p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.26Q8, C8⋊1(C4⋊C4), C8⋊C4⋊9C4, C2.1(C8⋊Q8), C4.4(C4⋊Q8), (C2×C8).10Q8, (C2×C8).108D4, C2.1(C8⋊3D4), C42⋊9C4.8C2, C42⋊8C4.9C2, C42.144(C2×C4), C2.1(C8.2D4), C23.756(C2×D4), (C22×C4).280D4, C22.30(C4⋊Q8), C2.8(C42⋊9C4), C22.30(C4⋊1D4), C22.66(C8⋊C22), (C22×C8).220C22, (C2×C42).259C22, C2.9(M4(2)⋊C4), (C22×C4).1346C23, C22.55(C8.C22), C4.36(C2×C4⋊C4), (C2×C8).63(C2×C4), (C2×C8⋊C4).4C2, (C2×C4.Q8).4C2, (C2×C4).48(C4⋊C4), (C2×C4).731(C2×D4), (C2×C4).196(C2×Q8), (C2×C2.D8).33C2, (C2×C4⋊C4).50C22, C22.105(C2×C4⋊C4), (C2×C4).545(C22×C4), SmallGroup(128,579)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.26Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c3 >
Subgroups: 236 in 128 conjugacy classes, 76 normal (18 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, C8⋊C4, C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C42⋊8C4, C42⋊9C4, C2×C8⋊C4, C2×C4.Q8, C2×C2.D8, C42.26Q8
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, C8⋊C22, C8.C22, C42⋊9C4, M4(2)⋊C4, C8⋊3D4, C8.2D4, C8⋊Q8, C42.26Q8
►Regular action on 128 points
Generators in S128
(1 104 57 29)(2 101 58 26)(3 98 59 31)(4 103 60 28)(5 100 61 25)(6 97 62 30)(7 102 63 27)(8 99 64 32)(9 107 114 127)(10 112 115 124)(11 109 116 121)(12 106 117 126)(13 111 118 123)(14 108 119 128)(15 105 120 125)(16 110 113 122)(17 70 74 56)(18 67 75 53)(19 72 76 50)(20 69 77 55)(21 66 78 52)(22 71 79 49)(23 68 80 54)(24 65 73 51)(33 41 93 85)(34 46 94 82)(35 43 95 87)(36 48 96 84)(37 45 89 81)(38 42 90 86)(39 47 91 83)(40 44 92 88)
(1 24 5 20)(2 17 6 21)(3 18 7 22)(4 19 8 23)(9 85 13 81)(10 86 14 82)(11 87 15 83)(12 88 16 84)(25 55 29 51)(26 56 30 52)(27 49 31 53)(28 50 32 54)(33 111 37 107)(34 112 38 108)(35 105 39 109)(36 106 40 110)(41 118 45 114)(42 119 46 115)(43 120 47 116)(44 113 48 117)(57 73 61 77)(58 74 62 78)(59 75 63 79)(60 76 64 80)(65 100 69 104)(66 101 70 97)(67 102 71 98)(68 103 72 99)(89 127 93 123)(90 128 94 124)(91 121 95 125)(92 122 96 126)
(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 91 22 127)(2 94 23 122)(3 89 24 125)(4 92 17 128)(5 95 18 123)(6 90 19 126)(7 93 20 121)(8 96 21 124)(9 29 83 49)(10 32 84 52)(11 27 85 55)(12 30 86 50)(13 25 87 53)(14 28 88 56)(15 31 81 51)(16 26 82 54)(33 77 109 63)(34 80 110 58)(35 75 111 61)(36 78 112 64)(37 73 105 59)(38 76 106 62)(39 79 107 57)(40 74 108 60)(41 69 116 102)(42 72 117 97)(43 67 118 100)(44 70 119 103)(45 65 120 98)(46 68 113 101)(47 71 114 104)(48 66 115 99)
G:=sub<Sym(128)| (1,104,57,29)(2,101,58,26)(3,98,59,31)(4,103,60,28)(5,100,61,25)(6,97,62,30)(7,102,63,27)(8,99,64,32)(9,107,114,127)(10,112,115,124)(11,109,116,121)(12,106,117,126)(13,111,118,123)(14,108,119,128)(15,105,120,125)(16,110,113,122)(17,70,74,56)(18,67,75,53)(19,72,76,50)(20,69,77,55)(21,66,78,52)(22,71,79,49)(23,68,80,54)(24,65,73,51)(33,41,93,85)(34,46,94,82)(35,43,95,87)(36,48,96,84)(37,45,89,81)(38,42,90,86)(39,47,91,83)(40,44,92,88), (1,24,5,20)(2,17,6,21)(3,18,7,22)(4,19,8,23)(9,85,13,81)(10,86,14,82)(11,87,15,83)(12,88,16,84)(25,55,29,51)(26,56,30,52)(27,49,31,53)(28,50,32,54)(33,111,37,107)(34,112,38,108)(35,105,39,109)(36,106,40,110)(41,118,45,114)(42,119,46,115)(43,120,47,116)(44,113,48,117)(57,73,61,77)(58,74,62,78)(59,75,63,79)(60,76,64,80)(65,100,69,104)(66,101,70,97)(67,102,71,98)(68,103,72,99)(89,127,93,123)(90,128,94,124)(91,121,95,125)(92,122,96,126), (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), (1,91,22,127)(2,94,23,122)(3,89,24,125)(4,92,17,128)(5,95,18,123)(6,90,19,126)(7,93,20,121)(8,96,21,124)(9,29,83,49)(10,32,84,52)(11,27,85,55)(12,30,86,50)(13,25,87,53)(14,28,88,56)(15,31,81,51)(16,26,82,54)(33,77,109,63)(34,80,110,58)(35,75,111,61)(36,78,112,64)(37,73,105,59)(38,76,106,62)(39,79,107,57)(40,74,108,60)(41,69,116,102)(42,72,117,97)(43,67,118,100)(44,70,119,103)(45,65,120,98)(46,68,113,101)(47,71,114,104)(48,66,115,99)>;
G:=Group( (1,104,57,29)(2,101,58,26)(3,98,59,31)(4,103,60,28)(5,100,61,25)(6,97,62,30)(7,102,63,27)(8,99,64,32)(9,107,114,127)(10,112,115,124)(11,109,116,121)(12,106,117,126)(13,111,118,123)(14,108,119,128)(15,105,120,125)(16,110,113,122)(17,70,74,56)(18,67,75,53)(19,72,76,50)(20,69,77,55)(21,66,78,52)(22,71,79,49)(23,68,80,54)(24,65,73,51)(33,41,93,85)(34,46,94,82)(35,43,95,87)(36,48,96,84)(37,45,89,81)(38,42,90,86)(39,47,91,83)(40,44,92,88), (1,24,5,20)(2,17,6,21)(3,18,7,22)(4,19,8,23)(9,85,13,81)(10,86,14,82)(11,87,15,83)(12,88,16,84)(25,55,29,51)(26,56,30,52)(27,49,31,53)(28,50,32,54)(33,111,37,107)(34,112,38,108)(35,105,39,109)(36,106,40,110)(41,118,45,114)(42,119,46,115)(43,120,47,116)(44,113,48,117)(57,73,61,77)(58,74,62,78)(59,75,63,79)(60,76,64,80)(65,100,69,104)(66,101,70,97)(67,102,71,98)(68,103,72,99)(89,127,93,123)(90,128,94,124)(91,121,95,125)(92,122,96,126), (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), (1,91,22,127)(2,94,23,122)(3,89,24,125)(4,92,17,128)(5,95,18,123)(6,90,19,126)(7,93,20,121)(8,96,21,124)(9,29,83,49)(10,32,84,52)(11,27,85,55)(12,30,86,50)(13,25,87,53)(14,28,88,56)(15,31,81,51)(16,26,82,54)(33,77,109,63)(34,80,110,58)(35,75,111,61)(36,78,112,64)(37,73,105,59)(38,76,106,62)(39,79,107,57)(40,74,108,60)(41,69,116,102)(42,72,117,97)(43,67,118,100)(44,70,119,103)(45,65,120,98)(46,68,113,101)(47,71,114,104)(48,66,115,99) );
G=PermutationGroup([[(1,104,57,29),(2,101,58,26),(3,98,59,31),(4,103,60,28),(5,100,61,25),(6,97,62,30),(7,102,63,27),(8,99,64,32),(9,107,114,127),(10,112,115,124),(11,109,116,121),(12,106,117,126),(13,111,118,123),(14,108,119,128),(15,105,120,125),(16,110,113,122),(17,70,74,56),(18,67,75,53),(19,72,76,50),(20,69,77,55),(21,66,78,52),(22,71,79,49),(23,68,80,54),(24,65,73,51),(33,41,93,85),(34,46,94,82),(35,43,95,87),(36,48,96,84),(37,45,89,81),(38,42,90,86),(39,47,91,83),(40,44,92,88)], [(1,24,5,20),(2,17,6,21),(3,18,7,22),(4,19,8,23),(9,85,13,81),(10,86,14,82),(11,87,15,83),(12,88,16,84),(25,55,29,51),(26,56,30,52),(27,49,31,53),(28,50,32,54),(33,111,37,107),(34,112,38,108),(35,105,39,109),(36,106,40,110),(41,118,45,114),(42,119,46,115),(43,120,47,116),(44,113,48,117),(57,73,61,77),(58,74,62,78),(59,75,63,79),(60,76,64,80),(65,100,69,104),(66,101,70,97),(67,102,71,98),(68,103,72,99),(89,127,93,123),(90,128,94,124),(91,121,95,125),(92,122,96,126)], [(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)], [(1,91,22,127),(2,94,23,122),(3,89,24,125),(4,92,17,128),(5,95,18,123),(6,90,19,126),(7,93,20,121),(8,96,21,124),(9,29,83,49),(10,32,84,52),(11,27,85,55),(12,30,86,50),(13,25,87,53),(14,28,88,56),(15,31,81,51),(16,26,82,54),(33,77,109,63),(34,80,110,58),(35,75,111,61),(36,78,112,64),(37,73,105,59),(38,76,106,62),(39,79,107,57),(40,74,108,60),(41,69,116,102),(42,72,117,97),(43,67,118,100),(44,70,119,103),(45,65,120,98),(46,68,113,101),(47,71,114,104),(48,66,115,99)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | Q8 | D4 | C8⋊C22 | C8.C22 |
| kernel | C42.26Q8 | C42⋊8C4 | C42⋊9C4 | C2×C8⋊C4 | C2×C4.Q8 | C2×C2.D8 | C8⋊C4 | C42 | C2×C8 | C2×C8 | C22×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 4 | 4 | 2 | 2 | 2 |
Matrix representation of C42.26Q8 ►in GL8(𝔽17)
| 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 13 | 13 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 | 15 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 4 | 4 | 0 |
| 0 | 0 | 0 | 0 | 8 | 2 | 4 | 13 |
| 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 9 | 15 | 13 | 4 |
| 0 | 0 | 0 | 0 | 0 | 2 | 2 | 13 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 15 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 12 | 13 | 16 |
| 0 | 0 | 0 | 0 | 8 | 3 | 14 | 4 |
| 0 | 0 | 0 | 0 | 1 | 12 | 16 | 4 |
| 0 | 0 | 0 | 0 | 7 | 8 | 1 | 2 |
G:=sub<GL(8,GF(17))| [4,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,13,0,15,0,0,0,0,1,13,0,15,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,16,8,0,0,0,0,16,0,4,2,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,15,2,1,0,0,0,0,0,13,2,0,0,0,0,0,1,4,13,0],[15,16,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,8,1,7,0,0,0,0,12,3,12,8,0,0,0,0,13,14,16,1,0,0,0,0,16,4,4,2] >;
C42.26Q8 in GAP, Magma, Sage, TeX
C_4^2._{26}Q_8 % in TeX
G:=Group("C4^2.26Q8"); // GroupNames label
G:=SmallGroup(128,579);
// by ID
G=gap.SmallGroup(128,579);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,723,100,2019,248]);
// 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,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations