p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8).23Q8, C4⋊C4.111D4, C2.10(C8⋊Q8), C2.5(C8⋊3Q8), (C2×C4).41SD16, (C22×C4).320D4, C23.934(C2×D4), C22.52(C4⋊Q8), C4.12(C22⋊Q8), C42⋊9C4.17C2, C2.17(C4⋊SD16), C4.20(C42.C2), C2.17(D4.D4), C22.4Q16.44C2, (C2×C42).385C22, (C22×C8).327C22, C22.106(C2×SD16), C22.257(C4⋊D4), C22.154(C8⋊C22), (C22×C4).1468C23, C2.9(C23.46D4), C2.9(C23.47D4), C22.143(C8.C22), C22.7C42.36C2, C23.65C23.21C2, C2.6(C23.81C23), C22.124(C22.D4), (C2×C4).221(C2×Q8), (C2×C4.Q8).25C2, (C2×C4).1064(C2×D4), (C2×C4).887(C4○D4), (C2×C4⋊C4).155C22, SmallGroup(128,816)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2.(C8⋊3Q8)
G = < a,b,c,d | a2=b8=c4=1, d2=c2, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=ab3, dcd-1=ac-1 >
Subgroups: 232 in 115 conjugacy classes, 54 normal (28 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4.Q8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.7C42, C22.4Q16, C42⋊9C4, C23.65C23, C2×C4.Q8, C2.(C8⋊3Q8)
Quotients: C1, C2, C22, D4, Q8, C23, SD16, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C2×SD16, C8⋊C22, C8.C22, C23.81C23, C4⋊SD16, D4.D4, C23.46D4, C23.47D4, C8⋊3Q8, C8⋊Q8, C2.(C8⋊3Q8)
►Regular action on 128 points
Generators in S128
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 39)(10 40)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 89)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(57 97)(58 98)(59 99)(60 100)(61 101)(62 102)(63 103)(64 104)(65 76)(66 77)(67 78)(68 79)(69 80)(70 73)(71 74)(72 75)(81 127)(82 128)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(105 116)(106 117)(107 118)(108 119)(109 120)(110 113)(111 114)(112 115)
(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 98 90 121)(2 59 91 84)(3 100 92 123)(4 61 93 86)(5 102 94 125)(6 63 95 88)(7 104 96 127)(8 57 89 82)(9 105 28 77)(10 117 29 67)(11 107 30 79)(12 119 31 69)(13 109 32 73)(14 113 25 71)(15 111 26 75)(16 115 27 65)(17 83 49 58)(18 122 50 99)(19 85 51 60)(20 124 52 101)(21 87 53 62)(22 126 54 103)(23 81 55 64)(24 128 56 97)(33 118 46 68)(34 108 47 80)(35 120 48 70)(36 110 41 74)(37 114 42 72)(38 112 43 76)(39 116 44 66)(40 106 45 78)
(1 75 90 111)(2 67 91 117)(3 73 92 109)(4 65 93 115)(5 79 94 107)(6 71 95 113)(7 77 96 105)(8 69 89 119)(9 64 28 81)(10 99 29 122)(11 62 30 87)(12 97 31 128)(13 60 32 85)(14 103 25 126)(15 58 26 83)(16 101 27 124)(17 114 49 72)(18 106 50 78)(19 120 51 70)(20 112 52 76)(21 118 53 68)(22 110 54 74)(23 116 55 66)(24 108 56 80)(33 102 46 125)(34 57 47 82)(35 100 48 123)(36 63 41 88)(37 98 42 121)(38 61 43 86)(39 104 44 127)(40 59 45 84)
G:=sub<Sym(128)| (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,89)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(65,76)(66,77)(67,78)(68,79)(69,80)(70,73)(71,74)(72,75)(81,127)(82,128)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(105,116)(106,117)(107,118)(108,119)(109,120)(110,113)(111,114)(112,115), (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), (1,98,90,121)(2,59,91,84)(3,100,92,123)(4,61,93,86)(5,102,94,125)(6,63,95,88)(7,104,96,127)(8,57,89,82)(9,105,28,77)(10,117,29,67)(11,107,30,79)(12,119,31,69)(13,109,32,73)(14,113,25,71)(15,111,26,75)(16,115,27,65)(17,83,49,58)(18,122,50,99)(19,85,51,60)(20,124,52,101)(21,87,53,62)(22,126,54,103)(23,81,55,64)(24,128,56,97)(33,118,46,68)(34,108,47,80)(35,120,48,70)(36,110,41,74)(37,114,42,72)(38,112,43,76)(39,116,44,66)(40,106,45,78), (1,75,90,111)(2,67,91,117)(3,73,92,109)(4,65,93,115)(5,79,94,107)(6,71,95,113)(7,77,96,105)(8,69,89,119)(9,64,28,81)(10,99,29,122)(11,62,30,87)(12,97,31,128)(13,60,32,85)(14,103,25,126)(15,58,26,83)(16,101,27,124)(17,114,49,72)(18,106,50,78)(19,120,51,70)(20,112,52,76)(21,118,53,68)(22,110,54,74)(23,116,55,66)(24,108,56,80)(33,102,46,125)(34,57,47,82)(35,100,48,123)(36,63,41,88)(37,98,42,121)(38,61,43,86)(39,104,44,127)(40,59,45,84)>;
G:=Group( (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,89)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(65,76)(66,77)(67,78)(68,79)(69,80)(70,73)(71,74)(72,75)(81,127)(82,128)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(105,116)(106,117)(107,118)(108,119)(109,120)(110,113)(111,114)(112,115), (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), (1,98,90,121)(2,59,91,84)(3,100,92,123)(4,61,93,86)(5,102,94,125)(6,63,95,88)(7,104,96,127)(8,57,89,82)(9,105,28,77)(10,117,29,67)(11,107,30,79)(12,119,31,69)(13,109,32,73)(14,113,25,71)(15,111,26,75)(16,115,27,65)(17,83,49,58)(18,122,50,99)(19,85,51,60)(20,124,52,101)(21,87,53,62)(22,126,54,103)(23,81,55,64)(24,128,56,97)(33,118,46,68)(34,108,47,80)(35,120,48,70)(36,110,41,74)(37,114,42,72)(38,112,43,76)(39,116,44,66)(40,106,45,78), (1,75,90,111)(2,67,91,117)(3,73,92,109)(4,65,93,115)(5,79,94,107)(6,71,95,113)(7,77,96,105)(8,69,89,119)(9,64,28,81)(10,99,29,122)(11,62,30,87)(12,97,31,128)(13,60,32,85)(14,103,25,126)(15,58,26,83)(16,101,27,124)(17,114,49,72)(18,106,50,78)(19,120,51,70)(20,112,52,76)(21,118,53,68)(22,110,54,74)(23,116,55,66)(24,108,56,80)(33,102,46,125)(34,57,47,82)(35,100,48,123)(36,63,41,88)(37,98,42,121)(38,61,43,86)(39,104,44,127)(40,59,45,84) );
G=PermutationGroup([[(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,39),(10,40),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,90),(18,91),(19,92),(20,93),(21,94),(22,95),(23,96),(24,89),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(57,97),(58,98),(59,99),(60,100),(61,101),(62,102),(63,103),(64,104),(65,76),(66,77),(67,78),(68,79),(69,80),(70,73),(71,74),(72,75),(81,127),(82,128),(83,121),(84,122),(85,123),(86,124),(87,125),(88,126),(105,116),(106,117),(107,118),(108,119),(109,120),(110,113),(111,114),(112,115)], [(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)], [(1,98,90,121),(2,59,91,84),(3,100,92,123),(4,61,93,86),(5,102,94,125),(6,63,95,88),(7,104,96,127),(8,57,89,82),(9,105,28,77),(10,117,29,67),(11,107,30,79),(12,119,31,69),(13,109,32,73),(14,113,25,71),(15,111,26,75),(16,115,27,65),(17,83,49,58),(18,122,50,99),(19,85,51,60),(20,124,52,101),(21,87,53,62),(22,126,54,103),(23,81,55,64),(24,128,56,97),(33,118,46,68),(34,108,47,80),(35,120,48,70),(36,110,41,74),(37,114,42,72),(38,112,43,76),(39,116,44,66),(40,106,45,78)], [(1,75,90,111),(2,67,91,117),(3,73,92,109),(4,65,93,115),(5,79,94,107),(6,71,95,113),(7,77,96,105),(8,69,89,119),(9,64,28,81),(10,99,29,122),(11,62,30,87),(12,97,31,128),(13,60,32,85),(14,103,25,126),(15,58,26,83),(16,101,27,124),(17,114,49,72),(18,106,50,78),(19,120,51,70),(20,112,52,76),(21,118,53,68),(22,110,54,74),(23,116,55,66),(24,108,56,80),(33,102,46,125),(34,57,47,82),(35,100,48,123),(36,63,41,88),(37,98,42,121),(38,61,43,86),(39,104,44,127),(40,59,45,84)]])
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 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | SD16 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C2.(C8⋊3Q8) | C22.7C42 | C22.4Q16 | C42⋊9C4 | C23.65C23 | C2×C4.Q8 | C4⋊C4 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 8 | 6 | 1 | 1 |
Matrix representation of C2.(C8⋊3Q8) ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 10 | 4 | 0 | 0 | 0 | 0 |
| 13 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 1 | 11 | 0 | 0 | 0 | 0 |
| 6 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 2 | 0 | 0 |
| 0 | 0 | 10 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 8 | 0 | 0 |
| 0 | 0 | 10 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 11 |
| 0 | 0 | 0 | 0 | 11 | 13 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,13,0,0,0,0,4,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,5,0,0,0,0,12,12],[1,6,0,0,0,0,11,16,0,0,0,0,0,0,8,10,0,0,0,0,2,9,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,15,10,0,0,0,0,8,2,0,0,0,0,0,0,4,11,0,0,0,0,11,13] >;
C2.(C8⋊3Q8) in GAP, Magma, Sage, TeX
C_2.(C_8\rtimes_3Q_8)
% in TeX
G:=Group("C2.(C8:3Q8)"); // GroupNames label
G:=SmallGroup(128,816);
// by ID
G=gap.SmallGroup(128,816);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,64,422,387,394,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=c^2,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=a*b^3,d*c*d^-1=a*c^-1>;
// generators/relations