direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8.5Q8, C42.362D4, C42.716C23, (C2×C8).47Q8, C8.22(C2×Q8), C4.13(C4⋊Q8), C4.9(C22×Q8), C4⋊C4.97C23, (C4×C8).417C22, (C2×C8).562C23, (C2×C4).356C24, (C22×C4).567D4, C23.883(C2×D4), C22.45(C4⋊Q8), C22.99(C4○D8), C2.D8.177C22, C4.Q8.158C22, (C22×C8).568C22, C22.616(C22×D4), (C2×C42).1131C22, (C22×C4).1565C23, C42.C2.114C22, (C2×C4×C8).47C2, C2.26(C2×C4⋊Q8), C2.32(C2×C4○D8), (C2×C4).696(C2×D4), (C2×C4).245(C2×Q8), (C2×C2.D8).29C2, (C2×C4.Q8).34C2, (C2×C4⋊C4).629C22, (C2×C42.C2).33C2, SmallGroup(128,1890)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 276 in 180 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×20], C23, C42 [×4], C4⋊C4 [×8], C4⋊C4 [×20], C2×C8 [×12], C22×C4, C22×C4 [×2], C22×C4 [×4], C4×C8 [×4], C4.Q8 [×8], C2.D8 [×8], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C42.C2 [×8], C42.C2 [×4], C22×C8 [×2], C2×C4×C8, C2×C4.Q8 [×2], C2×C2.D8 [×2], C8.5Q8 [×8], C2×C42.C2 [×2], C2×C8.5Q8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C4○D8 [×4], C22×D4, C22×Q8 [×2], C8.5Q8 [×4], C2×C4⋊Q8, C2×C4○D8 [×2], C2×C8.5Q8
Generators and relations
G = < a,b,c,d | a2=b8=c4=1, d2=b4c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b4c-1 >
►Regular action on 128 points
Generators in S128
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(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 78)(34 79)(35 80)(36 73)(37 74)(38 75)(39 76)(40 77)(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 94)(82 95)(83 96)(84 89)(85 90)(86 91)(87 92)(88 93)(113 126)(114 127)(115 128)(116 121)(117 122)(118 123)(119 124)(120 125)
(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 55)(10 79 17 56)(11 80 18 49)(12 73 19 50)(13 74 20 51)(14 75 21 52)(15 76 22 53)(16 77 23 54)(25 70 120 89)(26 71 113 90)(27 72 114 91)(28 65 115 92)(29 66 116 93)(30 67 117 94)(31 68 118 95)(32 69 119 96)(81 103 108 122)(82 104 109 123)(83 97 110 124)(84 98 111 125)(85 99 112 126)(86 100 105 127)(87 101 106 128)(88 102 107 121)
(1 93 47 70)(2 96 48 65)(3 91 41 68)(4 94 42 71)(5 89 43 66)(6 92 44 69)(7 95 45 72)(8 90 46 67)(9 106 20 83)(10 109 21 86)(11 112 22 81)(12 107 23 84)(13 110 24 87)(14 105 17 82)(15 108 18 85)(16 111 19 88)(25 63 116 36)(26 58 117 39)(27 61 118 34)(28 64 119 37)(29 59 120 40)(30 62 113 35)(31 57 114 38)(32 60 115 33)(49 122 76 99)(50 125 77 102)(51 128 78 97)(52 123 79 100)(53 126 80 103)(54 121 73 98)(55 124 74 101)(56 127 75 104)
G:=sub<Sym(128)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(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)(33,78)(34,79)(35,80)(36,73)(37,74)(38,75)(39,76)(40,77)(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)(81,94)(82,95)(83,96)(84,89)(85,90)(86,91)(87,92)(88,93)(113,126)(114,127)(115,128)(116,121)(117,122)(118,123)(119,124)(120,125), (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)(9,78,24,55)(10,79,17,56)(11,80,18,49)(12,73,19,50)(13,74,20,51)(14,75,21,52)(15,76,22,53)(16,77,23,54)(25,70,120,89)(26,71,113,90)(27,72,114,91)(28,65,115,92)(29,66,116,93)(30,67,117,94)(31,68,118,95)(32,69,119,96)(81,103,108,122)(82,104,109,123)(83,97,110,124)(84,98,111,125)(85,99,112,126)(86,100,105,127)(87,101,106,128)(88,102,107,121), (1,93,47,70)(2,96,48,65)(3,91,41,68)(4,94,42,71)(5,89,43,66)(6,92,44,69)(7,95,45,72)(8,90,46,67)(9,106,20,83)(10,109,21,86)(11,112,22,81)(12,107,23,84)(13,110,24,87)(14,105,17,82)(15,108,18,85)(16,111,19,88)(25,63,116,36)(26,58,117,39)(27,61,118,34)(28,64,119,37)(29,59,120,40)(30,62,113,35)(31,57,114,38)(32,60,115,33)(49,122,76,99)(50,125,77,102)(51,128,78,97)(52,123,79,100)(53,126,80,103)(54,121,73,98)(55,124,74,101)(56,127,75,104)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(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)(33,78)(34,79)(35,80)(36,73)(37,74)(38,75)(39,76)(40,77)(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)(81,94)(82,95)(83,96)(84,89)(85,90)(86,91)(87,92)(88,93)(113,126)(114,127)(115,128)(116,121)(117,122)(118,123)(119,124)(120,125), (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)(9,78,24,55)(10,79,17,56)(11,80,18,49)(12,73,19,50)(13,74,20,51)(14,75,21,52)(15,76,22,53)(16,77,23,54)(25,70,120,89)(26,71,113,90)(27,72,114,91)(28,65,115,92)(29,66,116,93)(30,67,117,94)(31,68,118,95)(32,69,119,96)(81,103,108,122)(82,104,109,123)(83,97,110,124)(84,98,111,125)(85,99,112,126)(86,100,105,127)(87,101,106,128)(88,102,107,121), (1,93,47,70)(2,96,48,65)(3,91,41,68)(4,94,42,71)(5,89,43,66)(6,92,44,69)(7,95,45,72)(8,90,46,67)(9,106,20,83)(10,109,21,86)(11,112,22,81)(12,107,23,84)(13,110,24,87)(14,105,17,82)(15,108,18,85)(16,111,19,88)(25,63,116,36)(26,58,117,39)(27,61,118,34)(28,64,119,37)(29,59,120,40)(30,62,113,35)(31,57,114,38)(32,60,115,33)(49,122,76,99)(50,125,77,102)(51,128,78,97)(52,123,79,100)(53,126,80,103)(54,121,73,98)(55,124,74,101)(56,127,75,104) );
G=PermutationGroup([(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(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),(33,78),(34,79),(35,80),(36,73),(37,74),(38,75),(39,76),(40,77),(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),(81,94),(82,95),(83,96),(84,89),(85,90),(86,91),(87,92),(88,93),(113,126),(114,127),(115,128),(116,121),(117,122),(118,123),(119,124),(120,125)], [(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),(9,78,24,55),(10,79,17,56),(11,80,18,49),(12,73,19,50),(13,74,20,51),(14,75,21,52),(15,76,22,53),(16,77,23,54),(25,70,120,89),(26,71,113,90),(27,72,114,91),(28,65,115,92),(29,66,116,93),(30,67,117,94),(31,68,118,95),(32,69,119,96),(81,103,108,122),(82,104,109,123),(83,97,110,124),(84,98,111,125),(85,99,112,126),(86,100,105,127),(87,101,106,128),(88,102,107,121)], [(1,93,47,70),(2,96,48,65),(3,91,41,68),(4,94,42,71),(5,89,43,66),(6,92,44,69),(7,95,45,72),(8,90,46,67),(9,106,20,83),(10,109,21,86),(11,112,22,81),(12,107,23,84),(13,110,24,87),(14,105,17,82),(15,108,18,85),(16,111,19,88),(25,63,116,36),(26,58,117,39),(27,61,118,34),(28,64,119,37),(29,59,120,40),(30,62,113,35),(31,57,114,38),(32,60,115,33),(49,122,76,99),(50,125,77,102),(51,128,78,97),(52,123,79,100),(53,126,80,103),(54,121,73,98),(55,124,74,101),(56,127,75,104)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 5 | 5 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,5,5,0,0,0,12,5],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,13],[16,0,0,0,0,0,12,12,0,0,0,12,5,0,0,0,0,0,1,0,0,0,0,0,16] >;
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 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | C4○D8 |
| kernel | C2×C8.5Q8 | C2×C4×C8 | C2×C4.Q8 | C2×C2.D8 | C8.5Q8 | C2×C42.C2 | C42 | C2×C8 | C22×C4 | C22 |
| # reps | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 8 | 2 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_8._5Q_8
% in TeX
G:=Group("C2xC8.5Q8"); // GroupNames label
G:=SmallGroup(128,1890);
// by ID
G=gap.SmallGroup(128,1890);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,568,758,184,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^4*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^4*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀