(19,75,27,43)(20,80,28,48)(21,77,29,45)(22,74,30,42)(23,79,31,47)(24,76,32,44)(33,109,104,85)(34,106,97,82)(35,111,98,87)(36,108,99,84)(37,105,100,81)(38,110,101,86)(39,107,102,83)(40,112,103,88)(65,117,125,93)(66,114,126,90)(67,119,127,95)(68,116,128,92)(69,113,121,89)(70,118,122,94)(71,115,123,91)(72,120,124,96), (1,17,5,21)(2,18,6,22)(3,19,7,23)(4,20,8,24)(9,46,13,42)(10,47,14,43)(11,48,15,44)(12,41,16,45)(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)(65,98,69,102)(66,99,70,103)(67,100,71,104)(68,101,72,97)(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)(9,66,11,68,13,70,15,72)(10,98,12,100,14,102,16,104)(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)(41,71,43,65,45,67,47,69)(42,103,44,97,46,99,48,101)(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([[(1,63,55,12),(2,60,56,9),(3,57,49,14),(4,62,50,11),(5,59,51,16),(6,64,52,13),(7,61,53,10),(8,58,54,15),(17,73,25,41),(18,78,26,46),(19,75,27,43),(20,80,28,48),(21,77,29,45),(22,74,30,42),(23,79,31,47),(24,76,32,44),(33,109,104,85),(34,106,97,82),(35,111,98,87),(36,108,99,84),(37,105,100,81),(38,110,101,86),(39,107,102,83),(40,112,103,88),(65,117,125,93),(66,114,126,90),(67,119,127,95),(68,116,128,92),(69,113,121,89),(70,118,122,94),(71,115,123,91),(72,120,124,96)], [(1,17,5,21),(2,18,6,22),(3,19,7,23),(4,20,8,24),(9,46,13,42),(10,47,14,43),(11,48,15,44),(12,41,16,45),(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),(65,98,69,102),(66,99,70,103),(67,100,71,104),(68,101,72,97),(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),(9,66,11,68,13,70,15,72),(10,98,12,100,14,102,16,104),(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),(41,71,43,65,45,67,47,69),(42,103,44,97,46,99,48,101),(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)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | Q8 | M4(2) | C4○D4 | C8○D4 |
| kernel | C42.27Q8 | C22.7C42 | C4×C4⋊C4 | C2×C8⋊C4 | C2×C4⋊C8 | C2.C42 | C2×C4⋊C4 | C42 | C42 | C2×C8 | C2×C8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 1 | 3 | 4 | 4 | 2 | 2 | 2 | 2 | 8 | 4 | 8 |
Matrix representation of C42.27Q8 ►in GL6(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 8 | 0 | 0 |
| 0 | 0 | 13 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 0 | 0 | 0 | 0 | 7 | 11 |
| 7 | 10 | 0 | 0 | 0 | 0 |
| 12 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 11 | 0 | 0 |
| 0 | 0 | 3 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 0 | 0 | 0 | 0 | 15 | 0 |
G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,13,0,0,0,0,8,4,0,0,0,0,0,0,6,7,0,0,0,0,10,11],[7,12,0,0,0,0,10,10,0,0,0,0,0,0,6,3,0,0,0,0,11,11,0,0,0,0,0,0,0,15,0,0,0,0,15,0] >;
C42.27Q8 in GAP, Magma, Sage, TeX
C_4^2._{27}Q_8 % in TeX
G:=Group("C4^2.27Q8"); // GroupNames label
G:=SmallGroup(128,672);
// by ID
G=gap.SmallGroup(128,672);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,723,100,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=c^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations