p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C42⋊19Q8, C43.19C2, C42.348D4, C23.768C24, C4⋊1(C4⋊Q8), C4.10(C4⋊1D4), C42⋊9C4.42C2, (C22×C4).270C23, C22.474(C22×D4), C22.186(C22×Q8), (C2×C42).1099C22, (C22×Q8).254C22, C2.24(C2×C4⋊Q8), (C2×C4⋊Q8).41C2, (C2×C4).838(C2×D4), C2.19(C2×C4⋊1D4), (C2×C4).234(C2×Q8), (C2×C4⋊C4).570C22, SmallGroup(128,1600)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊19Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 516 in 324 conjugacy classes, 180 normal (6 characteristic)
C1, C2, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×C42, C2×C4⋊C4, C4⋊Q8, C22×Q8, C43, C42⋊9C4, C2×C4⋊Q8, C42⋊19Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C4⋊1D4, C4⋊Q8, C22×D4, C22×Q8, C2×C4⋊1D4, C2×C4⋊Q8, C42⋊19Q8
►Regular action on 128 points
Generators in S128
(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 22 52)(2 99 23 49)(3 100 24 50)(4 97 21 51)(5 114 92 83)(6 115 89 84)(7 116 90 81)(8 113 91 82)(9 64 20 32)(10 61 17 29)(11 62 18 30)(12 63 19 31)(13 74 53 37)(14 75 54 38)(15 76 55 39)(16 73 56 40)(25 80 57 36)(26 77 58 33)(27 78 59 34)(28 79 60 35)(41 67 111 124)(42 68 112 121)(43 65 109 122)(44 66 110 123)(45 108 103 118)(46 105 104 119)(47 106 101 120)(48 107 102 117)(69 86 127 93)(70 87 128 94)(71 88 125 95)(72 85 126 96)
(1 27 12 16)(2 28 9 13)(3 25 10 14)(4 26 11 15)(5 45 127 42)(6 46 128 43)(7 47 125 44)(8 48 126 41)(17 54 24 57)(18 55 21 58)(19 56 22 59)(20 53 23 60)(29 38 50 36)(30 39 51 33)(31 40 52 34)(32 37 49 35)(61 75 100 80)(62 76 97 77)(63 73 98 78)(64 74 99 79)(65 115 105 94)(66 116 106 95)(67 113 107 96)(68 114 108 93)(69 112 92 103)(70 109 89 104)(71 110 90 101)(72 111 91 102)(81 120 88 123)(82 117 85 124)(83 118 86 121)(84 119 87 122)
(1 95 12 116)(2 94 9 115)(3 93 10 114)(4 96 11 113)(5 100 127 61)(6 99 128 64)(7 98 125 63)(8 97 126 62)(13 65 28 105)(14 68 25 108)(15 67 26 107)(16 66 27 106)(17 83 24 86)(18 82 21 85)(19 81 22 88)(20 84 23 87)(29 92 50 69)(30 91 51 72)(31 90 52 71)(32 89 49 70)(33 102 39 111)(34 101 40 110)(35 104 37 109)(36 103 38 112)(41 77 48 76)(42 80 45 75)(43 79 46 74)(44 78 47 73)(53 122 60 119)(54 121 57 118)(55 124 58 117)(56 123 59 120)
G:=sub<Sym(128)| (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), (1,98,22,52)(2,99,23,49)(3,100,24,50)(4,97,21,51)(5,114,92,83)(6,115,89,84)(7,116,90,81)(8,113,91,82)(9,64,20,32)(10,61,17,29)(11,62,18,30)(12,63,19,31)(13,74,53,37)(14,75,54,38)(15,76,55,39)(16,73,56,40)(25,80,57,36)(26,77,58,33)(27,78,59,34)(28,79,60,35)(41,67,111,124)(42,68,112,121)(43,65,109,122)(44,66,110,123)(45,108,103,118)(46,105,104,119)(47,106,101,120)(48,107,102,117)(69,86,127,93)(70,87,128,94)(71,88,125,95)(72,85,126,96), (1,27,12,16)(2,28,9,13)(3,25,10,14)(4,26,11,15)(5,45,127,42)(6,46,128,43)(7,47,125,44)(8,48,126,41)(17,54,24,57)(18,55,21,58)(19,56,22,59)(20,53,23,60)(29,38,50,36)(30,39,51,33)(31,40,52,34)(32,37,49,35)(61,75,100,80)(62,76,97,77)(63,73,98,78)(64,74,99,79)(65,115,105,94)(66,116,106,95)(67,113,107,96)(68,114,108,93)(69,112,92,103)(70,109,89,104)(71,110,90,101)(72,111,91,102)(81,120,88,123)(82,117,85,124)(83,118,86,121)(84,119,87,122), (1,95,12,116)(2,94,9,115)(3,93,10,114)(4,96,11,113)(5,100,127,61)(6,99,128,64)(7,98,125,63)(8,97,126,62)(13,65,28,105)(14,68,25,108)(15,67,26,107)(16,66,27,106)(17,83,24,86)(18,82,21,85)(19,81,22,88)(20,84,23,87)(29,92,50,69)(30,91,51,72)(31,90,52,71)(32,89,49,70)(33,102,39,111)(34,101,40,110)(35,104,37,109)(36,103,38,112)(41,77,48,76)(42,80,45,75)(43,79,46,74)(44,78,47,73)(53,122,60,119)(54,121,57,118)(55,124,58,117)(56,123,59,120)>;
G:=Group( (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), (1,98,22,52)(2,99,23,49)(3,100,24,50)(4,97,21,51)(5,114,92,83)(6,115,89,84)(7,116,90,81)(8,113,91,82)(9,64,20,32)(10,61,17,29)(11,62,18,30)(12,63,19,31)(13,74,53,37)(14,75,54,38)(15,76,55,39)(16,73,56,40)(25,80,57,36)(26,77,58,33)(27,78,59,34)(28,79,60,35)(41,67,111,124)(42,68,112,121)(43,65,109,122)(44,66,110,123)(45,108,103,118)(46,105,104,119)(47,106,101,120)(48,107,102,117)(69,86,127,93)(70,87,128,94)(71,88,125,95)(72,85,126,96), (1,27,12,16)(2,28,9,13)(3,25,10,14)(4,26,11,15)(5,45,127,42)(6,46,128,43)(7,47,125,44)(8,48,126,41)(17,54,24,57)(18,55,21,58)(19,56,22,59)(20,53,23,60)(29,38,50,36)(30,39,51,33)(31,40,52,34)(32,37,49,35)(61,75,100,80)(62,76,97,77)(63,73,98,78)(64,74,99,79)(65,115,105,94)(66,116,106,95)(67,113,107,96)(68,114,108,93)(69,112,92,103)(70,109,89,104)(71,110,90,101)(72,111,91,102)(81,120,88,123)(82,117,85,124)(83,118,86,121)(84,119,87,122), (1,95,12,116)(2,94,9,115)(3,93,10,114)(4,96,11,113)(5,100,127,61)(6,99,128,64)(7,98,125,63)(8,97,126,62)(13,65,28,105)(14,68,25,108)(15,67,26,107)(16,66,27,106)(17,83,24,86)(18,82,21,85)(19,81,22,88)(20,84,23,87)(29,92,50,69)(30,91,51,72)(31,90,52,71)(32,89,49,70)(33,102,39,111)(34,101,40,110)(35,104,37,109)(36,103,38,112)(41,77,48,76)(42,80,45,75)(43,79,46,74)(44,78,47,73)(53,122,60,119)(54,121,57,118)(55,124,58,117)(56,123,59,120) );
G=PermutationGroup([[(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)], [(1,98,22,52),(2,99,23,49),(3,100,24,50),(4,97,21,51),(5,114,92,83),(6,115,89,84),(7,116,90,81),(8,113,91,82),(9,64,20,32),(10,61,17,29),(11,62,18,30),(12,63,19,31),(13,74,53,37),(14,75,54,38),(15,76,55,39),(16,73,56,40),(25,80,57,36),(26,77,58,33),(27,78,59,34),(28,79,60,35),(41,67,111,124),(42,68,112,121),(43,65,109,122),(44,66,110,123),(45,108,103,118),(46,105,104,119),(47,106,101,120),(48,107,102,117),(69,86,127,93),(70,87,128,94),(71,88,125,95),(72,85,126,96)], [(1,27,12,16),(2,28,9,13),(3,25,10,14),(4,26,11,15),(5,45,127,42),(6,46,128,43),(7,47,125,44),(8,48,126,41),(17,54,24,57),(18,55,21,58),(19,56,22,59),(20,53,23,60),(29,38,50,36),(30,39,51,33),(31,40,52,34),(32,37,49,35),(61,75,100,80),(62,76,97,77),(63,73,98,78),(64,74,99,79),(65,115,105,94),(66,116,106,95),(67,113,107,96),(68,114,108,93),(69,112,92,103),(70,109,89,104),(71,110,90,101),(72,111,91,102),(81,120,88,123),(82,117,85,124),(83,118,86,121),(84,119,87,122)], [(1,95,12,116),(2,94,9,115),(3,93,10,114),(4,96,11,113),(5,100,127,61),(6,99,128,64),(7,98,125,63),(8,97,126,62),(13,65,28,105),(14,68,25,108),(15,67,26,107),(16,66,27,106),(17,83,24,86),(18,82,21,85),(19,81,22,88),(20,84,23,87),(29,92,50,69),(30,91,51,72),(31,90,52,71),(32,89,49,70),(33,102,39,111),(34,101,40,110),(35,104,37,109),(36,103,38,112),(41,77,48,76),(42,80,45,75),(43,79,46,74),(44,78,47,73),(53,122,60,119),(54,121,57,118),(55,124,58,117),(56,123,59,120)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4AB | 4AC | ··· | 4AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | D4 | Q8 |
| kernel | C42⋊19Q8 | C43 | C42⋊9C4 | C2×C4⋊Q8 | C42 | C42 |
| # reps | 1 | 1 | 8 | 6 | 12 | 16 |
Matrix representation of C42⋊19Q8 ►in GL6(𝔽5)
| 0 | 4 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(5))| [0,1,0,0,0,0,4,0,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,1,2,0,0,0,0,0,0,4,0,0,0,0,0,0,1] >;
C42⋊19Q8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{19}Q_8 % in TeX
G:=Group("C4^2:19Q8"); // GroupNames label
G:=SmallGroup(128,1600);
// by ID
G=gap.SmallGroup(128,1600);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,184,2019,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations