p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.60Q8, (C4×C8)⋊18C4, C4.3(C4⋊Q8), C8.19(C4⋊C4), (C2×C8).43Q8, (C2×C8).275D4, C42⋊8C4.8C2, C42.325(C2×C4), C23.755(C2×D4), (C22×C4).548D4, C22.29(C4⋊Q8), C2.1(C8.5Q8), C2.7(C42⋊9C4), C2.1(C8.12D4), C22.47(C4○D8), C22.29(C4⋊1D4), (C22×C8).525C22, (C2×C42).1061C22, (C22×C4).1345C23, C2.10(C23.25D4), (C2×C4×C8).44C2, C4.35(C2×C4⋊C4), (C2×C2.D8).3C2, (C2×C4).89(C4⋊C4), (C2×C8).234(C2×C4), (C2×C4).730(C2×D4), (C2×C4).195(C2×Q8), (C2×C4.Q8).14C2, (C2×C4⋊C4).49C22, C22.104(C2×C4⋊C4), (C2×C4).544(C22×C4), SmallGroup(128,578)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.60Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=c3 >
Subgroups: 220 in 124 conjugacy classes, 76 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2.C42, C4×C8, C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C22×C8, C42⋊8C4, C2×C4×C8, C2×C4.Q8, C2×C2.D8, C42.60Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C4⋊1D4, C4⋊Q8, C4○D8, C42⋊9C4, C23.25D4, C8.12D4, C8.5Q8, C42.60Q8
►Regular action on 128 points
Generators in S128
(1 101 61 52)(2 102 62 53)(3 103 63 54)(4 104 64 55)(5 97 57 56)(6 98 58 49)(7 99 59 50)(8 100 60 51)(9 95 84 124)(10 96 85 125)(11 89 86 126)(12 90 87 127)(13 91 88 128)(14 92 81 121)(15 93 82 122)(16 94 83 123)(17 27 67 74)(18 28 68 75)(19 29 69 76)(20 30 70 77)(21 31 71 78)(22 32 72 79)(23 25 65 80)(24 26 66 73)(33 46 120 105)(34 47 113 106)(35 48 114 107)(36 41 115 108)(37 42 116 109)(38 43 117 110)(39 44 118 111)(40 45 119 112)
(1 29 5 25)(2 30 6 26)(3 31 7 27)(4 32 8 28)(9 42 13 46)(10 43 14 47)(11 44 15 48)(12 45 16 41)(17 54 21 50)(18 55 22 51)(19 56 23 52)(20 49 24 53)(33 124 37 128)(34 125 38 121)(35 126 39 122)(36 127 40 123)(57 80 61 76)(58 73 62 77)(59 74 63 78)(60 75 64 79)(65 101 69 97)(66 102 70 98)(67 103 71 99)(68 104 72 100)(81 106 85 110)(82 107 86 111)(83 108 87 112)(84 109 88 105)(89 118 93 114)(90 119 94 115)(91 120 95 116)(92 113 96 117)
(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)
(1 113 27 94)(2 116 28 89)(3 119 29 92)(4 114 30 95)(5 117 31 90)(6 120 32 93)(7 115 25 96)(8 118 26 91)(9 100 48 66)(10 103 41 69)(11 98 42 72)(12 101 43 67)(13 104 44 70)(14 99 45 65)(15 102 46 68)(16 97 47 71)(17 87 52 110)(18 82 53 105)(19 85 54 108)(20 88 55 111)(21 83 56 106)(22 86 49 109)(23 81 50 112)(24 84 51 107)(33 79 122 58)(34 74 123 61)(35 77 124 64)(36 80 125 59)(37 75 126 62)(38 78 127 57)(39 73 128 60)(40 76 121 63)
G:=sub<Sym(128)| (1,101,61,52)(2,102,62,53)(3,103,63,54)(4,104,64,55)(5,97,57,56)(6,98,58,49)(7,99,59,50)(8,100,60,51)(9,95,84,124)(10,96,85,125)(11,89,86,126)(12,90,87,127)(13,91,88,128)(14,92,81,121)(15,93,82,122)(16,94,83,123)(17,27,67,74)(18,28,68,75)(19,29,69,76)(20,30,70,77)(21,31,71,78)(22,32,72,79)(23,25,65,80)(24,26,66,73)(33,46,120,105)(34,47,113,106)(35,48,114,107)(36,41,115,108)(37,42,116,109)(38,43,117,110)(39,44,118,111)(40,45,119,112), (1,29,5,25)(2,30,6,26)(3,31,7,27)(4,32,8,28)(9,42,13,46)(10,43,14,47)(11,44,15,48)(12,45,16,41)(17,54,21,50)(18,55,22,51)(19,56,23,52)(20,49,24,53)(33,124,37,128)(34,125,38,121)(35,126,39,122)(36,127,40,123)(57,80,61,76)(58,73,62,77)(59,74,63,78)(60,75,64,79)(65,101,69,97)(66,102,70,98)(67,103,71,99)(68,104,72,100)(81,106,85,110)(82,107,86,111)(83,108,87,112)(84,109,88,105)(89,118,93,114)(90,119,94,115)(91,120,95,116)(92,113,96,117), (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), (1,113,27,94)(2,116,28,89)(3,119,29,92)(4,114,30,95)(5,117,31,90)(6,120,32,93)(7,115,25,96)(8,118,26,91)(9,100,48,66)(10,103,41,69)(11,98,42,72)(12,101,43,67)(13,104,44,70)(14,99,45,65)(15,102,46,68)(16,97,47,71)(17,87,52,110)(18,82,53,105)(19,85,54,108)(20,88,55,111)(21,83,56,106)(22,86,49,109)(23,81,50,112)(24,84,51,107)(33,79,122,58)(34,74,123,61)(35,77,124,64)(36,80,125,59)(37,75,126,62)(38,78,127,57)(39,73,128,60)(40,76,121,63)>;
G:=Group( (1,101,61,52)(2,102,62,53)(3,103,63,54)(4,104,64,55)(5,97,57,56)(6,98,58,49)(7,99,59,50)(8,100,60,51)(9,95,84,124)(10,96,85,125)(11,89,86,126)(12,90,87,127)(13,91,88,128)(14,92,81,121)(15,93,82,122)(16,94,83,123)(17,27,67,74)(18,28,68,75)(19,29,69,76)(20,30,70,77)(21,31,71,78)(22,32,72,79)(23,25,65,80)(24,26,66,73)(33,46,120,105)(34,47,113,106)(35,48,114,107)(36,41,115,108)(37,42,116,109)(38,43,117,110)(39,44,118,111)(40,45,119,112), (1,29,5,25)(2,30,6,26)(3,31,7,27)(4,32,8,28)(9,42,13,46)(10,43,14,47)(11,44,15,48)(12,45,16,41)(17,54,21,50)(18,55,22,51)(19,56,23,52)(20,49,24,53)(33,124,37,128)(34,125,38,121)(35,126,39,122)(36,127,40,123)(57,80,61,76)(58,73,62,77)(59,74,63,78)(60,75,64,79)(65,101,69,97)(66,102,70,98)(67,103,71,99)(68,104,72,100)(81,106,85,110)(82,107,86,111)(83,108,87,112)(84,109,88,105)(89,118,93,114)(90,119,94,115)(91,120,95,116)(92,113,96,117), (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), (1,113,27,94)(2,116,28,89)(3,119,29,92)(4,114,30,95)(5,117,31,90)(6,120,32,93)(7,115,25,96)(8,118,26,91)(9,100,48,66)(10,103,41,69)(11,98,42,72)(12,101,43,67)(13,104,44,70)(14,99,45,65)(15,102,46,68)(16,97,47,71)(17,87,52,110)(18,82,53,105)(19,85,54,108)(20,88,55,111)(21,83,56,106)(22,86,49,109)(23,81,50,112)(24,84,51,107)(33,79,122,58)(34,74,123,61)(35,77,124,64)(36,80,125,59)(37,75,126,62)(38,78,127,57)(39,73,128,60)(40,76,121,63) );
G=PermutationGroup([[(1,101,61,52),(2,102,62,53),(3,103,63,54),(4,104,64,55),(5,97,57,56),(6,98,58,49),(7,99,59,50),(8,100,60,51),(9,95,84,124),(10,96,85,125),(11,89,86,126),(12,90,87,127),(13,91,88,128),(14,92,81,121),(15,93,82,122),(16,94,83,123),(17,27,67,74),(18,28,68,75),(19,29,69,76),(20,30,70,77),(21,31,71,78),(22,32,72,79),(23,25,65,80),(24,26,66,73),(33,46,120,105),(34,47,113,106),(35,48,114,107),(36,41,115,108),(37,42,116,109),(38,43,117,110),(39,44,118,111),(40,45,119,112)], [(1,29,5,25),(2,30,6,26),(3,31,7,27),(4,32,8,28),(9,42,13,46),(10,43,14,47),(11,44,15,48),(12,45,16,41),(17,54,21,50),(18,55,22,51),(19,56,23,52),(20,49,24,53),(33,124,37,128),(34,125,38,121),(35,126,39,122),(36,127,40,123),(57,80,61,76),(58,73,62,77),(59,74,63,78),(60,75,64,79),(65,101,69,97),(66,102,70,98),(67,103,71,99),(68,104,72,100),(81,106,85,110),(82,107,86,111),(83,108,87,112),(84,109,88,105),(89,118,93,114),(90,119,94,115),(91,120,95,116),(92,113,96,117)], [(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)], [(1,113,27,94),(2,116,28,89),(3,119,29,92),(4,114,30,95),(5,117,31,90),(6,120,32,93),(7,115,25,96),(8,118,26,91),(9,100,48,66),(10,103,41,69),(11,98,42,72),(12,101,43,67),(13,104,44,70),(14,99,45,65),(15,102,46,68),(16,97,47,71),(17,87,52,110),(18,82,53,105),(19,85,54,108),(20,88,55,111),(21,83,56,106),(22,86,49,109),(23,81,50,112),(24,84,51,107),(33,79,122,58),(34,74,123,61),(35,77,124,64),(36,80,125,59),(37,75,126,62),(38,78,127,57),(39,73,128,60),(40,76,121,63)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | Q8 | D4 | C4○D8 |
| kernel | C42.60Q8 | C42⋊8C4 | C2×C4×C8 | C2×C4.Q8 | C2×C2.D8 | C4×C8 | C42 | C2×C8 | C2×C8 | C22×C4 | C22 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 4 | 4 | 2 | 16 |
Matrix representation of C42.60Q8 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 4 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 5 | 12 | 0 | 0 |
| 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 5 | 5 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 11 | 13 | 0 | 0 |
| 0 | 13 | 6 | 0 | 0 |
| 0 | 0 | 0 | 6 | 4 |
| 0 | 0 | 0 | 4 | 11 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,13,0],[16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,5,5,0,0,0,12,5,0,0,0,0,0,5,5,0,0,0,12,5],[13,0,0,0,0,0,11,13,0,0,0,13,6,0,0,0,0,0,6,4,0,0,0,4,11] >;
C42.60Q8 in GAP, Magma, Sage, TeX
C_4^2._{60}Q_8 % in TeX
G:=Group("C4^2.60Q8"); // GroupNames label
G:=SmallGroup(128,578);
// by ID
G=gap.SmallGroup(128,578);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,436,2019,248]);
// 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,a*c=c*a,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations