direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8⋊2Q8, C42.363D4, C42.717C23, C8⋊5(C2×Q8), (C2×C8)⋊12Q8, (C2×C4).93D8, C4.20(C2×D8), C4.14(C4⋊Q8), (C2×C4).44Q16, C4.14(C2×Q16), C4⋊C4.98C23, C22.74(C2×D8), C2.12(C22×D8), C4.10(C22×Q8), (C2×C4).357C24, (C2×C8).563C23, (C4×C8).409C22, C23.884(C2×D4), (C22×C4).616D4, C4⋊Q8.283C22, C22.46(C4⋊Q8), C2.12(C22×Q16), C22.51(C2×Q16), C2.D8.178C22, (C22×C8).538C22, C22.617(C22×D4), (C2×C42).1132C22, (C22×C4).1566C23, (C2×C4×C8).39C2, C2.27(C2×C4⋊Q8), (C2×C4⋊Q8).50C2, (C2×C4).861(C2×D4), (C2×C4).246(C2×Q8), (C2×C2.D8).30C2, (C2×C4⋊C4).630C22, SmallGroup(128,1891)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 372 in 212 conjugacy classes, 132 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C4 [×8], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×16], C2×C4 [×16], Q8 [×16], C23, C42 [×4], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×12], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×16], C4×C8 [×4], C2.D8 [×16], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C4⋊Q8 [×8], C4⋊Q8 [×4], C22×C8 [×2], C22×Q8 [×2], C2×C4×C8, C2×C2.D8 [×4], C8⋊2Q8 [×8], C2×C4⋊Q8 [×2], C2×C8⋊2Q8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], D8 [×4], Q16 [×4], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C2×D8 [×6], C2×Q16 [×6], C22×D4, C22×Q8 [×2], C8⋊2Q8 [×4], C2×C4⋊Q8, C22×D8, C22×Q16, C2×C8⋊2Q8
Generators and relations
G = < a,b,c,d | a2=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
►Regular action on 128 points
Generators in S128
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(25 101)(26 102)(27 103)(28 104)(29 97)(30 98)(31 99)(32 100)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 73)(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 92)(82 93)(83 94)(84 95)(85 96)(86 89)(87 90)(88 91)(113 124)(114 125)(115 126)(116 127)(117 128)(118 121)(119 122)(120 123)
(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 59 43 36)(2 60 44 37)(3 61 45 38)(4 62 46 39)(5 63 47 40)(6 64 48 33)(7 57 41 34)(8 58 42 35)(9 78 24 53)(10 79 17 54)(11 80 18 55)(12 73 19 56)(13 74 20 49)(14 75 21 50)(15 76 22 51)(16 77 23 52)(25 65 119 94)(26 66 120 95)(27 67 113 96)(28 68 114 89)(29 69 115 90)(30 70 116 91)(31 71 117 92)(32 72 118 93)(81 99 110 128)(82 100 111 121)(83 101 112 122)(84 102 105 123)(85 103 106 124)(86 104 107 125)(87 97 108 126)(88 98 109 127)
(1 93 43 72)(2 92 44 71)(3 91 45 70)(4 90 46 69)(5 89 47 68)(6 96 48 67)(7 95 41 66)(8 94 42 65)(9 110 24 81)(10 109 17 88)(11 108 18 87)(12 107 19 86)(13 106 20 85)(14 105 21 84)(15 112 22 83)(16 111 23 82)(25 58 119 35)(26 57 120 34)(27 64 113 33)(28 63 114 40)(29 62 115 39)(30 61 116 38)(31 60 117 37)(32 59 118 36)(49 124 74 103)(50 123 75 102)(51 122 76 101)(52 121 77 100)(53 128 78 99)(54 127 79 98)(55 126 80 97)(56 125 73 104)
G:=sub<Sym(128)| (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(25,101)(26,102)(27,103)(28,104)(29,97)(30,98)(31,99)(32,100)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,73)(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)(81,92)(82,93)(83,94)(84,95)(85,96)(86,89)(87,90)(88,91)(113,124)(114,125)(115,126)(116,127)(117,128)(118,121)(119,122)(120,123), (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), (1,59,43,36)(2,60,44,37)(3,61,45,38)(4,62,46,39)(5,63,47,40)(6,64,48,33)(7,57,41,34)(8,58,42,35)(9,78,24,53)(10,79,17,54)(11,80,18,55)(12,73,19,56)(13,74,20,49)(14,75,21,50)(15,76,22,51)(16,77,23,52)(25,65,119,94)(26,66,120,95)(27,67,113,96)(28,68,114,89)(29,69,115,90)(30,70,116,91)(31,71,117,92)(32,72,118,93)(81,99,110,128)(82,100,111,121)(83,101,112,122)(84,102,105,123)(85,103,106,124)(86,104,107,125)(87,97,108,126)(88,98,109,127), (1,93,43,72)(2,92,44,71)(3,91,45,70)(4,90,46,69)(5,89,47,68)(6,96,48,67)(7,95,41,66)(8,94,42,65)(9,110,24,81)(10,109,17,88)(11,108,18,87)(12,107,19,86)(13,106,20,85)(14,105,21,84)(15,112,22,83)(16,111,23,82)(25,58,119,35)(26,57,120,34)(27,64,113,33)(28,63,114,40)(29,62,115,39)(30,61,116,38)(31,60,117,37)(32,59,118,36)(49,124,74,103)(50,123,75,102)(51,122,76,101)(52,121,77,100)(53,128,78,99)(54,127,79,98)(55,126,80,97)(56,125,73,104)>;
G:=Group( (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(25,101)(26,102)(27,103)(28,104)(29,97)(30,98)(31,99)(32,100)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,73)(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)(81,92)(82,93)(83,94)(84,95)(85,96)(86,89)(87,90)(88,91)(113,124)(114,125)(115,126)(116,127)(117,128)(118,121)(119,122)(120,123), (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), (1,59,43,36)(2,60,44,37)(3,61,45,38)(4,62,46,39)(5,63,47,40)(6,64,48,33)(7,57,41,34)(8,58,42,35)(9,78,24,53)(10,79,17,54)(11,80,18,55)(12,73,19,56)(13,74,20,49)(14,75,21,50)(15,76,22,51)(16,77,23,52)(25,65,119,94)(26,66,120,95)(27,67,113,96)(28,68,114,89)(29,69,115,90)(30,70,116,91)(31,71,117,92)(32,72,118,93)(81,99,110,128)(82,100,111,121)(83,101,112,122)(84,102,105,123)(85,103,106,124)(86,104,107,125)(87,97,108,126)(88,98,109,127), (1,93,43,72)(2,92,44,71)(3,91,45,70)(4,90,46,69)(5,89,47,68)(6,96,48,67)(7,95,41,66)(8,94,42,65)(9,110,24,81)(10,109,17,88)(11,108,18,87)(12,107,19,86)(13,106,20,85)(14,105,21,84)(15,112,22,83)(16,111,23,82)(25,58,119,35)(26,57,120,34)(27,64,113,33)(28,63,114,40)(29,62,115,39)(30,61,116,38)(31,60,117,37)(32,59,118,36)(49,124,74,103)(50,123,75,102)(51,122,76,101)(52,121,77,100)(53,128,78,99)(54,127,79,98)(55,126,80,97)(56,125,73,104) );
G=PermutationGroup([(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,44),(10,45),(11,46),(12,47),(13,48),(14,41),(15,42),(16,43),(25,101),(26,102),(27,103),(28,104),(29,97),(30,98),(31,99),(32,100),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,73),(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),(81,92),(82,93),(83,94),(84,95),(85,96),(86,89),(87,90),(88,91),(113,124),(114,125),(115,126),(116,127),(117,128),(118,121),(119,122),(120,123)], [(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)], [(1,59,43,36),(2,60,44,37),(3,61,45,38),(4,62,46,39),(5,63,47,40),(6,64,48,33),(7,57,41,34),(8,58,42,35),(9,78,24,53),(10,79,17,54),(11,80,18,55),(12,73,19,56),(13,74,20,49),(14,75,21,50),(15,76,22,51),(16,77,23,52),(25,65,119,94),(26,66,120,95),(27,67,113,96),(28,68,114,89),(29,69,115,90),(30,70,116,91),(31,71,117,92),(32,72,118,93),(81,99,110,128),(82,100,111,121),(83,101,112,122),(84,102,105,123),(85,103,106,124),(86,104,107,125),(87,97,108,126),(88,98,109,127)], [(1,93,43,72),(2,92,44,71),(3,91,45,70),(4,90,46,69),(5,89,47,68),(6,96,48,67),(7,95,41,66),(8,94,42,65),(9,110,24,81),(10,109,17,88),(11,108,18,87),(12,107,19,86),(13,106,20,85),(14,105,21,84),(15,112,22,83),(16,111,23,82),(25,58,119,35),(26,57,120,34),(27,64,113,33),(28,63,114,40),(29,62,115,39),(30,61,116,38),(31,60,117,37),(32,59,118,36),(49,124,74,103),(50,123,75,102),(51,122,76,101),(52,121,77,100),(53,128,78,99),(54,127,79,98),(55,126,80,97),(56,125,73,104)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 15 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 3 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 15 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 3 | 0 | 0 |
| 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 5 | 12 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,1,0,0,0,15,1,0,0,0,0,0,3,3,0,0,0,14,3],[1,0,0,0,0,0,16,1,0,0,0,15,1,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,9,1,0,0,0,3,8,0,0,0,0,0,5,5,0,0,0,5,12] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | D8 | Q16 |
| kernel | C2×C8⋊2Q8 | C2×C4×C8 | C2×C2.D8 | C8⋊2Q8 | C2×C4⋊Q8 | C42 | C2×C8 | C22×C4 | C2×C4 | C2×C4 |
| # reps | 1 | 1 | 4 | 8 | 2 | 2 | 8 | 2 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_8\rtimes_2Q_8
% in TeX
G:=Group("C2xC8:2Q8"); // GroupNames label
G:=SmallGroup(128,1891);
// by ID
G=gap.SmallGroup(128,1891);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,520,4037,124]);
// 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^-1,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀