p-group, metabelian, nilpotent (class 2), monomial
Aliases: (C2×C8).30Q8, (C2×C8).197D4, C2.6(C8⋊4Q8), C22.40(C4×Q8), C2.18(C8⋊9D4), C2.14(C8⋊6D4), C22.155(C4×D4), (C2×C4).19M4(2), (C2×C42).5C22, C22.53(C8○D4), C4.115(C22⋊Q8), (C22×C8).41C22, C4.37(C42.C2), C4.47(C42⋊2C2), C2.C42.22C4, C2.8(C42.6C4), C23.314(C22×C4), C22.67(C2×M4(2)), (C22×C4).1632C23, C22.82(C42⋊C2), C4.138(C22.D4), C22.7C42.44C2, C2.10(C42.7C22), C2.8(C23.63C23), (C2×C4⋊C8).55C2, (C2×C4⋊C4).57C4, (C4×C4⋊C4).13C2, (C2×C4).344(C2×Q8), (C2×C8⋊C4).30C2, (C2×C4).1533(C2×D4), (C2×C4).938(C4○D4), (C22×C4).122(C2×C4), SmallGroup(128,649)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8).Q8
G = < a,b,c,d | a2=b8=1, c4=b4, d2=b4c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=ab5, dcd-1=b6c3 >
Subgroups: 180 in 114 conjugacy classes, 60 normal (52 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C2.C42, C8⋊C4, C4⋊C8, C2×C42, C2×C4⋊C4, C22×C8, C22.7C42, C4×C4⋊C4, C2×C8⋊C4, C2×C4⋊C8, (C2×C8).Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C2×M4(2), C8○D4, C23.63C23, C42.6C4, C42.7C22, C8⋊9D4, C8⋊6D4, C8⋊4Q8, (C2×C8).Q8
►Regular action on 128 points
Generators in S128
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 73)(16 74)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 103)(26 104)(27 97)(28 98)(29 99)(30 100)(31 101)(32 102)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 89)(40 90)(49 107)(50 108)(51 109)(52 110)(53 111)(54 112)(55 105)(56 106)(57 118)(58 119)(59 120)(60 113)(61 114)(62 115)(63 116)(64 117)(65 122)(66 123)(67 124)(68 125)(69 126)(70 127)(71 128)(72 121)
(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 23 35 27 5 19 39 31)(2 20 36 32 6 24 40 28)(3 17 37 29 7 21 33 25)(4 22 38 26 8 18 34 30)(9 113 128 110 13 117 124 106)(10 118 121 107 14 114 125 111)(11 115 122 112 15 119 126 108)(12 120 123 109 16 116 127 105)(41 96 104 88 45 92 100 84)(42 93 97 85 46 89 101 81)(43 90 98 82 47 94 102 86)(44 95 99 87 48 91 103 83)(49 80 61 68 53 76 57 72)(50 77 62 65 54 73 58 69)(51 74 63 70 55 78 59 66)(52 79 64 67 56 75 60 71)
(1 60 39 56 5 64 35 52)(2 118 40 111 6 114 36 107)(3 62 33 50 7 58 37 54)(4 120 34 105 8 116 38 109)(9 99 124 44 13 103 128 48)(10 26 125 22 14 30 121 18)(11 101 126 46 15 97 122 42)(12 28 127 24 16 32 123 20)(17 79 25 71 21 75 29 67)(19 73 27 65 23 77 31 69)(41 80 100 72 45 76 104 68)(43 74 102 66 47 78 98 70)(49 82 57 90 53 86 61 94)(51 84 59 92 55 88 63 96)(81 113 89 106 85 117 93 110)(83 115 91 108 87 119 95 112)
G:=sub<Sym(128)| (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,73)(16,74)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,103)(26,104)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,89)(40,90)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,105)(56,106)(57,118)(58,119)(59,120)(60,113)(61,114)(62,115)(63,116)(64,117)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,121), (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), (1,23,35,27,5,19,39,31)(2,20,36,32,6,24,40,28)(3,17,37,29,7,21,33,25)(4,22,38,26,8,18,34,30)(9,113,128,110,13,117,124,106)(10,118,121,107,14,114,125,111)(11,115,122,112,15,119,126,108)(12,120,123,109,16,116,127,105)(41,96,104,88,45,92,100,84)(42,93,97,85,46,89,101,81)(43,90,98,82,47,94,102,86)(44,95,99,87,48,91,103,83)(49,80,61,68,53,76,57,72)(50,77,62,65,54,73,58,69)(51,74,63,70,55,78,59,66)(52,79,64,67,56,75,60,71), (1,60,39,56,5,64,35,52)(2,118,40,111,6,114,36,107)(3,62,33,50,7,58,37,54)(4,120,34,105,8,116,38,109)(9,99,124,44,13,103,128,48)(10,26,125,22,14,30,121,18)(11,101,126,46,15,97,122,42)(12,28,127,24,16,32,123,20)(17,79,25,71,21,75,29,67)(19,73,27,65,23,77,31,69)(41,80,100,72,45,76,104,68)(43,74,102,66,47,78,98,70)(49,82,57,90,53,86,61,94)(51,84,59,92,55,88,63,96)(81,113,89,106,85,117,93,110)(83,115,91,108,87,119,95,112)>;
G:=Group( (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,73)(16,74)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,103)(26,104)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,89)(40,90)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,105)(56,106)(57,118)(58,119)(59,120)(60,113)(61,114)(62,115)(63,116)(64,117)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,121), (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), (1,23,35,27,5,19,39,31)(2,20,36,32,6,24,40,28)(3,17,37,29,7,21,33,25)(4,22,38,26,8,18,34,30)(9,113,128,110,13,117,124,106)(10,118,121,107,14,114,125,111)(11,115,122,112,15,119,126,108)(12,120,123,109,16,116,127,105)(41,96,104,88,45,92,100,84)(42,93,97,85,46,89,101,81)(43,90,98,82,47,94,102,86)(44,95,99,87,48,91,103,83)(49,80,61,68,53,76,57,72)(50,77,62,65,54,73,58,69)(51,74,63,70,55,78,59,66)(52,79,64,67,56,75,60,71), (1,60,39,56,5,64,35,52)(2,118,40,111,6,114,36,107)(3,62,33,50,7,58,37,54)(4,120,34,105,8,116,38,109)(9,99,124,44,13,103,128,48)(10,26,125,22,14,30,121,18)(11,101,126,46,15,97,122,42)(12,28,127,24,16,32,123,20)(17,79,25,71,21,75,29,67)(19,73,27,65,23,77,31,69)(41,80,100,72,45,76,104,68)(43,74,102,66,47,78,98,70)(49,82,57,90,53,86,61,94)(51,84,59,92,55,88,63,96)(81,113,89,106,85,117,93,110)(83,115,91,108,87,119,95,112) );
G=PermutationGroup([[(1,81),(2,82),(3,83),(4,84),(5,85),(6,86),(7,87),(8,88),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,73),(16,74),(17,44),(18,45),(19,46),(20,47),(21,48),(22,41),(23,42),(24,43),(25,103),(26,104),(27,97),(28,98),(29,99),(30,100),(31,101),(32,102),(33,91),(34,92),(35,93),(36,94),(37,95),(38,96),(39,89),(40,90),(49,107),(50,108),(51,109),(52,110),(53,111),(54,112),(55,105),(56,106),(57,118),(58,119),(59,120),(60,113),(61,114),(62,115),(63,116),(64,117),(65,122),(66,123),(67,124),(68,125),(69,126),(70,127),(71,128),(72,121)], [(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)], [(1,23,35,27,5,19,39,31),(2,20,36,32,6,24,40,28),(3,17,37,29,7,21,33,25),(4,22,38,26,8,18,34,30),(9,113,128,110,13,117,124,106),(10,118,121,107,14,114,125,111),(11,115,122,112,15,119,126,108),(12,120,123,109,16,116,127,105),(41,96,104,88,45,92,100,84),(42,93,97,85,46,89,101,81),(43,90,98,82,47,94,102,86),(44,95,99,87,48,91,103,83),(49,80,61,68,53,76,57,72),(50,77,62,65,54,73,58,69),(51,74,63,70,55,78,59,66),(52,79,64,67,56,75,60,71)], [(1,60,39,56,5,64,35,52),(2,118,40,111,6,114,36,107),(3,62,33,50,7,58,37,54),(4,120,34,105,8,116,38,109),(9,99,124,44,13,103,128,48),(10,26,125,22,14,30,121,18),(11,101,126,46,15,97,122,42),(12,28,127,24,16,32,123,20),(17,79,25,71,21,75,29,67),(19,73,27,65,23,77,31,69),(41,80,100,72,45,76,104,68),(43,74,102,66,47,78,98,70),(49,82,57,90,53,86,61,94),(51,84,59,92,55,88,63,96),(81,113,89,106,85,117,93,110),(83,115,91,108,87,119,95,112)]])
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 |
| type | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | M4(2) | C4○D4 | C8○D4 |
| kernel | (C2×C8).Q8 | C22.7C42 | C4×C4⋊C4 | C2×C8⋊C4 | C2×C4⋊C8 | C2.C42 | C2×C4⋊C4 | C2×C8 | C2×C8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 8 |
Matrix representation of (C2×C8).Q8 ►in GL6(𝔽17)
| 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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 8 | 0 | 0 |
| 0 | 0 | 8 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 3 | 0 | 0 |
| 0 | 0 | 3 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 6 | 1 | 0 | 0 | 0 | 0 |
| 16 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 9 | 0 | 0 |
| 0 | 0 | 9 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,5,8,0,0,0,0,8,12,0,0,0,0,0,0,13,4,0,0,0,0,9,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,2,3,0,0,0,0,3,15,0,0,0,0,0,0,1,16,0,0,0,0,2,16],[6,16,0,0,0,0,1,11,0,0,0,0,0,0,12,9,0,0,0,0,9,5,0,0,0,0,0,0,13,0,0,0,0,0,9,4] >;
(C2×C8).Q8 in GAP, Magma, Sage, TeX
(C_2\times C_8).Q_8
% in TeX
G:=Group("(C2xC8).Q8"); // GroupNames label
G:=SmallGroup(128,649);
// by ID
G=gap.SmallGroup(128,649);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,723,58,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=b^4*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=a*b^5,d*c*d^-1=b^6*c^3>;
// generators/relations