p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.68(C4×D4), C4⋊C4.213D4, Q8⋊C4⋊7C4, (C2×C4).89SD16, C2.15(C4×SD16), C22.164(C4×D4), C23.782(C2×D4), (C22×C4).697D4, C2.9(Q16⋊C4), C2.3(D4.D4), C4.26(C4.4D4), C22.67(C4○D8), (C22×C8).50C22, C2.5(Q8.D4), C22.65(C2×SD16), C4.38(C42⋊C2), C4.31(C42⋊2C2), C22.4Q16.11C2, (C2×C42).298C22, (C22×Q8).27C22, C22.126(C4⋊D4), (C22×C4).1381C23, C2.6(C23.20D4), C2.5(C23.47D4), C22.74(C8.C22), C23.67C23.6C2, C22.7C42.34C2, C2.12(C24.C22), C22.94(C22.D4), (C4×C4⋊C4).18C2, C4⋊C4.151(C2×C4), (C2×C8).113(C2×C4), (C2×Q8).79(C2×C4), (C2×C4.Q8).18C2, (C2×C4).1011(C2×D4), (C2×Q8⋊C4).4C2, (C2×C4).577(C4○D4), (C2×C4⋊C4).775C22, (C2×C4).399(C22×C4), SmallGroup(128,659)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.68(C4×D4)
G = < a,b,c,d | a4=b4=c4=1, d2=a, bab-1=cac-1=a-1, ad=da, bc=cb, dbd-1=a-1b, dcd-1=a-1c-1 >
Subgroups: 244 in 127 conjugacy classes, 58 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, Q8⋊C4, C4.Q8, C2×C42, C2×C42, C2×C4⋊C4, C22×C8, C22×Q8, C22.7C42, C22.4Q16, C4×C4⋊C4, C23.67C23, C2×Q8⋊C4, C2×C4.Q8, C4.68(C4×D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, SD16, C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C2×SD16, C4○D8, C8.C22, C24.C22, C4×SD16, Q16⋊C4, D4.D4, Q8.D4, C23.47D4, C23.20D4, C4.68(C4×D4)
►Regular action on 128 points
Generators in S128
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)
(1 102 69 40)(2 97 70 35)(3 100 71 38)(4 103 72 33)(5 98 65 36)(6 101 66 39)(7 104 67 34)(8 99 68 37)(9 32 50 82)(10 27 51 85)(11 30 52 88)(12 25 53 83)(13 28 54 86)(14 31 55 81)(15 26 56 84)(16 29 49 87)(17 57 77 111)(18 60 78 106)(19 63 79 109)(20 58 80 112)(21 61 73 107)(22 64 74 110)(23 59 75 105)(24 62 76 108)(41 113 93 127)(42 116 94 122)(43 119 95 125)(44 114 96 128)(45 117 89 123)(46 120 90 126)(47 115 91 121)(48 118 92 124)
(1 13 115 111)(2 106 116 16)(3 11 117 109)(4 112 118 14)(5 9 119 107)(6 110 120 12)(7 15 113 105)(8 108 114 10)(17 102 28 91)(18 94 29 97)(19 100 30 89)(20 92 31 103)(21 98 32 95)(22 90 25 101)(23 104 26 93)(24 96 27 99)(33 80 48 81)(34 84 41 75)(35 78 42 87)(36 82 43 73)(37 76 44 85)(38 88 45 79)(39 74 46 83)(40 86 47 77)(49 70 60 122)(50 125 61 65)(51 68 62 128)(52 123 63 71)(53 66 64 126)(54 121 57 69)(55 72 58 124)(56 127 59 67)
(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)
G:=sub<Sym(128)| (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128), (1,102,69,40)(2,97,70,35)(3,100,71,38)(4,103,72,33)(5,98,65,36)(6,101,66,39)(7,104,67,34)(8,99,68,37)(9,32,50,82)(10,27,51,85)(11,30,52,88)(12,25,53,83)(13,28,54,86)(14,31,55,81)(15,26,56,84)(16,29,49,87)(17,57,77,111)(18,60,78,106)(19,63,79,109)(20,58,80,112)(21,61,73,107)(22,64,74,110)(23,59,75,105)(24,62,76,108)(41,113,93,127)(42,116,94,122)(43,119,95,125)(44,114,96,128)(45,117,89,123)(46,120,90,126)(47,115,91,121)(48,118,92,124), (1,13,115,111)(2,106,116,16)(3,11,117,109)(4,112,118,14)(5,9,119,107)(6,110,120,12)(7,15,113,105)(8,108,114,10)(17,102,28,91)(18,94,29,97)(19,100,30,89)(20,92,31,103)(21,98,32,95)(22,90,25,101)(23,104,26,93)(24,96,27,99)(33,80,48,81)(34,84,41,75)(35,78,42,87)(36,82,43,73)(37,76,44,85)(38,88,45,79)(39,74,46,83)(40,86,47,77)(49,70,60,122)(50,125,61,65)(51,68,62,128)(52,123,63,71)(53,66,64,126)(54,121,57,69)(55,72,58,124)(56,127,59,67), (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)>;
G:=Group( (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128), (1,102,69,40)(2,97,70,35)(3,100,71,38)(4,103,72,33)(5,98,65,36)(6,101,66,39)(7,104,67,34)(8,99,68,37)(9,32,50,82)(10,27,51,85)(11,30,52,88)(12,25,53,83)(13,28,54,86)(14,31,55,81)(15,26,56,84)(16,29,49,87)(17,57,77,111)(18,60,78,106)(19,63,79,109)(20,58,80,112)(21,61,73,107)(22,64,74,110)(23,59,75,105)(24,62,76,108)(41,113,93,127)(42,116,94,122)(43,119,95,125)(44,114,96,128)(45,117,89,123)(46,120,90,126)(47,115,91,121)(48,118,92,124), (1,13,115,111)(2,106,116,16)(3,11,117,109)(4,112,118,14)(5,9,119,107)(6,110,120,12)(7,15,113,105)(8,108,114,10)(17,102,28,91)(18,94,29,97)(19,100,30,89)(20,92,31,103)(21,98,32,95)(22,90,25,101)(23,104,26,93)(24,96,27,99)(33,80,48,81)(34,84,41,75)(35,78,42,87)(36,82,43,73)(37,76,44,85)(38,88,45,79)(39,74,46,83)(40,86,47,77)(49,70,60,122)(50,125,61,65)(51,68,62,128)(52,123,63,71)(53,66,64,126)(54,121,57,69)(55,72,58,124)(56,127,59,67), (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) );
G=PermutationGroup([[(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120),(121,123,125,127),(122,124,126,128)], [(1,102,69,40),(2,97,70,35),(3,100,71,38),(4,103,72,33),(5,98,65,36),(6,101,66,39),(7,104,67,34),(8,99,68,37),(9,32,50,82),(10,27,51,85),(11,30,52,88),(12,25,53,83),(13,28,54,86),(14,31,55,81),(15,26,56,84),(16,29,49,87),(17,57,77,111),(18,60,78,106),(19,63,79,109),(20,58,80,112),(21,61,73,107),(22,64,74,110),(23,59,75,105),(24,62,76,108),(41,113,93,127),(42,116,94,122),(43,119,95,125),(44,114,96,128),(45,117,89,123),(46,120,90,126),(47,115,91,121),(48,118,92,124)], [(1,13,115,111),(2,106,116,16),(3,11,117,109),(4,112,118,14),(5,9,119,107),(6,110,120,12),(7,15,113,105),(8,108,114,10),(17,102,28,91),(18,94,29,97),(19,100,30,89),(20,92,31,103),(21,98,32,95),(22,90,25,101),(23,104,26,93),(24,96,27,99),(33,80,48,81),(34,84,41,75),(35,78,42,87),(36,82,43,73),(37,76,44,85),(38,88,45,79),(39,74,46,83),(40,86,47,77),(49,70,60,122),(50,125,61,65),(51,68,62,128),(52,123,63,71),(53,66,64,126),(54,121,57,69),(55,72,58,124),(56,127,59,67)], [(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)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 4U | 4V | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | SD16 | C4○D4 | C4○D8 | C8.C22 |
| kernel | C4.68(C4×D4) | C22.7C42 | C22.4Q16 | C4×C4⋊C4 | C23.67C23 | C2×Q8⋊C4 | C2×C4.Q8 | Q8⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 8 | 2 | 2 | 4 | 8 | 4 | 2 |
Matrix representation of C4.68(C4×D4) ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 4 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 16 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 11 | 4 | 0 | 0 |
| 0 | 4 | 6 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 2 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,13,0,0,0,0,0,4],[4,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,4,0],[16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,16,0],[16,0,0,0,0,0,11,4,0,0,0,4,6,0,0,0,0,0,8,0,0,0,0,0,2] >;
C4.68(C4×D4) in GAP, Magma, Sage, TeX
C_4._{68}(C_4\times D_4) % in TeX
G:=Group("C4.68(C4xD4)"); // GroupNames label
G:=SmallGroup(128,659);
// by ID
G=gap.SmallGroup(128,659);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,58,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=a^-1*c^-1>;
// generators/relations