p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.Q8⋊10C4, C4⋊C4.13Q8, C4.25(C4×Q8), C2.13(C4×SD16), (C2×C4).88SD16, C2.5(Q8.Q8), C2.3(D4⋊2Q8), (C22×C4).693D4, C22.158(C4×D4), C23.778(C2×D4), C22.63(C4○D8), C22.4Q16.9C2, (C22×C8).44C22, C4.13(C42.C2), C22.63(C2×SD16), C4.15(C42⋊2C2), C4.24(C42⋊C2), C22.82(C8⋊C22), (C2×C42).293C22, C22.77(C22⋊Q8), C2.13(SD16⋊C4), (C22×C4).1376C23, C2.4(C23.19D4), C2.4(C23.47D4), C22.72(C8.C22), C22.7C42.33C2, C23.65C23.6C2, C22.90(C22.D4), C2.11(C23.63C23), (C4×C4⋊C4).16C2, C4⋊C4.78(C2×C4), (C2×C8).111(C2×C4), (C2×C4).271(C2×Q8), (C2×C4.Q8).17C2, (C2×C4).573(C4○D4), (C2×C4⋊C4).771C22, (C2×C4).394(C22×C4), SmallGroup(128,652)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.Q8⋊10C4
G = < a,b,c,d | a4=d4=1, b4=a2, c2=a-1b2, ab=ba, cac-1=a-1, ad=da, cbc-1=b3, dbd-1=a-1b3, cd=dc >
Subgroups: 220 in 115 conjugacy classes, 58 normal (44 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×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.7C42, C22.4Q16, C4×C4⋊C4, C23.65C23, C2×C4.Q8, C4.Q8⋊10C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C2×SD16, C4○D8, C8⋊C22, C8.C22, C23.63C23, C4×SD16, SD16⋊C4, D4⋊2Q8, Q8.Q8, C23.19D4, C23.47D4, C4.Q8⋊10C4
►Regular action on 128 points
Generators in S128
(1 58 5 62)(2 59 6 63)(3 60 7 64)(4 61 8 57)(9 107 13 111)(10 108 14 112)(11 109 15 105)(12 110 16 106)(17 99 21 103)(18 100 22 104)(19 101 23 97)(20 102 24 98)(25 91 29 95)(26 92 30 96)(27 93 31 89)(28 94 32 90)(33 78 37 74)(34 79 38 75)(35 80 39 76)(36 73 40 77)(41 86 45 82)(42 87 46 83)(43 88 47 84)(44 81 48 85)(49 70 53 66)(50 71 54 67)(51 72 55 68)(52 65 56 69)(113 125 117 121)(114 126 118 122)(115 127 119 123)(116 128 120 124)
(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)
(1 107 64 11)(2 110 57 14)(3 105 58 9)(4 108 59 12)(5 111 60 15)(6 106 61 10)(7 109 62 13)(8 112 63 16)(17 84 97 41)(18 87 98 44)(19 82 99 47)(20 85 100 42)(21 88 101 45)(22 83 102 48)(23 86 103 43)(24 81 104 46)(25 76 89 33)(26 79 90 36)(27 74 91 39)(28 77 92 34)(29 80 93 37)(30 75 94 40)(31 78 95 35)(32 73 96 38)(49 115 68 125)(50 118 69 128)(51 113 70 123)(52 116 71 126)(53 119 72 121)(54 114 65 124)(55 117 66 127)(56 120 67 122)
(1 74 123 99)(2 40 124 20)(3 76 125 101)(4 34 126 22)(5 78 127 103)(6 36 128 24)(7 80 121 97)(8 38 122 18)(9 25 68 88)(10 90 69 46)(11 27 70 82)(12 92 71 48)(13 29 72 84)(14 94 65 42)(15 31 66 86)(16 96 67 44)(17 62 37 119)(19 64 39 113)(21 58 33 115)(23 60 35 117)(26 50 81 106)(28 52 83 108)(30 54 85 110)(32 56 87 112)(41 109 93 53)(43 111 95 55)(45 105 89 49)(47 107 91 51)(57 75 114 100)(59 77 116 102)(61 79 118 104)(63 73 120 98)
G:=sub<Sym(128)| (1,58,5,62)(2,59,6,63)(3,60,7,64)(4,61,8,57)(9,107,13,111)(10,108,14,112)(11,109,15,105)(12,110,16,106)(17,99,21,103)(18,100,22,104)(19,101,23,97)(20,102,24,98)(25,91,29,95)(26,92,30,96)(27,93,31,89)(28,94,32,90)(33,78,37,74)(34,79,38,75)(35,80,39,76)(36,73,40,77)(41,86,45,82)(42,87,46,83)(43,88,47,84)(44,81,48,85)(49,70,53,66)(50,71,54,67)(51,72,55,68)(52,65,56,69)(113,125,117,121)(114,126,118,122)(115,127,119,123)(116,128,120,124), (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), (1,107,64,11)(2,110,57,14)(3,105,58,9)(4,108,59,12)(5,111,60,15)(6,106,61,10)(7,109,62,13)(8,112,63,16)(17,84,97,41)(18,87,98,44)(19,82,99,47)(20,85,100,42)(21,88,101,45)(22,83,102,48)(23,86,103,43)(24,81,104,46)(25,76,89,33)(26,79,90,36)(27,74,91,39)(28,77,92,34)(29,80,93,37)(30,75,94,40)(31,78,95,35)(32,73,96,38)(49,115,68,125)(50,118,69,128)(51,113,70,123)(52,116,71,126)(53,119,72,121)(54,114,65,124)(55,117,66,127)(56,120,67,122), (1,74,123,99)(2,40,124,20)(3,76,125,101)(4,34,126,22)(5,78,127,103)(6,36,128,24)(7,80,121,97)(8,38,122,18)(9,25,68,88)(10,90,69,46)(11,27,70,82)(12,92,71,48)(13,29,72,84)(14,94,65,42)(15,31,66,86)(16,96,67,44)(17,62,37,119)(19,64,39,113)(21,58,33,115)(23,60,35,117)(26,50,81,106)(28,52,83,108)(30,54,85,110)(32,56,87,112)(41,109,93,53)(43,111,95,55)(45,105,89,49)(47,107,91,51)(57,75,114,100)(59,77,116,102)(61,79,118,104)(63,73,120,98)>;
G:=Group( (1,58,5,62)(2,59,6,63)(3,60,7,64)(4,61,8,57)(9,107,13,111)(10,108,14,112)(11,109,15,105)(12,110,16,106)(17,99,21,103)(18,100,22,104)(19,101,23,97)(20,102,24,98)(25,91,29,95)(26,92,30,96)(27,93,31,89)(28,94,32,90)(33,78,37,74)(34,79,38,75)(35,80,39,76)(36,73,40,77)(41,86,45,82)(42,87,46,83)(43,88,47,84)(44,81,48,85)(49,70,53,66)(50,71,54,67)(51,72,55,68)(52,65,56,69)(113,125,117,121)(114,126,118,122)(115,127,119,123)(116,128,120,124), (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), (1,107,64,11)(2,110,57,14)(3,105,58,9)(4,108,59,12)(5,111,60,15)(6,106,61,10)(7,109,62,13)(8,112,63,16)(17,84,97,41)(18,87,98,44)(19,82,99,47)(20,85,100,42)(21,88,101,45)(22,83,102,48)(23,86,103,43)(24,81,104,46)(25,76,89,33)(26,79,90,36)(27,74,91,39)(28,77,92,34)(29,80,93,37)(30,75,94,40)(31,78,95,35)(32,73,96,38)(49,115,68,125)(50,118,69,128)(51,113,70,123)(52,116,71,126)(53,119,72,121)(54,114,65,124)(55,117,66,127)(56,120,67,122), (1,74,123,99)(2,40,124,20)(3,76,125,101)(4,34,126,22)(5,78,127,103)(6,36,128,24)(7,80,121,97)(8,38,122,18)(9,25,68,88)(10,90,69,46)(11,27,70,82)(12,92,71,48)(13,29,72,84)(14,94,65,42)(15,31,66,86)(16,96,67,44)(17,62,37,119)(19,64,39,113)(21,58,33,115)(23,60,35,117)(26,50,81,106)(28,52,83,108)(30,54,85,110)(32,56,87,112)(41,109,93,53)(43,111,95,55)(45,105,89,49)(47,107,91,51)(57,75,114,100)(59,77,116,102)(61,79,118,104)(63,73,120,98) );
G=PermutationGroup([[(1,58,5,62),(2,59,6,63),(3,60,7,64),(4,61,8,57),(9,107,13,111),(10,108,14,112),(11,109,15,105),(12,110,16,106),(17,99,21,103),(18,100,22,104),(19,101,23,97),(20,102,24,98),(25,91,29,95),(26,92,30,96),(27,93,31,89),(28,94,32,90),(33,78,37,74),(34,79,38,75),(35,80,39,76),(36,73,40,77),(41,86,45,82),(42,87,46,83),(43,88,47,84),(44,81,48,85),(49,70,53,66),(50,71,54,67),(51,72,55,68),(52,65,56,69),(113,125,117,121),(114,126,118,122),(115,127,119,123),(116,128,120,124)], [(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)], [(1,107,64,11),(2,110,57,14),(3,105,58,9),(4,108,59,12),(5,111,60,15),(6,106,61,10),(7,109,62,13),(8,112,63,16),(17,84,97,41),(18,87,98,44),(19,82,99,47),(20,85,100,42),(21,88,101,45),(22,83,102,48),(23,86,103,43),(24,81,104,46),(25,76,89,33),(26,79,90,36),(27,74,91,39),(28,77,92,34),(29,80,93,37),(30,75,94,40),(31,78,95,35),(32,73,96,38),(49,115,68,125),(50,118,69,128),(51,113,70,123),(52,116,71,126),(53,119,72,121),(54,114,65,124),(55,117,66,127),(56,120,67,122)], [(1,74,123,99),(2,40,124,20),(3,76,125,101),(4,34,126,22),(5,78,127,103),(6,36,128,24),(7,80,121,97),(8,38,122,18),(9,25,68,88),(10,90,69,46),(11,27,70,82),(12,92,71,48),(13,29,72,84),(14,94,65,42),(15,31,66,86),(16,96,67,44),(17,62,37,119),(19,64,39,113),(21,58,33,115),(23,60,35,117),(26,50,81,106),(28,52,83,108),(30,54,85,110),(32,56,87,112),(41,109,93,53),(43,111,95,55),(45,105,89,49),(47,107,91,51),(57,75,114,100),(59,77,116,102),(61,79,118,104),(63,73,120,98)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 4U | 4V | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | SD16 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | C4.Q8⋊10C4 | C22.7C42 | C22.4Q16 | C4×C4⋊C4 | C23.65C23 | C2×C4.Q8 | C4.Q8 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 3 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 8 | 4 | 1 | 1 |
Matrix representation of C4.Q8⋊10C4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 10 | 16 | 0 | 0 | 0 | 0 |
| 16 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 0 | 0 | 0 | 0 | 12 | 10 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 12 |
| 0 | 0 | 0 | 0 | 13 | 10 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[10,16,0,0,0,0,16,7,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,12,0,0,0,0,10,10],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,7,13,0,0,0,0,12,10],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;
C4.Q8⋊10C4 in GAP, Magma, Sage, TeX
C_4.Q_8\rtimes_{10}C_4 % in TeX
G:=Group("C4.Q8:10C4"); // GroupNames label
G:=SmallGroup(128,652);
// by ID
G=gap.SmallGroup(128,652);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,680,422,58,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=d^4=1,b^4=a^2,c^2=a^-1*b^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^3,d*b*d^-1=a^-1*b^3,c*d=d*c>;
// generators/relations