direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C8⋊4Q8, C42.689C23, C8⋊7(C2×Q8), (C2×C8)⋊15Q8, C4○(C8⋊4Q8), C4.51(C4×Q8), (C4×Q8).25C4, C22.33(C4×Q8), C4.63(C22×Q8), C4⋊C8.357C22, C42.285(C2×C4), (C2×C8).423C23, (C2×C4).656C24, (C4×C8).436C22, (C2×C4).52M4(2), C4.13(C2×M4(2)), (C22×Q8).30C4, C22.44(C8○D4), (C4×Q8).276C22, C8⋊C4.157C22, C23.294(C22×C4), (C22×C8).443C22, (C2×C42).770C22, C22.182(C23×C4), C2.13(C22×M4(2)), C22.64(C2×M4(2)), (C22×C4).1651C23, (C2×C4×C8).68C2, C2.23(C2×C4×Q8), (C2×C4⋊C4).73C4, (C2×C4⋊C8).57C2, (C2×C4×Q8).43C2, (C2×C4)○(C8⋊4Q8), C2.20(C2×C8○D4), C4⋊C4.223(C2×C4), C4.307(C2×C4○D4), (C2×C4).358(C2×Q8), (C2×C8⋊C4).38C2, (C2×Q8).208(C2×C4), (C2×C4).959(C4○D4), (C2×C4).270(C22×C4), (C22×C4).418(C2×C4), SmallGroup(128,1691)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8⋊4Q8
G = < a,b,c,d | a2=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=c-1 >
Subgroups: 244 in 200 conjugacy classes, 156 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22×C8, C22×C8, C22×Q8, C2×C4×C8, C2×C8⋊C4, C2×C4⋊C8, C2×C4⋊C8, C8⋊4Q8, C2×C4×Q8, C2×C8⋊4Q8
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, M4(2), C22×C4, C2×Q8, C4○D4, C24, C4×Q8, C2×M4(2), C8○D4, C23×C4, C22×Q8, C2×C4○D4, C8⋊4Q8, C2×C4×Q8, C22×M4(2), C2×C8○D4, C2×C8⋊4Q8
►Regular action on 128 points
Generators in S128
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 103)(26 104)(27 97)(28 98)(29 99)(30 100)(31 101)(32 102)(33 77)(34 78)(35 79)(36 80)(37 73)(38 74)(39 75)(40 76)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 105)(81 96)(82 89)(83 90)(84 91)(85 92)(86 93)(87 94)(88 95)(113 128)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(120 127)
(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 38 43 63)(2 39 44 64)(3 40 45 57)(4 33 46 58)(5 34 47 59)(6 35 48 60)(7 36 41 61)(8 37 42 62)(9 73 18 53)(10 74 19 54)(11 75 20 55)(12 76 21 56)(13 77 22 49)(14 78 23 50)(15 79 24 51)(16 80 17 52)(25 67 118 89)(26 68 119 90)(27 69 120 91)(28 70 113 92)(29 71 114 93)(30 72 115 94)(31 65 116 95)(32 66 117 96)(81 102 107 124)(82 103 108 125)(83 104 109 126)(84 97 110 127)(85 98 111 128)(86 99 112 121)(87 100 105 122)(88 101 106 123)
(1 72 43 94)(2 69 44 91)(3 66 45 96)(4 71 46 93)(5 68 47 90)(6 65 48 95)(7 70 41 92)(8 67 42 89)(9 108 18 82)(10 105 19 87)(11 110 20 84)(12 107 21 81)(13 112 22 86)(14 109 23 83)(15 106 24 88)(16 111 17 85)(25 62 118 37)(26 59 119 34)(27 64 120 39)(28 61 113 36)(29 58 114 33)(30 63 115 38)(31 60 116 35)(32 57 117 40)(49 121 77 99)(50 126 78 104)(51 123 79 101)(52 128 80 98)(53 125 73 103)(54 122 74 100)(55 127 75 97)(56 124 76 102)
G:=sub<Sym(128)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,103)(26,104)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(33,77)(34,78)(35,79)(36,80)(37,73)(38,74)(39,75)(40,76)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,105)(81,96)(82,89)(83,90)(84,91)(85,92)(86,93)(87,94)(88,95)(113,128)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(120,127), (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), (1,38,43,63)(2,39,44,64)(3,40,45,57)(4,33,46,58)(5,34,47,59)(6,35,48,60)(7,36,41,61)(8,37,42,62)(9,73,18,53)(10,74,19,54)(11,75,20,55)(12,76,21,56)(13,77,22,49)(14,78,23,50)(15,79,24,51)(16,80,17,52)(25,67,118,89)(26,68,119,90)(27,69,120,91)(28,70,113,92)(29,71,114,93)(30,72,115,94)(31,65,116,95)(32,66,117,96)(81,102,107,124)(82,103,108,125)(83,104,109,126)(84,97,110,127)(85,98,111,128)(86,99,112,121)(87,100,105,122)(88,101,106,123), (1,72,43,94)(2,69,44,91)(3,66,45,96)(4,71,46,93)(5,68,47,90)(6,65,48,95)(7,70,41,92)(8,67,42,89)(9,108,18,82)(10,105,19,87)(11,110,20,84)(12,107,21,81)(13,112,22,86)(14,109,23,83)(15,106,24,88)(16,111,17,85)(25,62,118,37)(26,59,119,34)(27,64,120,39)(28,61,113,36)(29,58,114,33)(30,63,115,38)(31,60,116,35)(32,57,117,40)(49,121,77,99)(50,126,78,104)(51,123,79,101)(52,128,80,98)(53,125,73,103)(54,122,74,100)(55,127,75,97)(56,124,76,102)>;
G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,103)(26,104)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(33,77)(34,78)(35,79)(36,80)(37,73)(38,74)(39,75)(40,76)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,105)(81,96)(82,89)(83,90)(84,91)(85,92)(86,93)(87,94)(88,95)(113,128)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(120,127), (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), (1,38,43,63)(2,39,44,64)(3,40,45,57)(4,33,46,58)(5,34,47,59)(6,35,48,60)(7,36,41,61)(8,37,42,62)(9,73,18,53)(10,74,19,54)(11,75,20,55)(12,76,21,56)(13,77,22,49)(14,78,23,50)(15,79,24,51)(16,80,17,52)(25,67,118,89)(26,68,119,90)(27,69,120,91)(28,70,113,92)(29,71,114,93)(30,72,115,94)(31,65,116,95)(32,66,117,96)(81,102,107,124)(82,103,108,125)(83,104,109,126)(84,97,110,127)(85,98,111,128)(86,99,112,121)(87,100,105,122)(88,101,106,123), (1,72,43,94)(2,69,44,91)(3,66,45,96)(4,71,46,93)(5,68,47,90)(6,65,48,95)(7,70,41,92)(8,67,42,89)(9,108,18,82)(10,105,19,87)(11,110,20,84)(12,107,21,81)(13,112,22,86)(14,109,23,83)(15,106,24,88)(16,111,17,85)(25,62,118,37)(26,59,119,34)(27,64,120,39)(28,61,113,36)(29,58,114,33)(30,63,115,38)(31,60,116,35)(32,57,117,40)(49,121,77,99)(50,126,78,104)(51,123,79,101)(52,128,80,98)(53,125,73,103)(54,122,74,100)(55,127,75,97)(56,124,76,102) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,103),(26,104),(27,97),(28,98),(29,99),(30,100),(31,101),(32,102),(33,77),(34,78),(35,79),(36,80),(37,73),(38,74),(39,75),(40,76),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,57),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,105),(81,96),(82,89),(83,90),(84,91),(85,92),(86,93),(87,94),(88,95),(113,128),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(120,127)], [(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)], [(1,38,43,63),(2,39,44,64),(3,40,45,57),(4,33,46,58),(5,34,47,59),(6,35,48,60),(7,36,41,61),(8,37,42,62),(9,73,18,53),(10,74,19,54),(11,75,20,55),(12,76,21,56),(13,77,22,49),(14,78,23,50),(15,79,24,51),(16,80,17,52),(25,67,118,89),(26,68,119,90),(27,69,120,91),(28,70,113,92),(29,71,114,93),(30,72,115,94),(31,65,116,95),(32,66,117,96),(81,102,107,124),(82,103,108,125),(83,104,109,126),(84,97,110,127),(85,98,111,128),(86,99,112,121),(87,100,105,122),(88,101,106,123)], [(1,72,43,94),(2,69,44,91),(3,66,45,96),(4,71,46,93),(5,68,47,90),(6,65,48,95),(7,70,41,92),(8,67,42,89),(9,108,18,82),(10,105,19,87),(11,110,20,84),(12,107,21,81),(13,112,22,86),(14,109,23,83),(15,106,24,88),(16,111,17,85),(25,62,118,37),(26,59,119,34),(27,64,120,39),(28,61,113,36),(29,58,114,33),(30,63,115,38),(31,60,116,35),(32,57,117,40),(49,121,77,99),(50,126,78,104),(51,123,79,101),(52,128,80,98),(53,125,73,103),(54,122,74,100),(55,127,75,97),(56,124,76,102)]])
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 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8 | M4(2) | C4○D4 | C8○D4 |
| kernel | C2×C8⋊4Q8 | C2×C4×C8 | C2×C8⋊C4 | C2×C4⋊C8 | C8⋊4Q8 | C2×C4×Q8 | C2×C4⋊C4 | C4×Q8 | C22×Q8 | C2×C8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 3 | 8 | 1 | 6 | 8 | 2 | 4 | 8 | 4 | 8 |
Matrix representation of C2×C8⋊4Q8 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 4 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 15 | 6 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,0,13,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,1,0],[16,0,0,0,0,0,0,15,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,9,0,0,0,15,0,0,0,0,0,0,11,15,0,0,0,9,6] >;
C2×C8⋊4Q8 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes_4Q_8
% in TeX
G:=Group("C2xC8:4Q8"); // GroupNames label
G:=SmallGroup(128,1691);
// by ID
G=gap.SmallGroup(128,1691);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,1430,268,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=c^-1>;
// generators/relations