p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊C4⋊1C4, (C2×C8).331D4, C4.122(C4×D4), (C2×C4).29Q16, C2.10(C4×Q16), C2.4(C8⋊8D4), (C2×C4).61SD16, C2.16(C4×SD16), C23.784(C2×D4), C22.167(C4×D4), (C22×C4).553D4, C22.36(C2×Q16), C4.1(C42⋊2C2), C4.7(C42⋊C2), C2.3(C8.18D4), C22.69(C4○D8), C2.2(C4.SD16), C22.66(C2×SD16), C22.4Q16.13C2, (C22×C8).483C22, (C22×Q8).30C22, C22.130(C4⋊D4), (C22×C4).1384C23, (C2×C42).1064C22, C22.58(C4.4D4), C4.92(C22.D4), C23.65C23.8C2, C23.67C23.8C2, C2.18(C24.C22), C2.2(C42.78C22), (C2×C4×C8).21C2, C4⋊C4.81(C2×C4), (C2×C8).159(C2×C4), (C2×Q8).84(C2×C4), (C2×C4).1341(C2×D4), (C2×C4⋊C4).70C22, (C2×Q8⋊C4).6C2, (C2×C4).579(C4○D4), (C2×C4).402(C22×C4), SmallGroup(128,665)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2.(C8⋊8D4)
G = < a,b,c,d | a2=b8=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=ab3, dcd-1=c-1 >
Subgroups: 244 in 128 conjugacy classes, 60 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C4×C8, Q8⋊C4, Q8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C22.4Q16, C23.65C23, C23.67C23, C2×C4×C8, C2×Q8⋊C4, C2.(C8⋊8D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, SD16, Q16, C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C2×SD16, C2×Q16, C4○D8, C24.C22, C4×SD16, C4×Q16, C8⋊8D4, C8.18D4, C4.SD16, C42.78C22, C2.(C8⋊8D4)
►Regular action on 128 points
Generators in S128
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 127)(10 128)(11 121)(12 122)(13 123)(14 124)(15 125)(16 126)(17 55)(18 56)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)(57 104)(58 97)(59 98)(60 99)(61 100)(62 101)(63 102)(64 103)(65 76)(66 77)(67 78)(68 79)(69 80)(70 73)(71 74)(72 75)(81 106)(82 107)(83 108)(84 109)(85 110)(86 111)(87 112)(88 105)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 113)(96 114)
(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 39 63 120)(2 43 64 89)(3 37 57 118)(4 41 58 95)(5 35 59 116)(6 47 60 93)(7 33 61 114)(8 45 62 91)(9 70 87 21)(10 76 88 54)(11 68 81 19)(12 74 82 52)(13 66 83 17)(14 80 84 50)(15 72 85 23)(16 78 86 56)(18 126 67 111)(20 124 69 109)(22 122 71 107)(24 128 65 105)(25 42 100 96)(26 36 101 117)(27 48 102 94)(28 34 103 115)(29 46 104 92)(30 40 97 113)(31 44 98 90)(32 38 99 119)(49 121 79 106)(51 127 73 112)(53 125 75 110)(55 123 77 108)
(1 87 27 112)(2 107 28 82)(3 85 29 110)(4 105 30 88)(5 83 31 108)(6 111 32 86)(7 81 25 106)(8 109 26 84)(9 102 127 63)(10 58 128 97)(11 100 121 61)(12 64 122 103)(13 98 123 59)(14 62 124 101)(15 104 125 57)(16 60 126 99)(17 90 55 116)(18 119 56 93)(19 96 49 114)(20 117 50 91)(21 94 51 120)(22 115 52 89)(23 92 53 118)(24 113 54 95)(33 68 42 79)(34 74 43 71)(35 66 44 77)(36 80 45 69)(37 72 46 75)(38 78 47 67)(39 70 48 73)(40 76 41 65)
G:=sub<Sym(128)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,127)(10,128)(11,121)(12,122)(13,123)(14,124)(15,125)(16,126)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41)(57,104)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,76)(66,77)(67,78)(68,79)(69,80)(70,73)(71,74)(72,75)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(87,112)(88,105)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,113)(96,114), (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), (1,39,63,120)(2,43,64,89)(3,37,57,118)(4,41,58,95)(5,35,59,116)(6,47,60,93)(7,33,61,114)(8,45,62,91)(9,70,87,21)(10,76,88,54)(11,68,81,19)(12,74,82,52)(13,66,83,17)(14,80,84,50)(15,72,85,23)(16,78,86,56)(18,126,67,111)(20,124,69,109)(22,122,71,107)(24,128,65,105)(25,42,100,96)(26,36,101,117)(27,48,102,94)(28,34,103,115)(29,46,104,92)(30,40,97,113)(31,44,98,90)(32,38,99,119)(49,121,79,106)(51,127,73,112)(53,125,75,110)(55,123,77,108), (1,87,27,112)(2,107,28,82)(3,85,29,110)(4,105,30,88)(5,83,31,108)(6,111,32,86)(7,81,25,106)(8,109,26,84)(9,102,127,63)(10,58,128,97)(11,100,121,61)(12,64,122,103)(13,98,123,59)(14,62,124,101)(15,104,125,57)(16,60,126,99)(17,90,55,116)(18,119,56,93)(19,96,49,114)(20,117,50,91)(21,94,51,120)(22,115,52,89)(23,92,53,118)(24,113,54,95)(33,68,42,79)(34,74,43,71)(35,66,44,77)(36,80,45,69)(37,72,46,75)(38,78,47,67)(39,70,48,73)(40,76,41,65)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,127)(10,128)(11,121)(12,122)(13,123)(14,124)(15,125)(16,126)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41)(57,104)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,76)(66,77)(67,78)(68,79)(69,80)(70,73)(71,74)(72,75)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(87,112)(88,105)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,113)(96,114), (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), (1,39,63,120)(2,43,64,89)(3,37,57,118)(4,41,58,95)(5,35,59,116)(6,47,60,93)(7,33,61,114)(8,45,62,91)(9,70,87,21)(10,76,88,54)(11,68,81,19)(12,74,82,52)(13,66,83,17)(14,80,84,50)(15,72,85,23)(16,78,86,56)(18,126,67,111)(20,124,69,109)(22,122,71,107)(24,128,65,105)(25,42,100,96)(26,36,101,117)(27,48,102,94)(28,34,103,115)(29,46,104,92)(30,40,97,113)(31,44,98,90)(32,38,99,119)(49,121,79,106)(51,127,73,112)(53,125,75,110)(55,123,77,108), (1,87,27,112)(2,107,28,82)(3,85,29,110)(4,105,30,88)(5,83,31,108)(6,111,32,86)(7,81,25,106)(8,109,26,84)(9,102,127,63)(10,58,128,97)(11,100,121,61)(12,64,122,103)(13,98,123,59)(14,62,124,101)(15,104,125,57)(16,60,126,99)(17,90,55,116)(18,119,56,93)(19,96,49,114)(20,117,50,91)(21,94,51,120)(22,115,52,89)(23,92,53,118)(24,113,54,95)(33,68,42,79)(34,74,43,71)(35,66,44,77)(36,80,45,69)(37,72,46,75)(38,78,47,67)(39,70,48,73)(40,76,41,65) );
G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,127),(10,128),(11,121),(12,122),(13,123),(14,124),(15,125),(16,126),(17,55),(18,56),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,41),(57,104),(58,97),(59,98),(60,99),(61,100),(62,101),(63,102),(64,103),(65,76),(66,77),(67,78),(68,79),(69,80),(70,73),(71,74),(72,75),(81,106),(82,107),(83,108),(84,109),(85,110),(86,111),(87,112),(88,105),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,113),(96,114)], [(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)], [(1,39,63,120),(2,43,64,89),(3,37,57,118),(4,41,58,95),(5,35,59,116),(6,47,60,93),(7,33,61,114),(8,45,62,91),(9,70,87,21),(10,76,88,54),(11,68,81,19),(12,74,82,52),(13,66,83,17),(14,80,84,50),(15,72,85,23),(16,78,86,56),(18,126,67,111),(20,124,69,109),(22,122,71,107),(24,128,65,105),(25,42,100,96),(26,36,101,117),(27,48,102,94),(28,34,103,115),(29,46,104,92),(30,40,97,113),(31,44,98,90),(32,38,99,119),(49,121,79,106),(51,127,73,112),(53,125,75,110),(55,123,77,108)], [(1,87,27,112),(2,107,28,82),(3,85,29,110),(4,105,30,88),(5,83,31,108),(6,111,32,86),(7,81,25,106),(8,109,26,84),(9,102,127,63),(10,58,128,97),(11,100,121,61),(12,64,122,103),(13,98,123,59),(14,62,124,101),(15,104,125,57),(16,60,126,99),(17,90,55,116),(18,119,56,93),(19,96,49,114),(20,117,50,91),(21,94,51,120),(22,115,52,89),(23,92,53,118),(24,113,54,95),(33,68,42,79),(34,74,43,71),(35,66,44,77),(36,80,45,69),(37,72,46,75),(38,78,47,67),(39,70,48,73),(40,76,41,65)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | SD16 | Q16 | C4○D4 | C4○D8 |
| kernel | C2.(C8⋊8D4) | C22.4Q16 | C23.65C23 | C23.67C23 | C2×C4×C8 | C2×Q8⋊C4 | Q8⋊C4 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C2.(C8⋊8D4) ►in 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 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 5 | 12 | 0 | 0 |
| 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 0 | 3 | 14 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 13 | 6 | 0 | 0 |
| 0 | 6 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 4 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 13 | 6 | 0 | 0 |
| 0 | 6 | 4 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 4 |
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],[13,0,0,0,0,0,5,5,0,0,0,12,5,0,0,0,0,0,14,3,0,0,0,14,14],[16,0,0,0,0,0,13,6,0,0,0,6,4,0,0,0,0,0,0,4,0,0,0,4,0],[4,0,0,0,0,0,13,6,0,0,0,6,4,0,0,0,0,0,13,0,0,0,0,0,4] >;
C2.(C8⋊8D4) in GAP, Magma, Sage, TeX
C_2.(C_8\rtimes_8D_4)
% in TeX
G:=Group("C2.(C8:8D4)"); // GroupNames label
G:=SmallGroup(128,665);
// by ID
G=gap.SmallGroup(128,665);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,58,2019,248,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a*b^3,d*c*d^-1=c^-1>;
// generators/relations