p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.38Q8, C23.506C24, C22.2862+ 1+4, C22.2092- 1+4, C42⋊4C4.23C2, C42⋊5C4.11C2, (C2×C42).593C22, (C22×C4).125C23, C22.127(C22×Q8), C23.81C23.24C2, C2.C42.551C22, C23.65C23.65C2, C23.63C23.35C2, C2.38(C23.37C23), C2.76(C22.46C24), C2.17(C23.41C23), C2.80(C22.47C24), (C4×C4⋊C4).77C2, (C2×C4).128(C2×Q8), (C2×C4).164(C4○D4), (C2×C4⋊C4).345C22, C22.382(C2×C4○D4), SmallGroup(128,1338)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.38Q8
G = < a,b,c,d | a4=b4=c4=1, d2=a2c2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 308 in 190 conjugacy classes, 100 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2.C42, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C42⋊4C4, C4×C4⋊C4, C42⋊5C4, C23.63C23, C23.65C23, C23.81C23, C42.38Q8
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, C22.46C24, C22.47C24, C42.38Q8
►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 57 25 17)(2 58 26 18)(3 59 27 19)(4 60 28 20)(5 35 77 82)(6 36 78 83)(7 33 79 84)(8 34 80 81)(9 50 100 72)(10 51 97 69)(11 52 98 70)(12 49 99 71)(13 29 53 47)(14 30 54 48)(15 31 55 45)(16 32 56 46)(21 65 61 43)(22 66 62 44)(23 67 63 41)(24 68 64 42)(37 126 103 116)(38 127 104 113)(39 128 101 114)(40 125 102 115)(73 123 93 109)(74 124 94 110)(75 121 95 111)(76 122 96 112)(85 105 89 119)(86 106 90 120)(87 107 91 117)(88 108 92 118)
(1 21 13 51)(2 62 14 70)(3 23 15 49)(4 64 16 72)(5 88 37 110)(6 89 38 121)(7 86 39 112)(8 91 40 123)(9 60 42 32)(10 17 43 47)(11 58 44 30)(12 19 41 45)(18 66 48 98)(20 68 46 100)(22 54 52 26)(24 56 50 28)(25 61 53 69)(27 63 55 71)(29 97 57 65)(31 99 59 67)(33 106 128 76)(34 117 125 93)(35 108 126 74)(36 119 127 95)(73 81 107 115)(75 83 105 113)(77 92 103 124)(78 85 104 111)(79 90 101 122)(80 87 102 109)(82 118 116 94)(84 120 114 96)
(1 107 15 75)(2 106 16 74)(3 105 13 73)(4 108 14 76)(5 100 39 66)(6 99 40 65)(7 98 37 68)(8 97 38 67)(9 101 44 77)(10 104 41 80)(11 103 42 79)(12 102 43 78)(17 85 45 109)(18 88 46 112)(19 87 47 111)(20 86 48 110)(21 81 49 113)(22 84 50 116)(23 83 51 115)(24 82 52 114)(25 117 55 95)(26 120 56 94)(27 119 53 93)(28 118 54 96)(29 121 59 91)(30 124 60 90)(31 123 57 89)(32 122 58 92)(33 72 126 62)(34 71 127 61)(35 70 128 64)(36 69 125 63)
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), (1,57,25,17)(2,58,26,18)(3,59,27,19)(4,60,28,20)(5,35,77,82)(6,36,78,83)(7,33,79,84)(8,34,80,81)(9,50,100,72)(10,51,97,69)(11,52,98,70)(12,49,99,71)(13,29,53,47)(14,30,54,48)(15,31,55,45)(16,32,56,46)(21,65,61,43)(22,66,62,44)(23,67,63,41)(24,68,64,42)(37,126,103,116)(38,127,104,113)(39,128,101,114)(40,125,102,115)(73,123,93,109)(74,124,94,110)(75,121,95,111)(76,122,96,112)(85,105,89,119)(86,106,90,120)(87,107,91,117)(88,108,92,118), (1,21,13,51)(2,62,14,70)(3,23,15,49)(4,64,16,72)(5,88,37,110)(6,89,38,121)(7,86,39,112)(8,91,40,123)(9,60,42,32)(10,17,43,47)(11,58,44,30)(12,19,41,45)(18,66,48,98)(20,68,46,100)(22,54,52,26)(24,56,50,28)(25,61,53,69)(27,63,55,71)(29,97,57,65)(31,99,59,67)(33,106,128,76)(34,117,125,93)(35,108,126,74)(36,119,127,95)(73,81,107,115)(75,83,105,113)(77,92,103,124)(78,85,104,111)(79,90,101,122)(80,87,102,109)(82,118,116,94)(84,120,114,96), (1,107,15,75)(2,106,16,74)(3,105,13,73)(4,108,14,76)(5,100,39,66)(6,99,40,65)(7,98,37,68)(8,97,38,67)(9,101,44,77)(10,104,41,80)(11,103,42,79)(12,102,43,78)(17,85,45,109)(18,88,46,112)(19,87,47,111)(20,86,48,110)(21,81,49,113)(22,84,50,116)(23,83,51,115)(24,82,52,114)(25,117,55,95)(26,120,56,94)(27,119,53,93)(28,118,54,96)(29,121,59,91)(30,124,60,90)(31,123,57,89)(32,122,58,92)(33,72,126,62)(34,71,127,61)(35,70,128,64)(36,69,125,63)>;
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), (1,57,25,17)(2,58,26,18)(3,59,27,19)(4,60,28,20)(5,35,77,82)(6,36,78,83)(7,33,79,84)(8,34,80,81)(9,50,100,72)(10,51,97,69)(11,52,98,70)(12,49,99,71)(13,29,53,47)(14,30,54,48)(15,31,55,45)(16,32,56,46)(21,65,61,43)(22,66,62,44)(23,67,63,41)(24,68,64,42)(37,126,103,116)(38,127,104,113)(39,128,101,114)(40,125,102,115)(73,123,93,109)(74,124,94,110)(75,121,95,111)(76,122,96,112)(85,105,89,119)(86,106,90,120)(87,107,91,117)(88,108,92,118), (1,21,13,51)(2,62,14,70)(3,23,15,49)(4,64,16,72)(5,88,37,110)(6,89,38,121)(7,86,39,112)(8,91,40,123)(9,60,42,32)(10,17,43,47)(11,58,44,30)(12,19,41,45)(18,66,48,98)(20,68,46,100)(22,54,52,26)(24,56,50,28)(25,61,53,69)(27,63,55,71)(29,97,57,65)(31,99,59,67)(33,106,128,76)(34,117,125,93)(35,108,126,74)(36,119,127,95)(73,81,107,115)(75,83,105,113)(77,92,103,124)(78,85,104,111)(79,90,101,122)(80,87,102,109)(82,118,116,94)(84,120,114,96), (1,107,15,75)(2,106,16,74)(3,105,13,73)(4,108,14,76)(5,100,39,66)(6,99,40,65)(7,98,37,68)(8,97,38,67)(9,101,44,77)(10,104,41,80)(11,103,42,79)(12,102,43,78)(17,85,45,109)(18,88,46,112)(19,87,47,111)(20,86,48,110)(21,81,49,113)(22,84,50,116)(23,83,51,115)(24,82,52,114)(25,117,55,95)(26,120,56,94)(27,119,53,93)(28,118,54,96)(29,121,59,91)(30,124,60,90)(31,123,57,89)(32,122,58,92)(33,72,126,62)(34,71,127,61)(35,70,128,64)(36,69,125,63) );
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)], [(1,57,25,17),(2,58,26,18),(3,59,27,19),(4,60,28,20),(5,35,77,82),(6,36,78,83),(7,33,79,84),(8,34,80,81),(9,50,100,72),(10,51,97,69),(11,52,98,70),(12,49,99,71),(13,29,53,47),(14,30,54,48),(15,31,55,45),(16,32,56,46),(21,65,61,43),(22,66,62,44),(23,67,63,41),(24,68,64,42),(37,126,103,116),(38,127,104,113),(39,128,101,114),(40,125,102,115),(73,123,93,109),(74,124,94,110),(75,121,95,111),(76,122,96,112),(85,105,89,119),(86,106,90,120),(87,107,91,117),(88,108,92,118)], [(1,21,13,51),(2,62,14,70),(3,23,15,49),(4,64,16,72),(5,88,37,110),(6,89,38,121),(7,86,39,112),(8,91,40,123),(9,60,42,32),(10,17,43,47),(11,58,44,30),(12,19,41,45),(18,66,48,98),(20,68,46,100),(22,54,52,26),(24,56,50,28),(25,61,53,69),(27,63,55,71),(29,97,57,65),(31,99,59,67),(33,106,128,76),(34,117,125,93),(35,108,126,74),(36,119,127,95),(73,81,107,115),(75,83,105,113),(77,92,103,124),(78,85,104,111),(79,90,101,122),(80,87,102,109),(82,118,116,94),(84,120,114,96)], [(1,107,15,75),(2,106,16,74),(3,105,13,73),(4,108,14,76),(5,100,39,66),(6,99,40,65),(7,98,37,68),(8,97,38,67),(9,101,44,77),(10,104,41,80),(11,103,42,79),(12,102,43,78),(17,85,45,109),(18,88,46,112),(19,87,47,111),(20,86,48,110),(21,81,49,113),(22,84,50,116),(23,83,51,115),(24,82,52,114),(25,117,55,95),(26,120,56,94),(27,119,53,93),(28,118,54,96),(29,121,59,91),(30,124,60,90),(31,123,57,89),(32,122,58,92),(33,72,126,62),(34,71,127,61),(35,70,128,64),(36,69,125,63)]])
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 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42.38Q8 | C42⋊4C4 | C4×C4⋊C4 | C42⋊5C4 | C23.63C23 | C23.65C23 | C23.81C23 | C42 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 16 | 1 | 1 |
Matrix representation of C42.38Q8 ►in GL6(𝔽5)
| 3 | 3 | 0 | 0 | 0 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 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 | 4 | 0 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 2 | 0 |
G:=sub<GL(6,GF(5))| [3,4,0,0,0,0,3,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[2,0,0,0,0,0,2,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0],[1,3,0,0,0,0,1,4,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,3,0] >;
C42.38Q8 in GAP, Magma, Sage, TeX
C_4^2._{38}Q_8 % in TeX
G:=Group("C4^2.38Q8"); // GroupNames label
G:=SmallGroup(128,1338);
// by ID
G=gap.SmallGroup(128,1338);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723,184,675,248]);
// 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*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations