p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.7Q8, C42.28D4, C8⋊C4⋊10C4, (C2×C4).7C42, C22.33C4≀C2, C42.35(C2×C4), C42⋊8C4.2C2, (C22×C4).639D4, C2.9(C42⋊6C4), (C2×C42).132C22, C2.6(C22.C42), C23.215(C22⋊C4), C22.24(C4.D4), C22.15(C4.10D4), C2.2(C42.2C22), C2.3(C42.C22), C22.45(C2.C42), (C2×C4⋊C4).4C4, (C2×C4).20(C4⋊C4), (C2×C8⋊C4).15C2, (C22×C4).150(C2×C4), (C2×C4).300(C22⋊C4), SmallGroup(128,27)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.7Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=ac3 >
Subgroups: 168 in 90 conjugacy classes, 40 normal (14 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2.C42, C8⋊C4, C8⋊C4, C2×C42, C2×C4⋊C4, C22×C8, C42⋊8C4, C2×C8⋊C4, C42.7Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C4.D4, C4.10D4, C4≀C2, C42.C22, C42.2C22, C42⋊6C4, C22.C42, C42.7Q8
►Regular action on 128 points
Generators in S128
(1 57 51 47)(2 62 52 44)(3 59 53 41)(4 64 54 46)(5 61 55 43)(6 58 56 48)(7 63 49 45)(8 60 50 42)(9 24 82 26)(10 21 83 31)(11 18 84 28)(12 23 85 25)(13 20 86 30)(14 17 87 27)(15 22 88 32)(16 19 81 29)(33 68 90 73)(34 65 91 78)(35 70 92 75)(36 67 93 80)(37 72 94 77)(38 69 95 74)(39 66 96 79)(40 71 89 76)(97 116 122 111)(98 113 123 108)(99 118 124 105)(100 115 125 110)(101 120 126 107)(102 117 127 112)(103 114 128 109)(104 119 121 106)
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 44 13 48)(10 45 14 41)(11 46 15 42)(12 47 16 43)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(33 125 37 121)(34 126 38 122)(35 127 39 123)(36 128 40 124)(57 81 61 85)(58 82 62 86)(59 83 63 87)(60 84 64 88)(65 107 69 111)(66 108 70 112)(67 109 71 105)(68 110 72 106)(73 115 77 119)(74 116 78 120)(75 117 79 113)(76 118 80 114)(89 99 93 103)(90 100 94 104)(91 101 95 97)(92 102 96 98)
(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 33 17 123)(2 71 18 111)(3 92 19 100)(4 78 20 118)(5 37 21 127)(6 67 22 107)(7 96 23 104)(8 74 24 114)(9 99 42 91)(10 113 43 73)(11 126 44 36)(12 110 45 70)(13 103 46 95)(14 117 47 77)(15 122 48 40)(16 106 41 66)(25 121 49 39)(26 109 50 69)(27 98 51 90)(28 116 52 76)(29 125 53 35)(30 105 54 65)(31 102 55 94)(32 120 56 80)(34 82 124 60)(38 86 128 64)(57 72 87 112)(58 89 88 97)(59 79 81 119)(61 68 83 108)(62 93 84 101)(63 75 85 115)
G:=sub<Sym(128)| (1,57,51,47)(2,62,52,44)(3,59,53,41)(4,64,54,46)(5,61,55,43)(6,58,56,48)(7,63,49,45)(8,60,50,42)(9,24,82,26)(10,21,83,31)(11,18,84,28)(12,23,85,25)(13,20,86,30)(14,17,87,27)(15,22,88,32)(16,19,81,29)(33,68,90,73)(34,65,91,78)(35,70,92,75)(36,67,93,80)(37,72,94,77)(38,69,95,74)(39,66,96,79)(40,71,89,76)(97,116,122,111)(98,113,123,108)(99,118,124,105)(100,115,125,110)(101,120,126,107)(102,117,127,112)(103,114,128,109)(104,119,121,106), (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,44,13,48)(10,45,14,41)(11,46,15,42)(12,47,16,43)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,125,37,121)(34,126,38,122)(35,127,39,123)(36,128,40,124)(57,81,61,85)(58,82,62,86)(59,83,63,87)(60,84,64,88)(65,107,69,111)(66,108,70,112)(67,109,71,105)(68,110,72,106)(73,115,77,119)(74,116,78,120)(75,117,79,113)(76,118,80,114)(89,99,93,103)(90,100,94,104)(91,101,95,97)(92,102,96,98), (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), (1,33,17,123)(2,71,18,111)(3,92,19,100)(4,78,20,118)(5,37,21,127)(6,67,22,107)(7,96,23,104)(8,74,24,114)(9,99,42,91)(10,113,43,73)(11,126,44,36)(12,110,45,70)(13,103,46,95)(14,117,47,77)(15,122,48,40)(16,106,41,66)(25,121,49,39)(26,109,50,69)(27,98,51,90)(28,116,52,76)(29,125,53,35)(30,105,54,65)(31,102,55,94)(32,120,56,80)(34,82,124,60)(38,86,128,64)(57,72,87,112)(58,89,88,97)(59,79,81,119)(61,68,83,108)(62,93,84,101)(63,75,85,115)>;
G:=Group( (1,57,51,47)(2,62,52,44)(3,59,53,41)(4,64,54,46)(5,61,55,43)(6,58,56,48)(7,63,49,45)(8,60,50,42)(9,24,82,26)(10,21,83,31)(11,18,84,28)(12,23,85,25)(13,20,86,30)(14,17,87,27)(15,22,88,32)(16,19,81,29)(33,68,90,73)(34,65,91,78)(35,70,92,75)(36,67,93,80)(37,72,94,77)(38,69,95,74)(39,66,96,79)(40,71,89,76)(97,116,122,111)(98,113,123,108)(99,118,124,105)(100,115,125,110)(101,120,126,107)(102,117,127,112)(103,114,128,109)(104,119,121,106), (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,44,13,48)(10,45,14,41)(11,46,15,42)(12,47,16,43)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,125,37,121)(34,126,38,122)(35,127,39,123)(36,128,40,124)(57,81,61,85)(58,82,62,86)(59,83,63,87)(60,84,64,88)(65,107,69,111)(66,108,70,112)(67,109,71,105)(68,110,72,106)(73,115,77,119)(74,116,78,120)(75,117,79,113)(76,118,80,114)(89,99,93,103)(90,100,94,104)(91,101,95,97)(92,102,96,98), (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), (1,33,17,123)(2,71,18,111)(3,92,19,100)(4,78,20,118)(5,37,21,127)(6,67,22,107)(7,96,23,104)(8,74,24,114)(9,99,42,91)(10,113,43,73)(11,126,44,36)(12,110,45,70)(13,103,46,95)(14,117,47,77)(15,122,48,40)(16,106,41,66)(25,121,49,39)(26,109,50,69)(27,98,51,90)(28,116,52,76)(29,125,53,35)(30,105,54,65)(31,102,55,94)(32,120,56,80)(34,82,124,60)(38,86,128,64)(57,72,87,112)(58,89,88,97)(59,79,81,119)(61,68,83,108)(62,93,84,101)(63,75,85,115) );
G=PermutationGroup([[(1,57,51,47),(2,62,52,44),(3,59,53,41),(4,64,54,46),(5,61,55,43),(6,58,56,48),(7,63,49,45),(8,60,50,42),(9,24,82,26),(10,21,83,31),(11,18,84,28),(12,23,85,25),(13,20,86,30),(14,17,87,27),(15,22,88,32),(16,19,81,29),(33,68,90,73),(34,65,91,78),(35,70,92,75),(36,67,93,80),(37,72,94,77),(38,69,95,74),(39,66,96,79),(40,71,89,76),(97,116,122,111),(98,113,123,108),(99,118,124,105),(100,115,125,110),(101,120,126,107),(102,117,127,112),(103,114,128,109),(104,119,121,106)], [(1,19,5,23),(2,20,6,24),(3,21,7,17),(4,22,8,18),(9,44,13,48),(10,45,14,41),(11,46,15,42),(12,47,16,43),(25,51,29,55),(26,52,30,56),(27,53,31,49),(28,54,32,50),(33,125,37,121),(34,126,38,122),(35,127,39,123),(36,128,40,124),(57,81,61,85),(58,82,62,86),(59,83,63,87),(60,84,64,88),(65,107,69,111),(66,108,70,112),(67,109,71,105),(68,110,72,106),(73,115,77,119),(74,116,78,120),(75,117,79,113),(76,118,80,114),(89,99,93,103),(90,100,94,104),(91,101,95,97),(92,102,96,98)], [(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)], [(1,33,17,123),(2,71,18,111),(3,92,19,100),(4,78,20,118),(5,37,21,127),(6,67,22,107),(7,96,23,104),(8,74,24,114),(9,99,42,91),(10,113,43,73),(11,126,44,36),(12,110,45,70),(13,103,46,95),(14,117,47,77),(15,122,48,40),(16,106,41,66),(25,121,49,39),(26,109,50,69),(27,98,51,90),(28,116,52,76),(29,125,53,35),(30,105,54,65),(31,102,55,94),(32,120,56,80),(34,82,124,60),(38,86,128,64),(57,72,87,112),(58,89,88,97),(59,79,81,119),(61,68,83,108),(62,93,84,101),(63,75,85,115)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8P |
| 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 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | C4≀C2 | C4.D4 | C4.10D4 |
| kernel | C42.7Q8 | C42⋊8C4 | C2×C8⋊C4 | C8⋊C4 | C2×C4⋊C4 | C42 | C42 | C22×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 2 | 8 | 4 | 1 | 1 | 2 | 16 | 1 | 1 |
Matrix representation of C42.7Q8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 13 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 10 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 13 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 2 | 0 | 0 |
| 0 | 0 | 7 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 8 | 1 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 2 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 2 |
| 0 | 0 | 0 | 0 | 10 | 7 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,13,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,10,0,0,0,0,0,1],[1,13,0,0,0,0,0,16,0,0,0,0,0,0,13,7,0,0,0,0,2,4,0,0,0,0,0,0,13,8,0,0,0,0,0,1],[4,0,0,0,0,0,2,13,0,0,0,0,0,0,13,1,0,0,0,0,2,4,0,0,0,0,0,0,10,10,0,0,0,0,2,7] >;
C42.7Q8 in GAP, Magma, Sage, TeX
C_4^2._7Q_8
% in TeX
G:=Group("C4^2.7Q8"); // GroupNames label
G:=SmallGroup(128,27);
// by ID
G=gap.SmallGroup(128,27);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,3924,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=a*c^3>;
// generators/relations