p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊9Q8, C23.512C24, C22.2912+ 1+4, C22.2122- 1+4, C42⋊8C4.38C2, C42⋊4C4.24C2, (C2×C42).599C22, (C22×C4).850C23, C22.130(C22×Q8), (C22×Q8).150C22, C2.74(C22.45C24), C23.65C23.66C2, C2.C42.241C22, C23.78C23.14C2, C23.67C23.48C2, C23.63C23.36C2, C23.83C23.22C2, C23.81C23.25C2, C2.33(C22.49C24), C2.41(C23.37C23), C2.79(C22.46C24), C2.18(C23.41C23), (C2×C4).168(C2×Q8), (C2×C4).414(C4○D4), (C2×C4⋊C4).351C22, C22.388(C2×C4○D4), SmallGroup(128,1344)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊9Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 324 in 194 conjugacy classes, 100 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×Q8, C42⋊4C4, C42⋊8C4, C23.63C23, C23.65C23, C23.67C23, C23.78C23, C23.81C23, C23.83C23, C42⋊9Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C23.37C23, C23.41C23, C22.45C24, C22.46C24, C22.49C24, C42⋊9Q8
►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 9 53 16)(2 10 54 13)(3 11 55 14)(4 12 56 15)(5 85 125 84)(6 86 126 81)(7 87 127 82)(8 88 128 83)(17 61 24 98)(18 62 21 99)(19 63 22 100)(20 64 23 97)(25 48 32 44)(26 45 29 41)(27 46 30 42)(28 47 31 43)(33 109 40 114)(34 110 37 115)(35 111 38 116)(36 112 39 113)(49 70 58 65)(50 71 59 66)(51 72 60 67)(52 69 57 68)(73 123 78 118)(74 124 79 119)(75 121 80 120)(76 122 77 117)(89 105 94 104)(90 106 95 101)(91 107 96 102)(92 108 93 103)
(1 49 41 98)(2 59 42 62)(3 51 43 100)(4 57 44 64)(5 102 110 122)(6 108 111 118)(7 104 112 124)(8 106 109 120)(9 70 26 17)(10 66 27 21)(11 72 28 19)(12 68 25 23)(13 71 30 18)(14 67 31 22)(15 69 32 20)(16 65 29 24)(33 80 83 90)(34 76 84 96)(35 78 81 92)(36 74 82 94)(37 77 85 91)(38 73 86 93)(39 79 87 89)(40 75 88 95)(45 61 53 58)(46 99 54 50)(47 63 55 60)(48 97 56 52)(101 114 121 128)(103 116 123 126)(105 113 119 127)(107 115 117 125)
(1 105 41 119)(2 103 42 123)(3 107 43 117)(4 101 44 121)(5 63 110 60)(6 99 111 50)(7 61 112 58)(8 97 109 52)(9 91 26 77)(10 95 27 75)(11 89 28 79)(12 93 25 73)(13 90 30 80)(14 94 31 74)(15 92 32 78)(16 96 29 76)(17 37 70 85)(18 33 71 83)(19 39 72 87)(20 35 69 81)(21 40 66 88)(22 36 67 82)(23 38 68 86)(24 34 65 84)(45 124 53 104)(46 118 54 108)(47 122 55 102)(48 120 56 106)(49 127 98 113)(51 125 100 115)(57 128 64 114)(59 126 62 116)
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), (1,9,53,16)(2,10,54,13)(3,11,55,14)(4,12,56,15)(5,85,125,84)(6,86,126,81)(7,87,127,82)(8,88,128,83)(17,61,24,98)(18,62,21,99)(19,63,22,100)(20,64,23,97)(25,48,32,44)(26,45,29,41)(27,46,30,42)(28,47,31,43)(33,109,40,114)(34,110,37,115)(35,111,38,116)(36,112,39,113)(49,70,58,65)(50,71,59,66)(51,72,60,67)(52,69,57,68)(73,123,78,118)(74,124,79,119)(75,121,80,120)(76,122,77,117)(89,105,94,104)(90,106,95,101)(91,107,96,102)(92,108,93,103), (1,49,41,98)(2,59,42,62)(3,51,43,100)(4,57,44,64)(5,102,110,122)(6,108,111,118)(7,104,112,124)(8,106,109,120)(9,70,26,17)(10,66,27,21)(11,72,28,19)(12,68,25,23)(13,71,30,18)(14,67,31,22)(15,69,32,20)(16,65,29,24)(33,80,83,90)(34,76,84,96)(35,78,81,92)(36,74,82,94)(37,77,85,91)(38,73,86,93)(39,79,87,89)(40,75,88,95)(45,61,53,58)(46,99,54,50)(47,63,55,60)(48,97,56,52)(101,114,121,128)(103,116,123,126)(105,113,119,127)(107,115,117,125), (1,105,41,119)(2,103,42,123)(3,107,43,117)(4,101,44,121)(5,63,110,60)(6,99,111,50)(7,61,112,58)(8,97,109,52)(9,91,26,77)(10,95,27,75)(11,89,28,79)(12,93,25,73)(13,90,30,80)(14,94,31,74)(15,92,32,78)(16,96,29,76)(17,37,70,85)(18,33,71,83)(19,39,72,87)(20,35,69,81)(21,40,66,88)(22,36,67,82)(23,38,68,86)(24,34,65,84)(45,124,53,104)(46,118,54,108)(47,122,55,102)(48,120,56,106)(49,127,98,113)(51,125,100,115)(57,128,64,114)(59,126,62,116)>;
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), (1,9,53,16)(2,10,54,13)(3,11,55,14)(4,12,56,15)(5,85,125,84)(6,86,126,81)(7,87,127,82)(8,88,128,83)(17,61,24,98)(18,62,21,99)(19,63,22,100)(20,64,23,97)(25,48,32,44)(26,45,29,41)(27,46,30,42)(28,47,31,43)(33,109,40,114)(34,110,37,115)(35,111,38,116)(36,112,39,113)(49,70,58,65)(50,71,59,66)(51,72,60,67)(52,69,57,68)(73,123,78,118)(74,124,79,119)(75,121,80,120)(76,122,77,117)(89,105,94,104)(90,106,95,101)(91,107,96,102)(92,108,93,103), (1,49,41,98)(2,59,42,62)(3,51,43,100)(4,57,44,64)(5,102,110,122)(6,108,111,118)(7,104,112,124)(8,106,109,120)(9,70,26,17)(10,66,27,21)(11,72,28,19)(12,68,25,23)(13,71,30,18)(14,67,31,22)(15,69,32,20)(16,65,29,24)(33,80,83,90)(34,76,84,96)(35,78,81,92)(36,74,82,94)(37,77,85,91)(38,73,86,93)(39,79,87,89)(40,75,88,95)(45,61,53,58)(46,99,54,50)(47,63,55,60)(48,97,56,52)(101,114,121,128)(103,116,123,126)(105,113,119,127)(107,115,117,125), (1,105,41,119)(2,103,42,123)(3,107,43,117)(4,101,44,121)(5,63,110,60)(6,99,111,50)(7,61,112,58)(8,97,109,52)(9,91,26,77)(10,95,27,75)(11,89,28,79)(12,93,25,73)(13,90,30,80)(14,94,31,74)(15,92,32,78)(16,96,29,76)(17,37,70,85)(18,33,71,83)(19,39,72,87)(20,35,69,81)(21,40,66,88)(22,36,67,82)(23,38,68,86)(24,34,65,84)(45,124,53,104)(46,118,54,108)(47,122,55,102)(48,120,56,106)(49,127,98,113)(51,125,100,115)(57,128,64,114)(59,126,62,116) );
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)], [(1,9,53,16),(2,10,54,13),(3,11,55,14),(4,12,56,15),(5,85,125,84),(6,86,126,81),(7,87,127,82),(8,88,128,83),(17,61,24,98),(18,62,21,99),(19,63,22,100),(20,64,23,97),(25,48,32,44),(26,45,29,41),(27,46,30,42),(28,47,31,43),(33,109,40,114),(34,110,37,115),(35,111,38,116),(36,112,39,113),(49,70,58,65),(50,71,59,66),(51,72,60,67),(52,69,57,68),(73,123,78,118),(74,124,79,119),(75,121,80,120),(76,122,77,117),(89,105,94,104),(90,106,95,101),(91,107,96,102),(92,108,93,103)], [(1,49,41,98),(2,59,42,62),(3,51,43,100),(4,57,44,64),(5,102,110,122),(6,108,111,118),(7,104,112,124),(8,106,109,120),(9,70,26,17),(10,66,27,21),(11,72,28,19),(12,68,25,23),(13,71,30,18),(14,67,31,22),(15,69,32,20),(16,65,29,24),(33,80,83,90),(34,76,84,96),(35,78,81,92),(36,74,82,94),(37,77,85,91),(38,73,86,93),(39,79,87,89),(40,75,88,95),(45,61,53,58),(46,99,54,50),(47,63,55,60),(48,97,56,52),(101,114,121,128),(103,116,123,126),(105,113,119,127),(107,115,117,125)], [(1,105,41,119),(2,103,42,123),(3,107,43,117),(4,101,44,121),(5,63,110,60),(6,99,111,50),(7,61,112,58),(8,97,109,52),(9,91,26,77),(10,95,27,75),(11,89,28,79),(12,93,25,73),(13,90,30,80),(14,94,31,74),(15,92,32,78),(16,96,29,76),(17,37,70,85),(18,33,71,83),(19,39,72,87),(20,35,69,81),(21,40,66,88),(22,36,67,82),(23,38,68,86),(24,34,65,84),(45,124,53,104),(46,118,54,108),(47,122,55,102),(48,120,56,106),(49,127,98,113),(51,125,100,115),(57,128,64,114),(59,126,62,116)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4Z | 4AA | 4AB | 4AC | 4AD |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42⋊9Q8 | C42⋊4C4 | C42⋊8C4 | C23.63C23 | C23.65C23 | C23.67C23 | C23.78C23 | C23.81C23 | C23.83C23 | C42 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 4 | 16 | 1 | 1 |
Matrix representation of C42⋊9Q8 ►in GL6(𝔽5)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,1,0,0,0,0,0,2],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,3,0,0,0,0,0,1],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,3,2,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,1,2,0,0,0,0,0,0,1,0,0,0,0,0,4,4] >;
C42⋊9Q8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_9Q_8
% in TeX
G:=Group("C4^2:9Q8"); // GroupNames label
G:=SmallGroup(128,1344);
// by ID
G=gap.SmallGroup(128,1344);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,568,758,723,184,675,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations