p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4⋊C4.3Q8, (C2×C8).57D4, C4.36(C4⋊Q8), (C2×C4).21Q16, C2.17(C8⋊2D4), C2.9(C4.Q16), (C22×C4).322D4, C23.937(C2×D4), C2.12(D4.Q8), C22.63(C2×Q16), C4.23(C42.C2), C2.17(C8.18D4), C22.127(C4○D8), C22.4Q16.28C2, (C2×C42).388C22, (C22×C8).121C22, C22.260(C4⋊D4), C22.156(C8⋊C22), (C22×C4).1471C23, C22.114(C22⋊Q8), C4.116(C22.D4), C23.65C23.24C2, C2.9(C23.81C23), (C2×C4⋊C8).40C2, (C2×C4).288(C2×Q8), (C2×C2.D8).18C2, (C2×C4).1380(C2×D4), (C2×C4).630(C4○D4), (C2×C4⋊C4).158C22, SmallGroup(128,819)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4).21Q16
G = < a,b,c,d | a2=b4=c8=1, d2=b2c4, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=b-1, dcd-1=c-1 >
Subgroups: 224 in 112 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, 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⋊C8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.4Q16, C22.4Q16, C23.65C23, C2×C4⋊C8, C2×C2.D8, (C2×C4).21Q16
Quotients: C1, C2, C22, D4, Q8, C23, Q16, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C2×Q16, C4○D8, C8⋊C22, C23.81C23, C8.18D4, C8⋊2D4, C4.Q16, D4.Q8, (C2×C4).21Q16
►Regular action on 128 points
Generators in S128
(1 103)(2 104)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 126)(10 127)(11 128)(12 121)(13 122)(14 123)(15 124)(16 125)(17 47)(18 48)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 49)(32 50)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(65 77)(66 78)(67 79)(68 80)(69 73)(70 74)(71 75)(72 76)(81 107)(82 108)(83 109)(84 110)(85 111)(86 112)(87 105)(88 106)(89 119)(90 120)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)
(1 37 17 93)(2 94 18 38)(3 39 19 95)(4 96 20 40)(5 33 21 89)(6 90 22 34)(7 35 23 91)(8 92 24 36)(9 32 80 88)(10 81 73 25)(11 26 74 82)(12 83 75 27)(13 28 76 84)(14 85 77 29)(15 30 78 86)(16 87 79 31)(41 117 97 61)(42 62 98 118)(43 119 99 63)(44 64 100 120)(45 113 101 57)(46 58 102 114)(47 115 103 59)(48 60 104 116)(49 125 105 67)(50 68 106 126)(51 127 107 69)(52 70 108 128)(53 121 109 71)(54 72 110 122)(55 123 111 65)(56 66 112 124)
(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 29 21 81)(2 28 22 88)(3 27 23 87)(4 26 24 86)(5 25 17 85)(6 32 18 84)(7 31 19 83)(8 30 20 82)(9 116 76 64)(10 115 77 63)(11 114 78 62)(12 113 79 61)(13 120 80 60)(14 119 73 59)(15 118 74 58)(16 117 75 57)(33 127 93 65)(34 126 94 72)(35 125 95 71)(36 124 96 70)(37 123 89 69)(38 122 90 68)(39 121 91 67)(40 128 92 66)(41 109 101 49)(42 108 102 56)(43 107 103 55)(44 106 104 54)(45 105 97 53)(46 112 98 52)(47 111 99 51)(48 110 100 50)
G:=sub<Sym(128)| (1,103)(2,104)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,126)(10,127)(11,128)(12,121)(13,122)(14,123)(15,124)(16,125)(17,47)(18,48)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(65,77)(66,78)(67,79)(68,80)(69,73)(70,74)(71,75)(72,76)(81,107)(82,108)(83,109)(84,110)(85,111)(86,112)(87,105)(88,106)(89,119)(90,120)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118), (1,37,17,93)(2,94,18,38)(3,39,19,95)(4,96,20,40)(5,33,21,89)(6,90,22,34)(7,35,23,91)(8,92,24,36)(9,32,80,88)(10,81,73,25)(11,26,74,82)(12,83,75,27)(13,28,76,84)(14,85,77,29)(15,30,78,86)(16,87,79,31)(41,117,97,61)(42,62,98,118)(43,119,99,63)(44,64,100,120)(45,113,101,57)(46,58,102,114)(47,115,103,59)(48,60,104,116)(49,125,105,67)(50,68,106,126)(51,127,107,69)(52,70,108,128)(53,121,109,71)(54,72,110,122)(55,123,111,65)(56,66,112,124), (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), (1,29,21,81)(2,28,22,88)(3,27,23,87)(4,26,24,86)(5,25,17,85)(6,32,18,84)(7,31,19,83)(8,30,20,82)(9,116,76,64)(10,115,77,63)(11,114,78,62)(12,113,79,61)(13,120,80,60)(14,119,73,59)(15,118,74,58)(16,117,75,57)(33,127,93,65)(34,126,94,72)(35,125,95,71)(36,124,96,70)(37,123,89,69)(38,122,90,68)(39,121,91,67)(40,128,92,66)(41,109,101,49)(42,108,102,56)(43,107,103,55)(44,106,104,54)(45,105,97,53)(46,112,98,52)(47,111,99,51)(48,110,100,50)>;
G:=Group( (1,103)(2,104)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,126)(10,127)(11,128)(12,121)(13,122)(14,123)(15,124)(16,125)(17,47)(18,48)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(65,77)(66,78)(67,79)(68,80)(69,73)(70,74)(71,75)(72,76)(81,107)(82,108)(83,109)(84,110)(85,111)(86,112)(87,105)(88,106)(89,119)(90,120)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118), (1,37,17,93)(2,94,18,38)(3,39,19,95)(4,96,20,40)(5,33,21,89)(6,90,22,34)(7,35,23,91)(8,92,24,36)(9,32,80,88)(10,81,73,25)(11,26,74,82)(12,83,75,27)(13,28,76,84)(14,85,77,29)(15,30,78,86)(16,87,79,31)(41,117,97,61)(42,62,98,118)(43,119,99,63)(44,64,100,120)(45,113,101,57)(46,58,102,114)(47,115,103,59)(48,60,104,116)(49,125,105,67)(50,68,106,126)(51,127,107,69)(52,70,108,128)(53,121,109,71)(54,72,110,122)(55,123,111,65)(56,66,112,124), (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), (1,29,21,81)(2,28,22,88)(3,27,23,87)(4,26,24,86)(5,25,17,85)(6,32,18,84)(7,31,19,83)(8,30,20,82)(9,116,76,64)(10,115,77,63)(11,114,78,62)(12,113,79,61)(13,120,80,60)(14,119,73,59)(15,118,74,58)(16,117,75,57)(33,127,93,65)(34,126,94,72)(35,125,95,71)(36,124,96,70)(37,123,89,69)(38,122,90,68)(39,121,91,67)(40,128,92,66)(41,109,101,49)(42,108,102,56)(43,107,103,55)(44,106,104,54)(45,105,97,53)(46,112,98,52)(47,111,99,51)(48,110,100,50) );
G=PermutationGroup([[(1,103),(2,104),(3,97),(4,98),(5,99),(6,100),(7,101),(8,102),(9,126),(10,127),(11,128),(12,121),(13,122),(14,123),(15,124),(16,125),(17,47),(18,48),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,49),(32,50),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(65,77),(66,78),(67,79),(68,80),(69,73),(70,74),(71,75),(72,76),(81,107),(82,108),(83,109),(84,110),(85,111),(86,112),(87,105),(88,106),(89,119),(90,120),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118)], [(1,37,17,93),(2,94,18,38),(3,39,19,95),(4,96,20,40),(5,33,21,89),(6,90,22,34),(7,35,23,91),(8,92,24,36),(9,32,80,88),(10,81,73,25),(11,26,74,82),(12,83,75,27),(13,28,76,84),(14,85,77,29),(15,30,78,86),(16,87,79,31),(41,117,97,61),(42,62,98,118),(43,119,99,63),(44,64,100,120),(45,113,101,57),(46,58,102,114),(47,115,103,59),(48,60,104,116),(49,125,105,67),(50,68,106,126),(51,127,107,69),(52,70,108,128),(53,121,109,71),(54,72,110,122),(55,123,111,65),(56,66,112,124)], [(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)], [(1,29,21,81),(2,28,22,88),(3,27,23,87),(4,26,24,86),(5,25,17,85),(6,32,18,84),(7,31,19,83),(8,30,20,82),(9,116,76,64),(10,115,77,63),(11,114,78,62),(12,113,79,61),(13,120,80,60),(14,119,73,59),(15,118,74,58),(16,117,75,57),(33,127,93,65),(34,126,94,72),(35,125,95,71),(36,124,96,70),(37,123,89,69),(38,122,90,68),(39,121,91,67),(40,128,92,66),(41,109,101,49),(42,108,102,56),(43,107,103,55),(44,106,104,54),(45,105,97,53),(46,112,98,52),(47,111,99,51),(48,110,100,50)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D4 | D4 | Q16 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | (C2×C4).21Q16 | C22.4Q16 | C23.65C23 | C2×C4⋊C8 | C2×C2.D8 | C4⋊C4 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 3 | 2 | 1 | 1 | 4 | 2 | 2 | 4 | 6 | 4 | 2 |
Matrix representation of (C2×C4).21Q16 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 6 | 2 | 0 | 0 | 0 | 0 |
| 7 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 8 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 11 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 4 | 15 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 5 | 7 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,7,0,0,0,0,2,11,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,8,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,6,3,0,0,0,0,11,0],[4,0,0,0,0,0,15,13,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,5,0,0,0,0,7,7] >;
(C2×C4).21Q16 in GAP, Magma, Sage, TeX
(C_2\times C_4)._{21}Q_{16} % in TeX
G:=Group("(C2xC4).21Q16"); // GroupNames label
G:=SmallGroup(128,819);
// by ID
G=gap.SmallGroup(128,819);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,64,422,387,394,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=b^2*c^4,d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations