direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: Q8×C2×C8, C42.688C23, C8○(C8×Q8), C4○(C8×Q8), C8○2(C4×Q8), C4.50(C4×Q8), C2.5(C23×C8), (C4×Q8).39C4, C4.17(C22×C8), C22.32(C4×Q8), C4.62(C22×Q8), C4⋊C8.374C22, (C4×C8).380C22, (C2×C8).479C23, (C2×C4).655C24, C42.284(C2×C4), (C22×Q8).36C4, C22.43(C8○D4), C22.32(C22×C8), C22.41(C23×C4), (C4×Q8).329C22, C23.293(C22×C4), (C22×C8).594C22, (C2×C42).1116C22, (C22×C4).1650C23, C8○3(C2×C4⋊C8), C8○3(C2×C4⋊C4), C4⋊C4○4(C2×C8), (C2×C8)○(C8×Q8), (C2×C4)○(C8×Q8), C2.3(C2×C4×Q8), (C2×C4×C8).29C2, (C2×C8)○2(C4×Q8), (C2×C8)○4(C4⋊C8), C2.4(C2×C8○D4), (C2×C4⋊C8).61C2, (C2×C4⋊C4).85C4, (C2×C4×Q8).59C2, (C4×Q8)○(C22×C8), (C2×C4).56(C2×C8), C4⋊C4.248(C2×C4), C4.306(C2×C4○D4), (C2×C4).357(C2×Q8), (C2×Q8).225(C2×C4), (C2×C4).958(C4○D4), (C22×C4).386(C2×C4), (C2×C4).295(C22×C4), (C2×C8)○(C2×C4×Q8), (C2×C8)○3(C2×C4⋊C4), (C2×C8)○3(C2×C4⋊C8), (C22×C8)○(C2×C4×Q8), SmallGroup(128,1690)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8×C2×C8
G = < a,b,c,d | a2=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 244 in 220 conjugacy classes, 196 normal (15 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, C4×C8, C4⋊C8, C2×C42, C2×C4⋊C4, C4×Q8, C22×C8, C22×C8, C22×Q8, C2×C4×C8, C2×C4⋊C8, C8×Q8, C2×C4×Q8, Q8×C2×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, Q8, C23, C2×C8, C22×C4, C2×Q8, C4○D4, C24, C4×Q8, C22×C8, C8○D4, C23×C4, C22×Q8, C2×C4○D4, C8×Q8, C2×C4×Q8, C23×C8, C2×C8○D4, Q8×C2×C8
►Regular action on 128 points
Generators in S128
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(25 98)(26 99)(27 100)(28 101)(29 102)(30 103)(31 104)(32 97)(33 80)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(65 112)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(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 57 43 36)(2 58 44 37)(3 59 45 38)(4 60 46 39)(5 61 47 40)(6 62 48 33)(7 63 41 34)(8 64 42 35)(9 80 22 55)(10 73 23 56)(11 74 24 49)(12 75 17 50)(13 76 18 51)(14 77 19 52)(15 78 20 53)(16 79 21 54)(25 66 120 95)(26 67 113 96)(27 68 114 89)(28 69 115 90)(29 70 116 91)(30 71 117 92)(31 72 118 93)(32 65 119 94)(81 99 106 128)(82 100 107 121)(83 101 108 122)(84 102 109 123)(85 103 110 124)(86 104 111 125)(87 97 112 126)(88 98 105 127)
(1 96 43 67)(2 89 44 68)(3 90 45 69)(4 91 46 70)(5 92 47 71)(6 93 48 72)(7 94 41 65)(8 95 42 66)(9 111 22 86)(10 112 23 87)(11 105 24 88)(12 106 17 81)(13 107 18 82)(14 108 19 83)(15 109 20 84)(16 110 21 85)(25 64 120 35)(26 57 113 36)(27 58 114 37)(28 59 115 38)(29 60 116 39)(30 61 117 40)(31 62 118 33)(32 63 119 34)(49 127 74 98)(50 128 75 99)(51 121 76 100)(52 122 77 101)(53 123 78 102)(54 124 79 103)(55 125 80 104)(56 126 73 97)
G:=sub<Sym(128)| (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(25,98)(26,99)(27,100)(28,101)(29,102)(30,103)(31,104)(32,97)(33,80)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(65,112)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(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), (1,57,43,36)(2,58,44,37)(3,59,45,38)(4,60,46,39)(5,61,47,40)(6,62,48,33)(7,63,41,34)(8,64,42,35)(9,80,22,55)(10,73,23,56)(11,74,24,49)(12,75,17,50)(13,76,18,51)(14,77,19,52)(15,78,20,53)(16,79,21,54)(25,66,120,95)(26,67,113,96)(27,68,114,89)(28,69,115,90)(29,70,116,91)(30,71,117,92)(31,72,118,93)(32,65,119,94)(81,99,106,128)(82,100,107,121)(83,101,108,122)(84,102,109,123)(85,103,110,124)(86,104,111,125)(87,97,112,126)(88,98,105,127), (1,96,43,67)(2,89,44,68)(3,90,45,69)(4,91,46,70)(5,92,47,71)(6,93,48,72)(7,94,41,65)(8,95,42,66)(9,111,22,86)(10,112,23,87)(11,105,24,88)(12,106,17,81)(13,107,18,82)(14,108,19,83)(15,109,20,84)(16,110,21,85)(25,64,120,35)(26,57,113,36)(27,58,114,37)(28,59,115,38)(29,60,116,39)(30,61,117,40)(31,62,118,33)(32,63,119,34)(49,127,74,98)(50,128,75,99)(51,121,76,100)(52,122,77,101)(53,123,78,102)(54,124,79,103)(55,125,80,104)(56,126,73,97)>;
G:=Group( (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(25,98)(26,99)(27,100)(28,101)(29,102)(30,103)(31,104)(32,97)(33,80)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(65,112)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(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), (1,57,43,36)(2,58,44,37)(3,59,45,38)(4,60,46,39)(5,61,47,40)(6,62,48,33)(7,63,41,34)(8,64,42,35)(9,80,22,55)(10,73,23,56)(11,74,24,49)(12,75,17,50)(13,76,18,51)(14,77,19,52)(15,78,20,53)(16,79,21,54)(25,66,120,95)(26,67,113,96)(27,68,114,89)(28,69,115,90)(29,70,116,91)(30,71,117,92)(31,72,118,93)(32,65,119,94)(81,99,106,128)(82,100,107,121)(83,101,108,122)(84,102,109,123)(85,103,110,124)(86,104,111,125)(87,97,112,126)(88,98,105,127), (1,96,43,67)(2,89,44,68)(3,90,45,69)(4,91,46,70)(5,92,47,71)(6,93,48,72)(7,94,41,65)(8,95,42,66)(9,111,22,86)(10,112,23,87)(11,105,24,88)(12,106,17,81)(13,107,18,82)(14,108,19,83)(15,109,20,84)(16,110,21,85)(25,64,120,35)(26,57,113,36)(27,58,114,37)(28,59,115,38)(29,60,116,39)(30,61,117,40)(31,62,118,33)(32,63,119,34)(49,127,74,98)(50,128,75,99)(51,121,76,100)(52,122,77,101)(53,123,78,102)(54,124,79,103)(55,125,80,104)(56,126,73,97) );
G=PermutationGroup([[(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,48),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(25,98),(26,99),(27,100),(28,101),(29,102),(30,103),(31,104),(32,97),(33,80),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79),(49,64),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63),(65,112),(66,105),(67,106),(68,107),(69,108),(70,109),(71,110),(72,111),(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)], [(1,57,43,36),(2,58,44,37),(3,59,45,38),(4,60,46,39),(5,61,47,40),(6,62,48,33),(7,63,41,34),(8,64,42,35),(9,80,22,55),(10,73,23,56),(11,74,24,49),(12,75,17,50),(13,76,18,51),(14,77,19,52),(15,78,20,53),(16,79,21,54),(25,66,120,95),(26,67,113,96),(27,68,114,89),(28,69,115,90),(29,70,116,91),(30,71,117,92),(31,72,118,93),(32,65,119,94),(81,99,106,128),(82,100,107,121),(83,101,108,122),(84,102,109,123),(85,103,110,124),(86,104,111,125),(87,97,112,126),(88,98,105,127)], [(1,96,43,67),(2,89,44,68),(3,90,45,69),(4,91,46,70),(5,92,47,71),(6,93,48,72),(7,94,41,65),(8,95,42,66),(9,111,22,86),(10,112,23,87),(11,105,24,88),(12,106,17,81),(13,107,18,82),(14,108,19,83),(15,109,20,84),(16,110,21,85),(25,64,120,35),(26,57,113,36),(27,58,114,37),(28,59,115,38),(29,60,116,39),(30,61,117,40),(31,62,118,33),(32,63,119,34),(49,127,74,98),(50,128,75,99),(51,121,76,100),(52,122,77,101),(53,123,78,102),(54,124,79,103),(55,125,80,104),(56,126,73,97)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AF | 8A | ··· | 8P | 8Q | ··· | 8AN |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | Q8 | C4○D4 | C8○D4 |
| kernel | Q8×C2×C8 | C2×C4×C8 | C2×C4⋊C8 | C8×Q8 | C2×C4×Q8 | C2×C4⋊C4 | C4×Q8 | C22×Q8 | C2×Q8 | C2×C8 | C2×C4 | C22 |
| # reps | 1 | 3 | 3 | 8 | 1 | 6 | 8 | 2 | 32 | 4 | 4 | 8 |
Matrix representation of Q8×C2×C8 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 2 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[2,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0] >;
Q8×C2×C8 in GAP, Magma, Sage, TeX
Q_8\times C_2\times C_8
% in TeX
G:=Group("Q8xC2xC8"); // GroupNames label
G:=SmallGroup(128,1690);
// by ID
G=gap.SmallGroup(128,1690);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,268,124]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations