p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4).19Q16, (C2×Q8).108D4, (C2×C4).40SD16, (C22×C4).318D4, C23.927(C2×D4), C22.60(C2×Q16), C42⋊9C4.15C2, C2.16(C4⋊SD16), C2.16(C4⋊2Q16), (C22×C8).85C22, C2.7(C4.SD16), C4.38(C42⋊2C2), C22.4Q16.25C2, (C2×C42).381C22, C22.104(C2×SD16), (C22×Q8).78C22, C22.248(C4⋊D4), C22.149(C8⋊C22), (C22×C4).1461C23, C2.7(C23.47D4), C4.27(C22.D4), C2.7(C23.48D4), C2.7(C23.11D4), C22.94(C4.4D4), C22.138(C8.C22), C22.7C42.13C2, C23.67C23.19C2, C2.10(C42.28C22), C22.117(C22.D4), (C2×C4).1060(C2×D4), (C2×C4).885(C4○D4), (C2×C4⋊C4).146C22, (C2×Q8⋊C4).16C2, SmallGroup(128,804)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4).19Q16
G = < a,b,c,d | a2=b4=c8=1, d2=ac4, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=ac-1 >
Subgroups: 248 in 119 conjugacy classes, 50 normal (28 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C22.7C42, C22.4Q16, C42⋊9C4, C23.67C23, C2×Q8⋊C4, (C2×C4).19Q16
Quotients: C1, C2, C22, D4, C23, SD16, Q16, C2×D4, C4○D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C2×SD16, C2×Q16, C8⋊C22, C8.C22, C23.11D4, C4⋊SD16, C4⋊2Q16, C23.47D4, C23.48D4, C4.SD16, C42.28C22, (C2×C4).19Q16
►Regular action on 128 points
Generators in S128
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 121)(10 122)(11 123)(12 124)(13 125)(14 126)(15 127)(16 128)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 97)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(65 74)(66 75)(67 76)(68 77)(69 78)(70 79)(71 80)(72 73)(81 95)(82 96)(83 89)(84 90)(85 91)(86 92)(87 93)(88 94)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(111 113)(112 114)
(1 108 95 61)(2 55 96 119)(3 110 89 63)(4 49 90 113)(5 112 91 57)(6 51 92 115)(7 106 93 59)(8 53 94 117)(9 20 68 100)(10 44 69 27)(11 22 70 102)(12 46 71 29)(13 24 72 104)(14 48 65 31)(15 18 66 98)(16 42 67 25)(17 126 97 74)(19 128 99 76)(21 122 101 78)(23 124 103 80)(26 77 43 121)(28 79 45 123)(30 73 47 125)(32 75 41 127)(33 118 81 54)(34 62 82 109)(35 120 83 56)(36 64 84 111)(37 114 85 50)(38 58 86 105)(39 116 87 52)(40 60 88 107)
(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 20 37 30)(2 25 38 23)(3 18 39 28)(4 31 40 21)(5 24 33 26)(6 29 34 19)(7 22 35 32)(8 27 36 17)(9 114 125 108)(10 111 126 117)(11 120 127 106)(12 109 128 115)(13 118 121 112)(14 107 122 113)(15 116 123 110)(16 105 124 119)(41 93 102 83)(42 86 103 96)(43 91 104 81)(44 84 97 94)(45 89 98 87)(46 82 99 92)(47 95 100 85)(48 88 101 90)(49 65 60 78)(50 73 61 68)(51 71 62 76)(52 79 63 66)(53 69 64 74)(54 77 57 72)(55 67 58 80)(56 75 59 70)
G:=sub<Sym(128)| (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,127)(16,128)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(41,98)(42,99)(43,100)(44,101)(45,102)(46,103)(47,104)(48,97)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(71,80)(72,73)(81,95)(82,96)(83,89)(84,90)(85,91)(86,92)(87,93)(88,94)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(111,113)(112,114), (1,108,95,61)(2,55,96,119)(3,110,89,63)(4,49,90,113)(5,112,91,57)(6,51,92,115)(7,106,93,59)(8,53,94,117)(9,20,68,100)(10,44,69,27)(11,22,70,102)(12,46,71,29)(13,24,72,104)(14,48,65,31)(15,18,66,98)(16,42,67,25)(17,126,97,74)(19,128,99,76)(21,122,101,78)(23,124,103,80)(26,77,43,121)(28,79,45,123)(30,73,47,125)(32,75,41,127)(33,118,81,54)(34,62,82,109)(35,120,83,56)(36,64,84,111)(37,114,85,50)(38,58,86,105)(39,116,87,52)(40,60,88,107), (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), (1,20,37,30)(2,25,38,23)(3,18,39,28)(4,31,40,21)(5,24,33,26)(6,29,34,19)(7,22,35,32)(8,27,36,17)(9,114,125,108)(10,111,126,117)(11,120,127,106)(12,109,128,115)(13,118,121,112)(14,107,122,113)(15,116,123,110)(16,105,124,119)(41,93,102,83)(42,86,103,96)(43,91,104,81)(44,84,97,94)(45,89,98,87)(46,82,99,92)(47,95,100,85)(48,88,101,90)(49,65,60,78)(50,73,61,68)(51,71,62,76)(52,79,63,66)(53,69,64,74)(54,77,57,72)(55,67,58,80)(56,75,59,70)>;
G:=Group( (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,127)(16,128)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(41,98)(42,99)(43,100)(44,101)(45,102)(46,103)(47,104)(48,97)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(71,80)(72,73)(81,95)(82,96)(83,89)(84,90)(85,91)(86,92)(87,93)(88,94)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(111,113)(112,114), (1,108,95,61)(2,55,96,119)(3,110,89,63)(4,49,90,113)(5,112,91,57)(6,51,92,115)(7,106,93,59)(8,53,94,117)(9,20,68,100)(10,44,69,27)(11,22,70,102)(12,46,71,29)(13,24,72,104)(14,48,65,31)(15,18,66,98)(16,42,67,25)(17,126,97,74)(19,128,99,76)(21,122,101,78)(23,124,103,80)(26,77,43,121)(28,79,45,123)(30,73,47,125)(32,75,41,127)(33,118,81,54)(34,62,82,109)(35,120,83,56)(36,64,84,111)(37,114,85,50)(38,58,86,105)(39,116,87,52)(40,60,88,107), (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), (1,20,37,30)(2,25,38,23)(3,18,39,28)(4,31,40,21)(5,24,33,26)(6,29,34,19)(7,22,35,32)(8,27,36,17)(9,114,125,108)(10,111,126,117)(11,120,127,106)(12,109,128,115)(13,118,121,112)(14,107,122,113)(15,116,123,110)(16,105,124,119)(41,93,102,83)(42,86,103,96)(43,91,104,81)(44,84,97,94)(45,89,98,87)(46,82,99,92)(47,95,100,85)(48,88,101,90)(49,65,60,78)(50,73,61,68)(51,71,62,76)(52,79,63,66)(53,69,64,74)(54,77,57,72)(55,67,58,80)(56,75,59,70) );
G=PermutationGroup([[(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,121),(10,122),(11,123),(12,124),(13,125),(14,126),(15,127),(16,128),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(41,98),(42,99),(43,100),(44,101),(45,102),(46,103),(47,104),(48,97),(49,64),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63),(65,74),(66,75),(67,76),(68,77),(69,78),(70,79),(71,80),(72,73),(81,95),(82,96),(83,89),(84,90),(85,91),(86,92),(87,93),(88,94),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(111,113),(112,114)], [(1,108,95,61),(2,55,96,119),(3,110,89,63),(4,49,90,113),(5,112,91,57),(6,51,92,115),(7,106,93,59),(8,53,94,117),(9,20,68,100),(10,44,69,27),(11,22,70,102),(12,46,71,29),(13,24,72,104),(14,48,65,31),(15,18,66,98),(16,42,67,25),(17,126,97,74),(19,128,99,76),(21,122,101,78),(23,124,103,80),(26,77,43,121),(28,79,45,123),(30,73,47,125),(32,75,41,127),(33,118,81,54),(34,62,82,109),(35,120,83,56),(36,64,84,111),(37,114,85,50),(38,58,86,105),(39,116,87,52),(40,60,88,107)], [(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)], [(1,20,37,30),(2,25,38,23),(3,18,39,28),(4,31,40,21),(5,24,33,26),(6,29,34,19),(7,22,35,32),(8,27,36,17),(9,114,125,108),(10,111,126,117),(11,120,127,106),(12,109,128,115),(13,118,121,112),(14,107,122,113),(15,116,123,110),(16,105,124,119),(41,93,102,83),(42,86,103,96),(43,91,104,81),(44,84,97,94),(45,89,98,87),(46,82,99,92),(47,95,100,85),(48,88,101,90),(49,65,60,78),(50,73,61,68),(51,71,62,76),(52,79,63,66),(53,69,64,74),(54,77,57,72),(55,67,58,80),(56,75,59,70)]])
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 | D4 | SD16 | Q16 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | (C2×C4).19Q16 | C22.7C42 | C22.4Q16 | C42⋊9C4 | C23.67C23 | C2×Q8⋊C4 | C22×C4 | C2×Q8 | C2×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 10 | 1 | 1 |
Matrix representation of (C2×C4).19Q16 ►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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 6 | 0 | 0 | 0 | 0 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 | 0 |
| 0 | 0 | 13 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 5 | 7 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 15 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 |
| 0 | 0 | 0 | 0 | 3 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,11,0,0,0,0,6,1,0,0,0,0,0,0,1,13,0,0,0,0,9,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,5,0,0,0,0,7,7],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,15,4,0,0,0,0,0,0,0,3,0,0,0,0,6,0] >;
(C2×C4).19Q16 in GAP, Magma, Sage, TeX
(C_2\times C_4)._{19}Q_{16} % in TeX
G:=Group("(C2xC4).19Q16"); // GroupNames label
G:=SmallGroup(128,804);
// by ID
G=gap.SmallGroup(128,804);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,456,422,387,58,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=a*c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=b^-1,d*c*d^-1=a*c^-1>;
// generators/relations