p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.37Q8, C23.471C24, C22.2542+ 1+4, C22.1922- 1+4, C42⋊8C4.36C2, C42⋊9C4.33C2, C4.33(C42.C2), (C2×C42).570C22, (C22×C4).545C23, C22.112(C22×Q8), C23.83C23.16C2, C23.65C23.57C2, C2.C42.207C22, C2.27(C22.49C24), C2.42(C22.50C24), C2.20(C22.53C24), C2.15(C23.41C23), (C4×C4⋊C4).70C2, (C2×C4).232(C2×Q8), C2.18(C2×C42.C2), (C2×C4).896(C4○D4), (C2×C4⋊C4).318C22, C22.347(C2×C4○D4), SmallGroup(128,1303)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.37Q8
G = < a,b,c,d | a4=b4=c4=1, d2=a2c2, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 308 in 190 conjugacy classes, 108 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C4×C4⋊C4, C42⋊8C4, C42⋊9C4, C23.65C23, C23.83C23, C42.37Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C42.C2, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C42.C2, C23.41C23, C22.49C24, C22.50C24, C22.53C24, C42.37Q8
►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 41 55 16)(2 42 56 13)(3 43 53 14)(4 44 54 15)(5 110 39 87)(6 111 40 88)(7 112 37 85)(8 109 38 86)(9 29 18 58)(10 30 19 59)(11 31 20 60)(12 32 17 57)(21 98 46 65)(22 99 47 66)(23 100 48 67)(24 97 45 68)(25 69 50 64)(26 70 51 61)(27 71 52 62)(28 72 49 63)(33 116 126 83)(34 113 127 84)(35 114 128 81)(36 115 125 82)(73 93 106 120)(74 94 107 117)(75 95 108 118)(76 96 105 119)(77 124 104 91)(78 121 101 92)(79 122 102 89)(80 123 103 90)
(1 46 9 72)(2 24 10 62)(3 48 11 70)(4 22 12 64)(5 76 113 89)(6 108 114 121)(7 74 115 91)(8 106 116 123)(13 68 59 52)(14 100 60 26)(15 66 57 50)(16 98 58 28)(17 69 54 47)(18 63 55 21)(19 71 56 45)(20 61 53 23)(25 44 99 32)(27 42 97 30)(29 49 41 65)(31 51 43 67)(33 80 86 93)(34 102 87 119)(35 78 88 95)(36 104 85 117)(37 107 82 124)(38 73 83 90)(39 105 84 122)(40 75 81 92)(77 112 94 125)(79 110 96 127)(101 111 118 128)(103 109 120 126)
(1 104 11 119)(2 78 12 93)(3 102 9 117)(4 80 10 95)(5 65 115 51)(6 99 116 27)(7 67 113 49)(8 97 114 25)(13 121 57 106)(14 89 58 74)(15 123 59 108)(16 91 60 76)(17 120 56 101)(18 94 53 79)(19 118 54 103)(20 96 55 77)(21 36 61 87)(22 126 62 111)(23 34 63 85)(24 128 64 109)(26 39 98 82)(28 37 100 84)(29 107 43 122)(30 75 44 90)(31 105 41 124)(32 73 42 92)(33 71 88 47)(35 69 86 45)(38 68 81 50)(40 66 83 52)(46 125 70 110)(48 127 72 112)
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), (1,41,55,16)(2,42,56,13)(3,43,53,14)(4,44,54,15)(5,110,39,87)(6,111,40,88)(7,112,37,85)(8,109,38,86)(9,29,18,58)(10,30,19,59)(11,31,20,60)(12,32,17,57)(21,98,46,65)(22,99,47,66)(23,100,48,67)(24,97,45,68)(25,69,50,64)(26,70,51,61)(27,71,52,62)(28,72,49,63)(33,116,126,83)(34,113,127,84)(35,114,128,81)(36,115,125,82)(73,93,106,120)(74,94,107,117)(75,95,108,118)(76,96,105,119)(77,124,104,91)(78,121,101,92)(79,122,102,89)(80,123,103,90), (1,46,9,72)(2,24,10,62)(3,48,11,70)(4,22,12,64)(5,76,113,89)(6,108,114,121)(7,74,115,91)(8,106,116,123)(13,68,59,52)(14,100,60,26)(15,66,57,50)(16,98,58,28)(17,69,54,47)(18,63,55,21)(19,71,56,45)(20,61,53,23)(25,44,99,32)(27,42,97,30)(29,49,41,65)(31,51,43,67)(33,80,86,93)(34,102,87,119)(35,78,88,95)(36,104,85,117)(37,107,82,124)(38,73,83,90)(39,105,84,122)(40,75,81,92)(77,112,94,125)(79,110,96,127)(101,111,118,128)(103,109,120,126), (1,104,11,119)(2,78,12,93)(3,102,9,117)(4,80,10,95)(5,65,115,51)(6,99,116,27)(7,67,113,49)(8,97,114,25)(13,121,57,106)(14,89,58,74)(15,123,59,108)(16,91,60,76)(17,120,56,101)(18,94,53,79)(19,118,54,103)(20,96,55,77)(21,36,61,87)(22,126,62,111)(23,34,63,85)(24,128,64,109)(26,39,98,82)(28,37,100,84)(29,107,43,122)(30,75,44,90)(31,105,41,124)(32,73,42,92)(33,71,88,47)(35,69,86,45)(38,68,81,50)(40,66,83,52)(46,125,70,110)(48,127,72,112)>;
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), (1,41,55,16)(2,42,56,13)(3,43,53,14)(4,44,54,15)(5,110,39,87)(6,111,40,88)(7,112,37,85)(8,109,38,86)(9,29,18,58)(10,30,19,59)(11,31,20,60)(12,32,17,57)(21,98,46,65)(22,99,47,66)(23,100,48,67)(24,97,45,68)(25,69,50,64)(26,70,51,61)(27,71,52,62)(28,72,49,63)(33,116,126,83)(34,113,127,84)(35,114,128,81)(36,115,125,82)(73,93,106,120)(74,94,107,117)(75,95,108,118)(76,96,105,119)(77,124,104,91)(78,121,101,92)(79,122,102,89)(80,123,103,90), (1,46,9,72)(2,24,10,62)(3,48,11,70)(4,22,12,64)(5,76,113,89)(6,108,114,121)(7,74,115,91)(8,106,116,123)(13,68,59,52)(14,100,60,26)(15,66,57,50)(16,98,58,28)(17,69,54,47)(18,63,55,21)(19,71,56,45)(20,61,53,23)(25,44,99,32)(27,42,97,30)(29,49,41,65)(31,51,43,67)(33,80,86,93)(34,102,87,119)(35,78,88,95)(36,104,85,117)(37,107,82,124)(38,73,83,90)(39,105,84,122)(40,75,81,92)(77,112,94,125)(79,110,96,127)(101,111,118,128)(103,109,120,126), (1,104,11,119)(2,78,12,93)(3,102,9,117)(4,80,10,95)(5,65,115,51)(6,99,116,27)(7,67,113,49)(8,97,114,25)(13,121,57,106)(14,89,58,74)(15,123,59,108)(16,91,60,76)(17,120,56,101)(18,94,53,79)(19,118,54,103)(20,96,55,77)(21,36,61,87)(22,126,62,111)(23,34,63,85)(24,128,64,109)(26,39,98,82)(28,37,100,84)(29,107,43,122)(30,75,44,90)(31,105,41,124)(32,73,42,92)(33,71,88,47)(35,69,86,45)(38,68,81,50)(40,66,83,52)(46,125,70,110)(48,127,72,112) );
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)], [(1,41,55,16),(2,42,56,13),(3,43,53,14),(4,44,54,15),(5,110,39,87),(6,111,40,88),(7,112,37,85),(8,109,38,86),(9,29,18,58),(10,30,19,59),(11,31,20,60),(12,32,17,57),(21,98,46,65),(22,99,47,66),(23,100,48,67),(24,97,45,68),(25,69,50,64),(26,70,51,61),(27,71,52,62),(28,72,49,63),(33,116,126,83),(34,113,127,84),(35,114,128,81),(36,115,125,82),(73,93,106,120),(74,94,107,117),(75,95,108,118),(76,96,105,119),(77,124,104,91),(78,121,101,92),(79,122,102,89),(80,123,103,90)], [(1,46,9,72),(2,24,10,62),(3,48,11,70),(4,22,12,64),(5,76,113,89),(6,108,114,121),(7,74,115,91),(8,106,116,123),(13,68,59,52),(14,100,60,26),(15,66,57,50),(16,98,58,28),(17,69,54,47),(18,63,55,21),(19,71,56,45),(20,61,53,23),(25,44,99,32),(27,42,97,30),(29,49,41,65),(31,51,43,67),(33,80,86,93),(34,102,87,119),(35,78,88,95),(36,104,85,117),(37,107,82,124),(38,73,83,90),(39,105,84,122),(40,75,81,92),(77,112,94,125),(79,110,96,127),(101,111,118,128),(103,109,120,126)], [(1,104,11,119),(2,78,12,93),(3,102,9,117),(4,80,10,95),(5,65,115,51),(6,99,116,27),(7,67,113,49),(8,97,114,25),(13,121,57,106),(14,89,58,74),(15,123,59,108),(16,91,60,76),(17,120,56,101),(18,94,53,79),(19,118,54,103),(20,96,55,77),(21,36,61,87),(22,126,62,111),(23,34,63,85),(24,128,64,109),(26,39,98,82),(28,37,100,84),(29,107,43,122),(30,75,44,90),(31,105,41,124),(32,73,42,92),(33,71,88,47),(35,69,86,45),(38,68,81,50),(40,66,83,52),(46,125,70,110),(48,127,72,112)]])
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 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42.37Q8 | C4×C4⋊C4 | C42⋊8C4 | C42⋊9C4 | C23.65C23 | C23.83C23 | C42 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 4 | 1 | 4 | 4 | 4 | 16 | 1 | 1 |
Matrix representation of C42.37Q8 ►in GL6(𝔽5)
| 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 | 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 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,2,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[3,1,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,1,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,4] >;
C42.37Q8 in GAP, Magma, Sage, TeX
C_4^2._{37}Q_8 % in TeX
G:=Group("C4^2.37Q8"); // GroupNames label
G:=SmallGroup(128,1303);
// by ID
G=gap.SmallGroup(128,1303);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,568,758,723,268,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*c^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations