direct product, p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C2×C8⋊Q8, C42.251D4, C42.379C23, C8⋊4(C2×Q8), (C2×C8)⋊2Q8, C4.15(C4⋊Q8), C4.12(C22×Q8), C4⋊C4.100C23, (C2×C8).270C23, (C2×C4).359C24, (C22×C4).471D4, C23.885(C2×D4), C4⋊Q8.285C22, C22.47(C4⋊Q8), C2.D8.215C22, C8⋊C4.120C22, C4.Q8.131C22, (C22×C8).274C22, (C2×C42).858C22, C22.619(C22×D4), C22.126(C8⋊C22), (C22×C4).1568C23, C42.C2.116C22, C22.115(C8.C22), C2.29(C2×C4⋊Q8), (C2×C4⋊Q8).51C2, (C2×C4).520(C2×D4), C2.43(C2×C8⋊C22), (C2×C4).248(C2×Q8), (C2×C4.Q8).11C2, (C2×C2.D8).38C2, (C2×C8⋊C4).11C2, C2.43(C2×C8.C22), (C2×C4⋊C4).631C22, (C2×C42.C2).34C2, SmallGroup(128,1893)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 324 in 192 conjugacy classes, 116 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×12], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×20], Q8 [×8], C23, C42 [×4], C4⋊C4 [×8], C4⋊C4 [×16], C2×C8 [×12], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×8], C8⋊C4 [×4], C4.Q8 [×8], C2.D8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×3], C42.C2 [×4], C42.C2 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8, C2×C8⋊C4, C2×C4.Q8 [×2], C2×C2.D8 [×2], C8⋊Q8 [×8], C2×C42.C2, C2×C4⋊Q8, C2×C8⋊Q8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C8⋊C22 [×2], C8.C22 [×2], C22×D4, C22×Q8 [×2], C8⋊Q8 [×4], C2×C4⋊Q8, C2×C8⋊C22, C2×C8.C22, C2×C8⋊Q8
Generators and relations
G = < a,b,c,d | a2=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b3, dcd-1=c-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 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 73)(40 74)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 105)(72 106)(81 96)(82 89)(83 90)(84 91)(85 92)(86 93)(87 94)(88 95)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 121)
(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 46 35)(2 64 47 40)(3 61 48 37)(4 58 41 34)(5 63 42 39)(6 60 43 36)(7 57 44 33)(8 62 45 38)(9 76 22 53)(10 73 23 50)(11 78 24 55)(12 75 17 52)(13 80 18 49)(14 77 19 54)(15 74 20 51)(16 79 21 56)(25 72 120 89)(26 69 113 94)(27 66 114 91)(28 71 115 96)(29 68 116 93)(30 65 117 90)(31 70 118 95)(32 67 119 92)(81 100 105 124)(82 97 106 121)(83 102 107 126)(84 99 108 123)(85 104 109 128)(86 101 110 125)(87 98 111 122)(88 103 112 127)
(1 93 46 68)(2 96 47 71)(3 91 48 66)(4 94 41 69)(5 89 42 72)(6 92 43 67)(7 95 44 70)(8 90 45 65)(9 111 22 87)(10 106 23 82)(11 109 24 85)(12 112 17 88)(13 107 18 83)(14 110 19 86)(15 105 20 81)(16 108 21 84)(25 63 120 39)(26 58 113 34)(27 61 114 37)(28 64 115 40)(29 59 116 35)(30 62 117 38)(31 57 118 33)(32 60 119 36)(49 126 80 102)(50 121 73 97)(51 124 74 100)(52 127 75 103)(53 122 76 98)(54 125 77 101)(55 128 78 104)(56 123 79 99)
G:=sub<Sym(128)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,105)(72,106)(81,96)(82,89)(83,90)(84,91)(85,92)(86,93)(87,94)(88,95)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,121), (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), (1,59,46,35)(2,64,47,40)(3,61,48,37)(4,58,41,34)(5,63,42,39)(6,60,43,36)(7,57,44,33)(8,62,45,38)(9,76,22,53)(10,73,23,50)(11,78,24,55)(12,75,17,52)(13,80,18,49)(14,77,19,54)(15,74,20,51)(16,79,21,56)(25,72,120,89)(26,69,113,94)(27,66,114,91)(28,71,115,96)(29,68,116,93)(30,65,117,90)(31,70,118,95)(32,67,119,92)(81,100,105,124)(82,97,106,121)(83,102,107,126)(84,99,108,123)(85,104,109,128)(86,101,110,125)(87,98,111,122)(88,103,112,127), (1,93,46,68)(2,96,47,71)(3,91,48,66)(4,94,41,69)(5,89,42,72)(6,92,43,67)(7,95,44,70)(8,90,45,65)(9,111,22,87)(10,106,23,82)(11,109,24,85)(12,112,17,88)(13,107,18,83)(14,110,19,86)(15,105,20,81)(16,108,21,84)(25,63,120,39)(26,58,113,34)(27,61,114,37)(28,64,115,40)(29,59,116,35)(30,62,117,38)(31,57,118,33)(32,60,119,36)(49,126,80,102)(50,121,73,97)(51,124,74,100)(52,127,75,103)(53,122,76,98)(54,125,77,101)(55,128,78,104)(56,123,79,99)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,105)(72,106)(81,96)(82,89)(83,90)(84,91)(85,92)(86,93)(87,94)(88,95)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,121), (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), (1,59,46,35)(2,64,47,40)(3,61,48,37)(4,58,41,34)(5,63,42,39)(6,60,43,36)(7,57,44,33)(8,62,45,38)(9,76,22,53)(10,73,23,50)(11,78,24,55)(12,75,17,52)(13,80,18,49)(14,77,19,54)(15,74,20,51)(16,79,21,56)(25,72,120,89)(26,69,113,94)(27,66,114,91)(28,71,115,96)(29,68,116,93)(30,65,117,90)(31,70,118,95)(32,67,119,92)(81,100,105,124)(82,97,106,121)(83,102,107,126)(84,99,108,123)(85,104,109,128)(86,101,110,125)(87,98,111,122)(88,103,112,127), (1,93,46,68)(2,96,47,71)(3,91,48,66)(4,94,41,69)(5,89,42,72)(6,92,43,67)(7,95,44,70)(8,90,45,65)(9,111,22,87)(10,106,23,82)(11,109,24,85)(12,112,17,88)(13,107,18,83)(14,110,19,86)(15,105,20,81)(16,108,21,84)(25,63,120,39)(26,58,113,34)(27,61,114,37)(28,64,115,40)(29,59,116,35)(30,62,117,38)(31,57,118,33)(32,60,119,36)(49,126,80,102)(50,121,73,97)(51,124,74,100)(52,127,75,103)(53,122,76,98)(54,125,77,101)(55,128,78,104)(56,123,79,99) );
G=PermutationGroup([(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,73),(40,74),(49,62),(50,63),(51,64),(52,57),(53,58),(54,59),(55,60),(56,61),(65,107),(66,108),(67,109),(68,110),(69,111),(70,112),(71,105),(72,106),(81,96),(82,89),(83,90),(84,91),(85,92),(86,93),(87,94),(88,95),(113,122),(114,123),(115,124),(116,125),(117,126),(118,127),(119,128),(120,121)], [(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)], [(1,59,46,35),(2,64,47,40),(3,61,48,37),(4,58,41,34),(5,63,42,39),(6,60,43,36),(7,57,44,33),(8,62,45,38),(9,76,22,53),(10,73,23,50),(11,78,24,55),(12,75,17,52),(13,80,18,49),(14,77,19,54),(15,74,20,51),(16,79,21,56),(25,72,120,89),(26,69,113,94),(27,66,114,91),(28,71,115,96),(29,68,116,93),(30,65,117,90),(31,70,118,95),(32,67,119,92),(81,100,105,124),(82,97,106,121),(83,102,107,126),(84,99,108,123),(85,104,109,128),(86,101,110,125),(87,98,111,122),(88,103,112,127)], [(1,93,46,68),(2,96,47,71),(3,91,48,66),(4,94,41,69),(5,89,42,72),(6,92,43,67),(7,95,44,70),(8,90,45,65),(9,111,22,87),(10,106,23,82),(11,109,24,85),(12,112,17,88),(13,107,18,83),(14,110,19,86),(15,105,20,81),(16,108,21,84),(25,63,120,39),(26,58,113,34),(27,61,114,37),(28,64,115,40),(29,59,116,35),(30,62,117,38),(31,57,118,33),(32,60,119,36),(49,126,80,102),(50,121,73,97),(51,124,74,100),(52,127,75,103),(53,122,76,98),(54,125,77,101),(55,128,78,104),(56,123,79,99)])
Matrix representation ►G ⊆ GL8(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 15 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 6 | 9 | 4 |
| 0 | 0 | 0 | 0 | 11 | 4 | 13 | 9 |
| 0 | 0 | 0 | 0 | 4 | 8 | 13 | 11 |
| 0 | 0 | 0 | 0 | 9 | 4 | 6 | 13 |
| 12 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 | 2 | 11 |
| 0 | 0 | 0 | 0 | 7 | 14 | 11 | 15 |
| 0 | 0 | 0 | 0 | 15 | 6 | 10 | 3 |
| 0 | 0 | 0 | 0 | 6 | 2 | 3 | 7 |
G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[15,6,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,4,11,4,9,0,0,0,0,6,4,8,4,0,0,0,0,9,13,13,6,0,0,0,0,4,9,11,13],[12,13,0,0,0,0,0,0,15,5,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,3,7,15,6,0,0,0,0,7,14,6,2,0,0,0,0,2,11,10,3,0,0,0,0,11,15,3,7] >;
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | C8⋊C22 | C8.C22 |
| kernel | C2×C8⋊Q8 | C2×C8⋊C4 | C2×C4.Q8 | C2×C2.D8 | C8⋊Q8 | C2×C42.C2 | C2×C4⋊Q8 | C42 | C2×C8 | C22×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 8 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2\times C_8\rtimes Q_8
% in TeX
G:=Group("C2xC8:Q8"); // GroupNames label
G:=SmallGroup(128,1893);
// by ID
G=gap.SmallGroup(128,1893);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,723,184,4037,124]);
// 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,c*b*c^-1=b^5,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀