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, C4.13(C2×M4(2)), (C2×C4).52M4(2), (C22×Q8).30C4, C22.44(C8○D4), (C4×Q8).276C22, C8⋊C4.157C22, C22.182(C23×C4), (C2×C42).770C22, (C22×C8).443C22, C23.294(C22×C4), C22.64(C2×M4(2)), C2.13(C22×M4(2)), (C22×C4).1651C23, (C2×C4×C8).68C2, C2.23(C2×C4×Q8), (C2×C4⋊C8).57C2, (C2×C4⋊C4).73C4, (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), (C22×C4).418(C2×C4), (C2×C4).270(C22×C4), SmallGroup(128,1691)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 244 in 200 conjugacy classes, 156 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×6], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×20], C2×C4 [×10], Q8 [×8], C23, C42 [×12], C4⋊C4 [×12], C2×C8 [×12], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C4×C8 [×4], C8⋊C4 [×8], C4⋊C8 [×12], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×Q8 [×8], C22×C8 [×2], C22×C8 [×2], C22×Q8, C2×C4×C8, C2×C8⋊C4 [×2], C2×C4⋊C8, C2×C4⋊C8 [×2], C8⋊4Q8 [×8], C2×C4×Q8, C2×C8⋊4Q8
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], M4(2) [×4], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, C4×Q8 [×4], C2×M4(2) [×6], C8○D4 [×2], C23×C4, C22×Q8, C2×C4○D4, C8⋊4Q8 [×4], C2×C4×Q8, C22×M4(2), C2×C8○D4, C2×C8⋊4Q8
Generators and relations
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 >
►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 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(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 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)(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 38 45 63)(2 39 46 64)(3 40 47 57)(4 33 48 58)(5 34 41 59)(6 35 42 60)(7 36 43 61)(8 37 44 62)(9 73 18 51)(10 74 19 52)(11 75 20 53)(12 76 21 54)(13 77 22 55)(14 78 23 56)(15 79 24 49)(16 80 17 50)(25 67 120 89)(26 68 113 90)(27 69 114 91)(28 70 115 92)(29 71 116 93)(30 72 117 94)(31 65 118 95)(32 66 119 96)(81 100 105 122)(82 101 106 123)(83 102 107 124)(84 103 108 125)(85 104 109 126)(86 97 110 127)(87 98 111 128)(88 99 112 121)
(1 72 45 94)(2 69 46 91)(3 66 47 96)(4 71 48 93)(5 68 41 90)(6 65 42 95)(7 70 43 92)(8 67 44 89)(9 108 18 84)(10 105 19 81)(11 110 20 86)(12 107 21 83)(13 112 22 88)(14 109 23 85)(15 106 24 82)(16 111 17 87)(25 62 120 37)(26 59 113 34)(27 64 114 39)(28 61 115 36)(29 58 116 33)(30 63 117 38)(31 60 118 35)(32 57 119 40)(49 123 79 101)(50 128 80 98)(51 125 73 103)(52 122 74 100)(53 127 75 97)(54 124 76 102)(55 121 77 99)(56 126 78 104)
G:=sub<Sym(128)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(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)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59)(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), (1,38,45,63)(2,39,46,64)(3,40,47,57)(4,33,48,58)(5,34,41,59)(6,35,42,60)(7,36,43,61)(8,37,44,62)(9,73,18,51)(10,74,19,52)(11,75,20,53)(12,76,21,54)(13,77,22,55)(14,78,23,56)(15,79,24,49)(16,80,17,50)(25,67,120,89)(26,68,113,90)(27,69,114,91)(28,70,115,92)(29,71,116,93)(30,72,117,94)(31,65,118,95)(32,66,119,96)(81,100,105,122)(82,101,106,123)(83,102,107,124)(84,103,108,125)(85,104,109,126)(86,97,110,127)(87,98,111,128)(88,99,112,121), (1,72,45,94)(2,69,46,91)(3,66,47,96)(4,71,48,93)(5,68,41,90)(6,65,42,95)(7,70,43,92)(8,67,44,89)(9,108,18,84)(10,105,19,81)(11,110,20,86)(12,107,21,83)(13,112,22,88)(14,109,23,85)(15,106,24,82)(16,111,17,87)(25,62,120,37)(26,59,113,34)(27,64,114,39)(28,61,115,36)(29,58,116,33)(30,63,117,38)(31,60,118,35)(32,57,119,40)(49,123,79,101)(50,128,80,98)(51,125,73,103)(52,122,74,100)(53,127,75,97)(54,124,76,102)(55,121,77,99)(56,126,78,104)>;
G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(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)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59)(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), (1,38,45,63)(2,39,46,64)(3,40,47,57)(4,33,48,58)(5,34,41,59)(6,35,42,60)(7,36,43,61)(8,37,44,62)(9,73,18,51)(10,74,19,52)(11,75,20,53)(12,76,21,54)(13,77,22,55)(14,78,23,56)(15,79,24,49)(16,80,17,50)(25,67,120,89)(26,68,113,90)(27,69,114,91)(28,70,115,92)(29,71,116,93)(30,72,117,94)(31,65,118,95)(32,66,119,96)(81,100,105,122)(82,101,106,123)(83,102,107,124)(84,103,108,125)(85,104,109,126)(86,97,110,127)(87,98,111,128)(88,99,112,121), (1,72,45,94)(2,69,46,91)(3,66,47,96)(4,71,48,93)(5,68,41,90)(6,65,42,95)(7,70,43,92)(8,67,44,89)(9,108,18,84)(10,105,19,81)(11,110,20,86)(12,107,21,83)(13,112,22,88)(14,109,23,85)(15,106,24,82)(16,111,17,87)(25,62,120,37)(26,59,113,34)(27,64,114,39)(28,61,115,36)(29,58,116,33)(30,63,117,38)(31,60,118,35)(32,57,119,40)(49,123,79,101)(50,128,80,98)(51,125,73,103)(52,122,74,100)(53,127,75,97)(54,124,76,102)(55,121,77,99)(56,126,78,104) );
G=PermutationGroup([(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,41),(24,42),(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),(49,60),(50,61),(51,62),(52,63),(53,64),(54,57),(55,58),(56,59),(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)], [(1,38,45,63),(2,39,46,64),(3,40,47,57),(4,33,48,58),(5,34,41,59),(6,35,42,60),(7,36,43,61),(8,37,44,62),(9,73,18,51),(10,74,19,52),(11,75,20,53),(12,76,21,54),(13,77,22,55),(14,78,23,56),(15,79,24,49),(16,80,17,50),(25,67,120,89),(26,68,113,90),(27,69,114,91),(28,70,115,92),(29,71,116,93),(30,72,117,94),(31,65,118,95),(32,66,119,96),(81,100,105,122),(82,101,106,123),(83,102,107,124),(84,103,108,125),(85,104,109,126),(86,97,110,127),(87,98,111,128),(88,99,112,121)], [(1,72,45,94),(2,69,46,91),(3,66,47,96),(4,71,48,93),(5,68,41,90),(6,65,42,95),(7,70,43,92),(8,67,44,89),(9,108,18,84),(10,105,19,81),(11,110,20,86),(12,107,21,83),(13,112,22,88),(14,109,23,85),(15,106,24,82),(16,111,17,87),(25,62,120,37),(26,59,113,34),(27,64,114,39),(28,61,115,36),(29,58,116,33),(30,63,117,38),(31,60,118,35),(32,57,119,40),(49,123,79,101),(50,128,80,98),(51,125,73,103),(52,122,74,100),(53,127,75,97),(54,124,76,102),(55,121,77,99),(56,126,78,104)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀