p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8).60D4, (C2×Q8).10Q8, (C2×C4).22Q16, C2.18(C8⋊3D4), C2.9(C4⋊Q16), C23.942(C2×D4), (C22×C4).326D4, C4.42(C22⋊Q8), C22.64(C2×Q16), C42⋊9C4.18C2, C2.10(C4.Q16), C22.81(C4⋊1D4), (C22×C8).124C22, (C2×C42).393C22, C2.8(C23.4Q8), (C22×Q8).81C22, C22.160(C8⋊C22), (C22×C4).1476C23, C4.36(C22.D4), C22.119(C22⋊Q8), C2.10(C23.48D4), C22.7C42.26C2, C23.67C23.21C2, C22.129(C22.D4), (C2×C4).751(C2×D4), (C2×C4).293(C2×Q8), (C2×C2.D8).20C2, (C2×C4).782(C4○D4), (C2×C4⋊C4).165C22, (C2×Q8⋊C4).21C2, SmallGroup(128,827)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8).60D4
G = < a,b,c,d | a2=b8=1, c4=b4, d2=ab4, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=b4c3 >
Subgroups: 256 in 127 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C22.7C42, C42⋊9C4, C23.67C23, C2×Q8⋊C4, C2×C2.D8, (C2×C8).60D4
Quotients: C1, C2, C22, D4, Q8, C23, Q16, C2×D4, C2×Q8, C4○D4, C22⋊Q8, C22.D4, C4⋊1D4, C2×Q16, C8⋊C22, C23.4Q8, C4.Q16, C23.48D4, C4⋊Q16, C8⋊3D4, (C2×C8).60D4
►Regular action on 128 points
Generators in S128
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 121)(8 122)(9 62)(10 63)(11 64)(12 57)(13 58)(14 59)(15 60)(16 61)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(49 111)(50 112)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(65 119)(66 120)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(79 97)(80 98)(81 93)(82 94)(83 95)(84 96)(85 89)(86 90)(87 91)(88 92)
(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 37 58 86 5 33 62 82)(2 19 59 91 6 23 63 95)(3 39 60 88 7 35 64 84)(4 21 61 93 8 17 57 89)(9 94 123 18 13 90 127 22)(10 83 124 38 14 87 128 34)(11 96 125 20 15 92 121 24)(12 85 126 40 16 81 122 36)(25 107 101 119 29 111 97 115)(26 54 102 66 30 50 98 70)(27 109 103 113 31 105 99 117)(28 56 104 68 32 52 100 72)(41 53 75 65 45 49 79 69)(42 108 76 120 46 112 80 116)(43 55 77 67 47 51 73 71)(44 110 78 114 48 106 74 118)
(1 78 127 100)(2 77 128 99)(3 76 121 98)(4 75 122 97)(5 74 123 104)(6 73 124 103)(7 80 125 102)(8 79 126 101)(9 32 58 44)(10 31 59 43)(11 30 60 42)(12 29 61 41)(13 28 62 48)(14 27 63 47)(15 26 64 46)(16 25 57 45)(17 49 40 107)(18 56 33 106)(19 55 34 105)(20 54 35 112)(21 53 36 111)(22 52 37 110)(23 51 38 109)(24 50 39 108)(65 81 115 89)(66 88 116 96)(67 87 117 95)(68 86 118 94)(69 85 119 93)(70 84 120 92)(71 83 113 91)(72 82 114 90)
G:=sub<Sym(128)| (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,121)(8,122)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(49,111)(50,112)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(65,119)(66,120)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,97)(80,98)(81,93)(82,94)(83,95)(84,96)(85,89)(86,90)(87,91)(88,92), (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), (1,37,58,86,5,33,62,82)(2,19,59,91,6,23,63,95)(3,39,60,88,7,35,64,84)(4,21,61,93,8,17,57,89)(9,94,123,18,13,90,127,22)(10,83,124,38,14,87,128,34)(11,96,125,20,15,92,121,24)(12,85,126,40,16,81,122,36)(25,107,101,119,29,111,97,115)(26,54,102,66,30,50,98,70)(27,109,103,113,31,105,99,117)(28,56,104,68,32,52,100,72)(41,53,75,65,45,49,79,69)(42,108,76,120,46,112,80,116)(43,55,77,67,47,51,73,71)(44,110,78,114,48,106,74,118), (1,78,127,100)(2,77,128,99)(3,76,121,98)(4,75,122,97)(5,74,123,104)(6,73,124,103)(7,80,125,102)(8,79,126,101)(9,32,58,44)(10,31,59,43)(11,30,60,42)(12,29,61,41)(13,28,62,48)(14,27,63,47)(15,26,64,46)(16,25,57,45)(17,49,40,107)(18,56,33,106)(19,55,34,105)(20,54,35,112)(21,53,36,111)(22,52,37,110)(23,51,38,109)(24,50,39,108)(65,81,115,89)(66,88,116,96)(67,87,117,95)(68,86,118,94)(69,85,119,93)(70,84,120,92)(71,83,113,91)(72,82,114,90)>;
G:=Group( (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,121)(8,122)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(49,111)(50,112)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(65,119)(66,120)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,97)(80,98)(81,93)(82,94)(83,95)(84,96)(85,89)(86,90)(87,91)(88,92), (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), (1,37,58,86,5,33,62,82)(2,19,59,91,6,23,63,95)(3,39,60,88,7,35,64,84)(4,21,61,93,8,17,57,89)(9,94,123,18,13,90,127,22)(10,83,124,38,14,87,128,34)(11,96,125,20,15,92,121,24)(12,85,126,40,16,81,122,36)(25,107,101,119,29,111,97,115)(26,54,102,66,30,50,98,70)(27,109,103,113,31,105,99,117)(28,56,104,68,32,52,100,72)(41,53,75,65,45,49,79,69)(42,108,76,120,46,112,80,116)(43,55,77,67,47,51,73,71)(44,110,78,114,48,106,74,118), (1,78,127,100)(2,77,128,99)(3,76,121,98)(4,75,122,97)(5,74,123,104)(6,73,124,103)(7,80,125,102)(8,79,126,101)(9,32,58,44)(10,31,59,43)(11,30,60,42)(12,29,61,41)(13,28,62,48)(14,27,63,47)(15,26,64,46)(16,25,57,45)(17,49,40,107)(18,56,33,106)(19,55,34,105)(20,54,35,112)(21,53,36,111)(22,52,37,110)(23,51,38,109)(24,50,39,108)(65,81,115,89)(66,88,116,96)(67,87,117,95)(68,86,118,94)(69,85,119,93)(70,84,120,92)(71,83,113,91)(72,82,114,90) );
G=PermutationGroup([[(1,123),(2,124),(3,125),(4,126),(5,127),(6,128),(7,121),(8,122),(9,62),(10,63),(11,64),(12,57),(13,58),(14,59),(15,60),(16,61),(17,36),(18,37),(19,38),(20,39),(21,40),(22,33),(23,34),(24,35),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(49,111),(50,112),(51,105),(52,106),(53,107),(54,108),(55,109),(56,110),(65,119),(66,120),(67,113),(68,114),(69,115),(70,116),(71,117),(72,118),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104),(79,97),(80,98),(81,93),(82,94),(83,95),(84,96),(85,89),(86,90),(87,91),(88,92)], [(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)], [(1,37,58,86,5,33,62,82),(2,19,59,91,6,23,63,95),(3,39,60,88,7,35,64,84),(4,21,61,93,8,17,57,89),(9,94,123,18,13,90,127,22),(10,83,124,38,14,87,128,34),(11,96,125,20,15,92,121,24),(12,85,126,40,16,81,122,36),(25,107,101,119,29,111,97,115),(26,54,102,66,30,50,98,70),(27,109,103,113,31,105,99,117),(28,56,104,68,32,52,100,72),(41,53,75,65,45,49,79,69),(42,108,76,120,46,112,80,116),(43,55,77,67,47,51,73,71),(44,110,78,114,48,106,74,118)], [(1,78,127,100),(2,77,128,99),(3,76,121,98),(4,75,122,97),(5,74,123,104),(6,73,124,103),(7,80,125,102),(8,79,126,101),(9,32,58,44),(10,31,59,43),(11,30,60,42),(12,29,61,41),(13,28,62,48),(14,27,63,47),(15,26,64,46),(16,25,57,45),(17,49,40,107),(18,56,33,106),(19,55,34,105),(20,54,35,112),(21,53,36,111),(22,52,37,110),(23,51,38,109),(24,50,39,108),(65,81,115,89),(66,88,116,96),(67,87,117,95),(68,86,118,94),(69,85,119,93),(70,84,120,92),(71,83,113,91),(72,82,114,90)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | Q16 | C4○D4 | C8⋊C22 |
| kernel | (C2×C8).60D4 | C22.7C42 | C42⋊9C4 | C23.67C23 | C2×Q8⋊C4 | C2×C2.D8 | C2×C8 | C22×C4 | C2×Q8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 8 | 6 | 2 |
Matrix representation of (C2×C8).60D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 1 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 11 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 7 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 6 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 13 | 13 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 0 | 4 | 4 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,1,0,0,0,0,1,10,0,0,0,0,0,0,6,3,0,0,0,0,11,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,7,0,0,0,0,7,1,0,0,0,0,0,0,11,14,0,0,0,0,6,0,0,0,0,0,0,0,4,13,0,0,0,0,0,13],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,4,0,0,0,0,0,13,0,0,0,0,0,0,13,4,0,0,0,0,9,4] >;
(C2×C8).60D4 in GAP, Magma, Sage, TeX
(C_2\times C_8)._{60}D_4 % in TeX
G:=Group("(C2xC8).60D4"); // GroupNames label
G:=SmallGroup(128,827);
// by ID
G=gap.SmallGroup(128,827);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,176,422,387,352,1018,521,248,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=a*b^4,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b^-1,d*c*d^-1=b^4*c^3>;
// generators/relations