p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.58Q8, C8⋊6(C4⋊C4), (C4×C8)⋊21C4, C4⋊2(C4.Q8), C4.1(C4⋊Q8), (C2×C8).53Q8, (C2×C8).274D4, C2.1(C8⋊5D4), C2.1(C8⋊3Q8), (C2×C4).71SD16, C42⋊9C4.6C2, C42.323(C2×C4), C23.753(C2×D4), (C22×C4).577D4, C22.27(C4⋊Q8), C2.5(C42⋊9C4), C22.52(C2×SD16), C22.27(C4⋊1D4), (C22×C8).563C22, (C22×C4).1343C23, (C2×C42).1059C22, (C2×C4×C8).54C2, C4.33(C2×C4⋊C4), C2.8(C2×C4.Q8), (C2×C8).233(C2×C4), (C2×C4).728(C2×D4), (C2×C4).193(C2×Q8), (C2×C4.Q8).13C2, (C2×C4).131(C4⋊C4), (C2×C4⋊C4).47C22, C22.102(C2×C4⋊C4), (C2×C4).542(C22×C4), SmallGroup(128,576)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.58Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c3 >
Subgroups: 252 in 140 conjugacy classes, 92 normal (12 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, C4×C8, C4.Q8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C42⋊9C4, C2×C4×C8, C2×C4.Q8, C42.58Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C4.Q8, C2×C4⋊C4, C4⋊1D4, C4⋊Q8, C2×SD16, C42⋊9C4, C2×C4.Q8, C8⋊5D4, C8⋊3Q8, C42.58Q8
►Regular action on 128 points
Generators in S128
(1 100 61 29)(2 101 62 30)(3 102 63 31)(4 103 64 32)(5 104 57 25)(6 97 58 26)(7 98 59 27)(8 99 60 28)(9 91 83 127)(10 92 84 128)(11 93 85 121)(12 94 86 122)(13 95 87 123)(14 96 88 124)(15 89 81 125)(16 90 82 126)(17 56 72 80)(18 49 65 73)(19 50 66 74)(20 51 67 75)(21 52 68 76)(22 53 69 77)(23 54 70 78)(24 55 71 79)(33 41 110 117)(34 42 111 118)(35 43 112 119)(36 44 105 120)(37 45 106 113)(38 46 107 114)(39 47 108 115)(40 48 109 116)
(1 49 5 53)(2 50 6 54)(3 51 7 55)(4 52 8 56)(9 117 13 113)(10 118 14 114)(11 119 15 115)(12 120 16 116)(17 32 21 28)(18 25 22 29)(19 26 23 30)(20 27 24 31)(33 95 37 91)(34 96 38 92)(35 89 39 93)(36 90 40 94)(41 87 45 83)(42 88 46 84)(43 81 47 85)(44 82 48 86)(57 77 61 73)(58 78 62 74)(59 79 63 75)(60 80 64 76)(65 104 69 100)(66 97 70 101)(67 98 71 102)(68 99 72 103)(105 126 109 122)(106 127 110 123)(107 128 111 124)(108 121 112 125)
(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)
(1 108 55 127)(2 111 56 122)(3 106 49 125)(4 109 50 128)(5 112 51 123)(6 107 52 126)(7 110 53 121)(8 105 54 124)(9 29 115 24)(10 32 116 19)(11 27 117 22)(12 30 118 17)(13 25 119 20)(14 28 120 23)(15 31 113 18)(16 26 114 21)(33 77 93 59)(34 80 94 62)(35 75 95 57)(36 78 96 60)(37 73 89 63)(38 76 90 58)(39 79 91 61)(40 74 92 64)(41 69 85 98)(42 72 86 101)(43 67 87 104)(44 70 88 99)(45 65 81 102)(46 68 82 97)(47 71 83 100)(48 66 84 103)
G:=sub<Sym(128)| (1,100,61,29)(2,101,62,30)(3,102,63,31)(4,103,64,32)(5,104,57,25)(6,97,58,26)(7,98,59,27)(8,99,60,28)(9,91,83,127)(10,92,84,128)(11,93,85,121)(12,94,86,122)(13,95,87,123)(14,96,88,124)(15,89,81,125)(16,90,82,126)(17,56,72,80)(18,49,65,73)(19,50,66,74)(20,51,67,75)(21,52,68,76)(22,53,69,77)(23,54,70,78)(24,55,71,79)(33,41,110,117)(34,42,111,118)(35,43,112,119)(36,44,105,120)(37,45,106,113)(38,46,107,114)(39,47,108,115)(40,48,109,116), (1,49,5,53)(2,50,6,54)(3,51,7,55)(4,52,8,56)(9,117,13,113)(10,118,14,114)(11,119,15,115)(12,120,16,116)(17,32,21,28)(18,25,22,29)(19,26,23,30)(20,27,24,31)(33,95,37,91)(34,96,38,92)(35,89,39,93)(36,90,40,94)(41,87,45,83)(42,88,46,84)(43,81,47,85)(44,82,48,86)(57,77,61,73)(58,78,62,74)(59,79,63,75)(60,80,64,76)(65,104,69,100)(66,97,70,101)(67,98,71,102)(68,99,72,103)(105,126,109,122)(106,127,110,123)(107,128,111,124)(108,121,112,125), (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), (1,108,55,127)(2,111,56,122)(3,106,49,125)(4,109,50,128)(5,112,51,123)(6,107,52,126)(7,110,53,121)(8,105,54,124)(9,29,115,24)(10,32,116,19)(11,27,117,22)(12,30,118,17)(13,25,119,20)(14,28,120,23)(15,31,113,18)(16,26,114,21)(33,77,93,59)(34,80,94,62)(35,75,95,57)(36,78,96,60)(37,73,89,63)(38,76,90,58)(39,79,91,61)(40,74,92,64)(41,69,85,98)(42,72,86,101)(43,67,87,104)(44,70,88,99)(45,65,81,102)(46,68,82,97)(47,71,83,100)(48,66,84,103)>;
G:=Group( (1,100,61,29)(2,101,62,30)(3,102,63,31)(4,103,64,32)(5,104,57,25)(6,97,58,26)(7,98,59,27)(8,99,60,28)(9,91,83,127)(10,92,84,128)(11,93,85,121)(12,94,86,122)(13,95,87,123)(14,96,88,124)(15,89,81,125)(16,90,82,126)(17,56,72,80)(18,49,65,73)(19,50,66,74)(20,51,67,75)(21,52,68,76)(22,53,69,77)(23,54,70,78)(24,55,71,79)(33,41,110,117)(34,42,111,118)(35,43,112,119)(36,44,105,120)(37,45,106,113)(38,46,107,114)(39,47,108,115)(40,48,109,116), (1,49,5,53)(2,50,6,54)(3,51,7,55)(4,52,8,56)(9,117,13,113)(10,118,14,114)(11,119,15,115)(12,120,16,116)(17,32,21,28)(18,25,22,29)(19,26,23,30)(20,27,24,31)(33,95,37,91)(34,96,38,92)(35,89,39,93)(36,90,40,94)(41,87,45,83)(42,88,46,84)(43,81,47,85)(44,82,48,86)(57,77,61,73)(58,78,62,74)(59,79,63,75)(60,80,64,76)(65,104,69,100)(66,97,70,101)(67,98,71,102)(68,99,72,103)(105,126,109,122)(106,127,110,123)(107,128,111,124)(108,121,112,125), (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), (1,108,55,127)(2,111,56,122)(3,106,49,125)(4,109,50,128)(5,112,51,123)(6,107,52,126)(7,110,53,121)(8,105,54,124)(9,29,115,24)(10,32,116,19)(11,27,117,22)(12,30,118,17)(13,25,119,20)(14,28,120,23)(15,31,113,18)(16,26,114,21)(33,77,93,59)(34,80,94,62)(35,75,95,57)(36,78,96,60)(37,73,89,63)(38,76,90,58)(39,79,91,61)(40,74,92,64)(41,69,85,98)(42,72,86,101)(43,67,87,104)(44,70,88,99)(45,65,81,102)(46,68,82,97)(47,71,83,100)(48,66,84,103) );
G=PermutationGroup([[(1,100,61,29),(2,101,62,30),(3,102,63,31),(4,103,64,32),(5,104,57,25),(6,97,58,26),(7,98,59,27),(8,99,60,28),(9,91,83,127),(10,92,84,128),(11,93,85,121),(12,94,86,122),(13,95,87,123),(14,96,88,124),(15,89,81,125),(16,90,82,126),(17,56,72,80),(18,49,65,73),(19,50,66,74),(20,51,67,75),(21,52,68,76),(22,53,69,77),(23,54,70,78),(24,55,71,79),(33,41,110,117),(34,42,111,118),(35,43,112,119),(36,44,105,120),(37,45,106,113),(38,46,107,114),(39,47,108,115),(40,48,109,116)], [(1,49,5,53),(2,50,6,54),(3,51,7,55),(4,52,8,56),(9,117,13,113),(10,118,14,114),(11,119,15,115),(12,120,16,116),(17,32,21,28),(18,25,22,29),(19,26,23,30),(20,27,24,31),(33,95,37,91),(34,96,38,92),(35,89,39,93),(36,90,40,94),(41,87,45,83),(42,88,46,84),(43,81,47,85),(44,82,48,86),(57,77,61,73),(58,78,62,74),(59,79,63,75),(60,80,64,76),(65,104,69,100),(66,97,70,101),(67,98,71,102),(68,99,72,103),(105,126,109,122),(106,127,110,123),(107,128,111,124),(108,121,112,125)], [(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)], [(1,108,55,127),(2,111,56,122),(3,106,49,125),(4,109,50,128),(5,112,51,123),(6,107,52,126),(7,110,53,121),(8,105,54,124),(9,29,115,24),(10,32,116,19),(11,27,117,22),(12,30,118,17),(13,25,119,20),(14,28,120,23),(15,31,113,18),(16,26,114,21),(33,77,93,59),(34,80,94,62),(35,75,95,57),(36,78,96,60),(37,73,89,63),(38,76,90,58),(39,79,91,61),(40,74,92,64),(41,69,85,98),(42,72,86,101),(43,67,87,104),(44,70,88,99),(45,65,81,102),(46,68,82,97),(47,71,83,100),(48,66,84,103)]])
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 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | Q8 | D4 | SD16 |
| kernel | C42.58Q8 | C42⋊9C4 | C2×C4×C8 | C2×C4.Q8 | C4×C8 | C42 | C2×C8 | C2×C8 | C22×C4 | C2×C4 |
| # reps | 1 | 2 | 1 | 4 | 8 | 2 | 4 | 4 | 2 | 16 |
Matrix representation of C42.58Q8 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 1 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 2 | 0 | 0 |
| 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 |
| 0 | 5 | 7 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 6 | 3 | 0 | 0 |
| 0 | 16 | 11 | 0 | 0 |
| 0 | 0 | 0 | 9 | 3 |
| 0 | 0 | 0 | 1 | 8 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,1,0,0,0,15,1],[16,0,0,0,0,0,16,16,0,0,0,2,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,0,5,0,0,0,7,7,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,6,16,0,0,0,3,11,0,0,0,0,0,9,1,0,0,0,3,8] >;
C42.58Q8 in GAP, Magma, Sage, TeX
C_4^2._{58}Q_8 % in TeX
G:=Group("C4^2.58Q8"); // GroupNames label
G:=SmallGroup(128,576);
// by ID
G=gap.SmallGroup(128,576);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,100,2019,248]);
// 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*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations