direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4×C4.Q8, C8⋊3C42, C42.52Q8, (C4×C8)⋊20C4, C4.3(C4×Q8), C2.4(C4×SD16), C4.23(C2×C42), C22.95(C4×D4), C42.316(C2×C4), (C2×C4).130SD16, C23.734(C2×D4), (C22×C4).814D4, C4○4(C22.4Q16), C22.38(C4○D8), C22.43(C2×SD16), C4.47(C42⋊C2), C22.4Q16.54C2, (C22×C8).561C22, (C22×C4).1315C23, (C2×C42).1051C22, C2.3(C23.25D4), (C2×C4×C8).53C2, (C4×C4⋊C4).9C2, C2.13(C4×C4⋊C4), C2.3(C2×C4.Q8), C4⋊C4.146(C2×C4), (C2×C8).205(C2×C4), C22.58(C2×C4⋊C4), (C2×C4).184(C2×Q8), (C2×C4.Q8).35C2, (C2×C4).164(C4⋊C4), (C2×C4).546(C4○D4), (C2×C4⋊C4).750C22, (C2×C4).356(C22×C4), (C2×C4)○2(C22.4Q16), SmallGroup(128,506)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×C4.Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Subgroups: 228 in 140 conjugacy classes, 92 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4×C8, C4.Q8, C2×C42, C2×C42, C2×C4⋊C4, C22×C8, C22.4Q16, C4×C4⋊C4, C2×C4×C8, C2×C4.Q8, C4×C4.Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, C4.Q8, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×SD16, C4○D8, C4×C4⋊C4, C2×C4.Q8, C23.25D4, C4×SD16, C4×C4.Q8
►Regular action on 128 points
Generators in S128
(1 18 43 25)(2 19 44 26)(3 20 45 27)(4 21 46 28)(5 22 47 29)(6 23 48 30)(7 24 41 31)(8 17 42 32)(9 97 59 81)(10 98 60 82)(11 99 61 83)(12 100 62 84)(13 101 63 85)(14 102 64 86)(15 103 57 87)(16 104 58 88)(33 51 89 124)(34 52 90 125)(35 53 91 126)(36 54 92 127)(37 55 93 128)(38 56 94 121)(39 49 95 122)(40 50 96 123)(65 112 80 120)(66 105 73 113)(67 106 74 114)(68 107 75 115)(69 108 76 116)(70 109 77 117)(71 110 78 118)(72 111 79 119)
(1 75 5 79)(2 76 6 80)(3 77 7 73)(4 78 8 74)(9 51 13 55)(10 52 14 56)(11 53 15 49)(12 54 16 50)(17 114 21 118)(18 115 22 119)(19 116 23 120)(20 117 24 113)(25 107 29 111)(26 108 30 112)(27 109 31 105)(28 110 32 106)(33 85 37 81)(34 86 38 82)(35 87 39 83)(36 88 40 84)(41 66 45 70)(42 67 46 71)(43 68 47 72)(44 69 48 65)(57 122 61 126)(58 123 62 127)(59 124 63 128)(60 125 64 121)(89 101 93 97)(90 102 94 98)(91 103 95 99)(92 104 96 100)
(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 64 73 127)(2 59 74 122)(3 62 75 125)(4 57 76 128)(5 60 77 123)(6 63 78 126)(7 58 79 121)(8 61 80 124)(9 67 49 44)(10 70 50 47)(11 65 51 42)(12 68 52 45)(13 71 53 48)(14 66 54 43)(15 69 55 46)(16 72 56 41)(17 83 120 33)(18 86 113 36)(19 81 114 39)(20 84 115 34)(21 87 116 37)(22 82 117 40)(23 85 118 35)(24 88 119 38)(25 102 105 92)(26 97 106 95)(27 100 107 90)(28 103 108 93)(29 98 109 96)(30 101 110 91)(31 104 111 94)(32 99 112 89)
G:=sub<Sym(128)| (1,18,43,25)(2,19,44,26)(3,20,45,27)(4,21,46,28)(5,22,47,29)(6,23,48,30)(7,24,41,31)(8,17,42,32)(9,97,59,81)(10,98,60,82)(11,99,61,83)(12,100,62,84)(13,101,63,85)(14,102,64,86)(15,103,57,87)(16,104,58,88)(33,51,89,124)(34,52,90,125)(35,53,91,126)(36,54,92,127)(37,55,93,128)(38,56,94,121)(39,49,95,122)(40,50,96,123)(65,112,80,120)(66,105,73,113)(67,106,74,114)(68,107,75,115)(69,108,76,116)(70,109,77,117)(71,110,78,118)(72,111,79,119), (1,75,5,79)(2,76,6,80)(3,77,7,73)(4,78,8,74)(9,51,13,55)(10,52,14,56)(11,53,15,49)(12,54,16,50)(17,114,21,118)(18,115,22,119)(19,116,23,120)(20,117,24,113)(25,107,29,111)(26,108,30,112)(27,109,31,105)(28,110,32,106)(33,85,37,81)(34,86,38,82)(35,87,39,83)(36,88,40,84)(41,66,45,70)(42,67,46,71)(43,68,47,72)(44,69,48,65)(57,122,61,126)(58,123,62,127)(59,124,63,128)(60,125,64,121)(89,101,93,97)(90,102,94,98)(91,103,95,99)(92,104,96,100), (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), (1,64,73,127)(2,59,74,122)(3,62,75,125)(4,57,76,128)(5,60,77,123)(6,63,78,126)(7,58,79,121)(8,61,80,124)(9,67,49,44)(10,70,50,47)(11,65,51,42)(12,68,52,45)(13,71,53,48)(14,66,54,43)(15,69,55,46)(16,72,56,41)(17,83,120,33)(18,86,113,36)(19,81,114,39)(20,84,115,34)(21,87,116,37)(22,82,117,40)(23,85,118,35)(24,88,119,38)(25,102,105,92)(26,97,106,95)(27,100,107,90)(28,103,108,93)(29,98,109,96)(30,101,110,91)(31,104,111,94)(32,99,112,89)>;
G:=Group( (1,18,43,25)(2,19,44,26)(3,20,45,27)(4,21,46,28)(5,22,47,29)(6,23,48,30)(7,24,41,31)(8,17,42,32)(9,97,59,81)(10,98,60,82)(11,99,61,83)(12,100,62,84)(13,101,63,85)(14,102,64,86)(15,103,57,87)(16,104,58,88)(33,51,89,124)(34,52,90,125)(35,53,91,126)(36,54,92,127)(37,55,93,128)(38,56,94,121)(39,49,95,122)(40,50,96,123)(65,112,80,120)(66,105,73,113)(67,106,74,114)(68,107,75,115)(69,108,76,116)(70,109,77,117)(71,110,78,118)(72,111,79,119), (1,75,5,79)(2,76,6,80)(3,77,7,73)(4,78,8,74)(9,51,13,55)(10,52,14,56)(11,53,15,49)(12,54,16,50)(17,114,21,118)(18,115,22,119)(19,116,23,120)(20,117,24,113)(25,107,29,111)(26,108,30,112)(27,109,31,105)(28,110,32,106)(33,85,37,81)(34,86,38,82)(35,87,39,83)(36,88,40,84)(41,66,45,70)(42,67,46,71)(43,68,47,72)(44,69,48,65)(57,122,61,126)(58,123,62,127)(59,124,63,128)(60,125,64,121)(89,101,93,97)(90,102,94,98)(91,103,95,99)(92,104,96,100), (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), (1,64,73,127)(2,59,74,122)(3,62,75,125)(4,57,76,128)(5,60,77,123)(6,63,78,126)(7,58,79,121)(8,61,80,124)(9,67,49,44)(10,70,50,47)(11,65,51,42)(12,68,52,45)(13,71,53,48)(14,66,54,43)(15,69,55,46)(16,72,56,41)(17,83,120,33)(18,86,113,36)(19,81,114,39)(20,84,115,34)(21,87,116,37)(22,82,117,40)(23,85,118,35)(24,88,119,38)(25,102,105,92)(26,97,106,95)(27,100,107,90)(28,103,108,93)(29,98,109,96)(30,101,110,91)(31,104,111,94)(32,99,112,89) );
G=PermutationGroup([[(1,18,43,25),(2,19,44,26),(3,20,45,27),(4,21,46,28),(5,22,47,29),(6,23,48,30),(7,24,41,31),(8,17,42,32),(9,97,59,81),(10,98,60,82),(11,99,61,83),(12,100,62,84),(13,101,63,85),(14,102,64,86),(15,103,57,87),(16,104,58,88),(33,51,89,124),(34,52,90,125),(35,53,91,126),(36,54,92,127),(37,55,93,128),(38,56,94,121),(39,49,95,122),(40,50,96,123),(65,112,80,120),(66,105,73,113),(67,106,74,114),(68,107,75,115),(69,108,76,116),(70,109,77,117),(71,110,78,118),(72,111,79,119)], [(1,75,5,79),(2,76,6,80),(3,77,7,73),(4,78,8,74),(9,51,13,55),(10,52,14,56),(11,53,15,49),(12,54,16,50),(17,114,21,118),(18,115,22,119),(19,116,23,120),(20,117,24,113),(25,107,29,111),(26,108,30,112),(27,109,31,105),(28,110,32,106),(33,85,37,81),(34,86,38,82),(35,87,39,83),(36,88,40,84),(41,66,45,70),(42,67,46,71),(43,68,47,72),(44,69,48,65),(57,122,61,126),(58,123,62,127),(59,124,63,128),(60,125,64,121),(89,101,93,97),(90,102,94,98),(91,103,95,99),(92,104,96,100)], [(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)], [(1,64,73,127),(2,59,74,122),(3,62,75,125),(4,57,76,128),(5,60,77,123),(6,63,78,126),(7,58,79,121),(8,61,80,124),(9,67,49,44),(10,70,50,47),(11,65,51,42),(12,68,52,45),(13,71,53,48),(14,66,54,43),(15,69,55,46),(16,72,56,41),(17,83,120,33),(18,86,113,36),(19,81,114,39),(20,84,115,34),(21,87,116,37),(22,82,117,40),(23,85,118,35),(24,88,119,38),(25,102,105,92),(26,97,106,95),(27,100,107,90),(28,103,108,93),(29,98,109,96),(30,101,110,91),(31,104,111,94),(32,99,112,89)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AF | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | Q8 | D4 | SD16 | C4○D4 | C4○D8 |
| kernel | C4×C4.Q8 | C22.4Q16 | C4×C4⋊C4 | C2×C4×C8 | C2×C4.Q8 | C4×C8 | C4.Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 2 | 1 | 2 | 8 | 16 | 2 | 2 | 8 | 4 | 8 |
Matrix representation of C4×C4.Q8 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 5 | 12 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[16,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0],[16,0,0,0,0,16,0,0,0,0,12,5,0,0,12,12],[13,0,0,0,0,4,0,0,0,0,16,0,0,0,0,1] >;
C4×C4.Q8 in GAP, Magma, Sage, TeX
C_4\times C_4.Q_8
% in TeX
G:=Group("C4xC4.Q8"); // GroupNames label
G:=SmallGroup(128,506);
// by ID
G=gap.SmallGroup(128,506);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,142,1018,248,1411,172]);
// 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,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations