p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.327D4, (C2×Q8)⋊6C8, (C2×C8)⋊10Q8, C2.5(C8×Q8), C4.52(C4⋊Q8), C2.5(C8⋊4Q8), C22.27(C4×Q8), C4.13(C22⋊C8), (C2×C4).48M4(2), (C22×Q8).24C4, C4.92(C4.4D4), C22.35(C8○D4), C4.120(C22⋊Q8), (C22×C8).59C22, C22.45(C22×C8), (C2×C42).330C22, C23.274(C22×C4), C22.56(C2×M4(2)), (C22×C4).1639C23, C22.7C42.9C2, C2.3(C23.67C23), (C2×C4×C8).24C2, (C2×C4⋊C8).31C2, (C2×C4⋊C4).62C4, (C2×C4×Q8).16C2, (C2×C4).23(C2×C8), C2.21(C2×C22⋊C8), (C2×C4).347(C2×Q8), (C2×C4).1546(C2×D4), (C2×C4).945(C4○D4), (C22×C4).129(C2×C4), (C2×C4).261(C22⋊C4), C2.5((C22×C8)⋊C2), C22.128(C2×C22⋊C4), SmallGroup(128,716)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.327D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=a2b-1c3 >
Subgroups: 220 in 146 conjugacy classes, 80 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×8], C22 [×3], C22 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×16], C2×C4 [×12], Q8 [×8], C23, C42 [×4], C42 [×4], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×10], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C4×C8 [×2], C4⋊C8 [×2], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×Q8 [×4], C22×C8 [×4], C22×Q8, C22.7C42 [×4], C2×C4×C8, C2×C4⋊C8, C2×C4×Q8, C42.327D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C22⋊C8 [×4], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C8○D4 [×2], C23.67C23, C2×C22⋊C8, (C22×C8)⋊C2, C8×Q8 [×2], C8⋊4Q8 [×2], C42.327D4
►Regular action on 128 points
Generators in S128
(1 61 53 71)(2 62 54 72)(3 63 55 65)(4 64 56 66)(5 57 49 67)(6 58 50 68)(7 59 51 69)(8 60 52 70)(9 113 92 121)(10 114 93 122)(11 115 94 123)(12 116 95 124)(13 117 96 125)(14 118 89 126)(15 119 90 127)(16 120 91 128)(17 28 104 78)(18 29 97 79)(19 30 98 80)(20 31 99 73)(21 32 100 74)(22 25 101 75)(23 26 102 76)(24 27 103 77)(33 47 85 108)(34 48 86 109)(35 41 87 110)(36 42 88 111)(37 43 81 112)(38 44 82 105)(39 45 83 106)(40 46 84 107)
(1 31 5 27)(2 32 6 28)(3 25 7 29)(4 26 8 30)(9 112 13 108)(10 105 14 109)(11 106 15 110)(12 107 16 111)(17 72 21 68)(18 65 22 69)(19 66 23 70)(20 67 24 71)(33 113 37 117)(34 114 38 118)(35 115 39 119)(36 116 40 120)(41 94 45 90)(42 95 46 91)(43 96 47 92)(44 89 48 93)(49 77 53 73)(50 78 54 74)(51 79 55 75)(52 80 56 76)(57 103 61 99)(58 104 62 100)(59 97 63 101)(60 98 64 102)(81 125 85 121)(82 126 86 122)(83 127 87 123)(84 128 88 124)
(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)
(1 114 49 126)(2 81 50 33)(3 116 51 128)(4 83 52 35)(5 118 53 122)(6 85 54 37)(7 120 55 124)(8 87 56 39)(9 100 96 17)(10 67 89 61)(11 102 90 19)(12 69 91 63)(13 104 92 21)(14 71 93 57)(15 98 94 23)(16 65 95 59)(18 42 101 107)(20 44 103 109)(22 46 97 111)(24 48 99 105)(25 40 79 88)(26 127 80 115)(27 34 73 82)(28 121 74 117)(29 36 75 84)(30 123 76 119)(31 38 77 86)(32 125 78 113)(41 66 106 60)(43 68 108 62)(45 70 110 64)(47 72 112 58)
G:=sub<Sym(128)| (1,61,53,71)(2,62,54,72)(3,63,55,65)(4,64,56,66)(5,57,49,67)(6,58,50,68)(7,59,51,69)(8,60,52,70)(9,113,92,121)(10,114,93,122)(11,115,94,123)(12,116,95,124)(13,117,96,125)(14,118,89,126)(15,119,90,127)(16,120,91,128)(17,28,104,78)(18,29,97,79)(19,30,98,80)(20,31,99,73)(21,32,100,74)(22,25,101,75)(23,26,102,76)(24,27,103,77)(33,47,85,108)(34,48,86,109)(35,41,87,110)(36,42,88,111)(37,43,81,112)(38,44,82,105)(39,45,83,106)(40,46,84,107), (1,31,5,27)(2,32,6,28)(3,25,7,29)(4,26,8,30)(9,112,13,108)(10,105,14,109)(11,106,15,110)(12,107,16,111)(17,72,21,68)(18,65,22,69)(19,66,23,70)(20,67,24,71)(33,113,37,117)(34,114,38,118)(35,115,39,119)(36,116,40,120)(41,94,45,90)(42,95,46,91)(43,96,47,92)(44,89,48,93)(49,77,53,73)(50,78,54,74)(51,79,55,75)(52,80,56,76)(57,103,61,99)(58,104,62,100)(59,97,63,101)(60,98,64,102)(81,125,85,121)(82,126,86,122)(83,127,87,123)(84,128,88,124), (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), (1,114,49,126)(2,81,50,33)(3,116,51,128)(4,83,52,35)(5,118,53,122)(6,85,54,37)(7,120,55,124)(8,87,56,39)(9,100,96,17)(10,67,89,61)(11,102,90,19)(12,69,91,63)(13,104,92,21)(14,71,93,57)(15,98,94,23)(16,65,95,59)(18,42,101,107)(20,44,103,109)(22,46,97,111)(24,48,99,105)(25,40,79,88)(26,127,80,115)(27,34,73,82)(28,121,74,117)(29,36,75,84)(30,123,76,119)(31,38,77,86)(32,125,78,113)(41,66,106,60)(43,68,108,62)(45,70,110,64)(47,72,112,58)>;
G:=Group( (1,61,53,71)(2,62,54,72)(3,63,55,65)(4,64,56,66)(5,57,49,67)(6,58,50,68)(7,59,51,69)(8,60,52,70)(9,113,92,121)(10,114,93,122)(11,115,94,123)(12,116,95,124)(13,117,96,125)(14,118,89,126)(15,119,90,127)(16,120,91,128)(17,28,104,78)(18,29,97,79)(19,30,98,80)(20,31,99,73)(21,32,100,74)(22,25,101,75)(23,26,102,76)(24,27,103,77)(33,47,85,108)(34,48,86,109)(35,41,87,110)(36,42,88,111)(37,43,81,112)(38,44,82,105)(39,45,83,106)(40,46,84,107), (1,31,5,27)(2,32,6,28)(3,25,7,29)(4,26,8,30)(9,112,13,108)(10,105,14,109)(11,106,15,110)(12,107,16,111)(17,72,21,68)(18,65,22,69)(19,66,23,70)(20,67,24,71)(33,113,37,117)(34,114,38,118)(35,115,39,119)(36,116,40,120)(41,94,45,90)(42,95,46,91)(43,96,47,92)(44,89,48,93)(49,77,53,73)(50,78,54,74)(51,79,55,75)(52,80,56,76)(57,103,61,99)(58,104,62,100)(59,97,63,101)(60,98,64,102)(81,125,85,121)(82,126,86,122)(83,127,87,123)(84,128,88,124), (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), (1,114,49,126)(2,81,50,33)(3,116,51,128)(4,83,52,35)(5,118,53,122)(6,85,54,37)(7,120,55,124)(8,87,56,39)(9,100,96,17)(10,67,89,61)(11,102,90,19)(12,69,91,63)(13,104,92,21)(14,71,93,57)(15,98,94,23)(16,65,95,59)(18,42,101,107)(20,44,103,109)(22,46,97,111)(24,48,99,105)(25,40,79,88)(26,127,80,115)(27,34,73,82)(28,121,74,117)(29,36,75,84)(30,123,76,119)(31,38,77,86)(32,125,78,113)(41,66,106,60)(43,68,108,62)(45,70,110,64)(47,72,112,58) );
G=PermutationGroup([(1,61,53,71),(2,62,54,72),(3,63,55,65),(4,64,56,66),(5,57,49,67),(6,58,50,68),(7,59,51,69),(8,60,52,70),(9,113,92,121),(10,114,93,122),(11,115,94,123),(12,116,95,124),(13,117,96,125),(14,118,89,126),(15,119,90,127),(16,120,91,128),(17,28,104,78),(18,29,97,79),(19,30,98,80),(20,31,99,73),(21,32,100,74),(22,25,101,75),(23,26,102,76),(24,27,103,77),(33,47,85,108),(34,48,86,109),(35,41,87,110),(36,42,88,111),(37,43,81,112),(38,44,82,105),(39,45,83,106),(40,46,84,107)], [(1,31,5,27),(2,32,6,28),(3,25,7,29),(4,26,8,30),(9,112,13,108),(10,105,14,109),(11,106,15,110),(12,107,16,111),(17,72,21,68),(18,65,22,69),(19,66,23,70),(20,67,24,71),(33,113,37,117),(34,114,38,118),(35,115,39,119),(36,116,40,120),(41,94,45,90),(42,95,46,91),(43,96,47,92),(44,89,48,93),(49,77,53,73),(50,78,54,74),(51,79,55,75),(52,80,56,76),(57,103,61,99),(58,104,62,100),(59,97,63,101),(60,98,64,102),(81,125,85,121),(82,126,86,122),(83,127,87,123),(84,128,88,124)], [(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)], [(1,114,49,126),(2,81,50,33),(3,116,51,128),(4,83,52,35),(5,118,53,122),(6,85,54,37),(7,120,55,124),(8,87,56,39),(9,100,96,17),(10,67,89,61),(11,102,90,19),(12,69,91,63),(13,104,92,21),(14,71,93,57),(15,98,94,23),(16,65,95,59),(18,42,101,107),(20,44,103,109),(22,46,97,111),(24,48,99,105),(25,40,79,88),(26,127,80,115),(27,34,73,82),(28,121,74,117),(29,36,75,84),(30,123,76,119),(31,38,77,86),(32,125,78,113),(41,66,106,60),(43,68,108,62),(45,70,110,64),(47,72,112,58)])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4X | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | M4(2) | C4○D4 | C8○D4 |
| kernel | C42.327D4 | C22.7C42 | C2×C4×C8 | C2×C4⋊C8 | C2×C4×Q8 | C2×C4⋊C4 | C22×Q8 | C2×Q8 | C42 | C2×C8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 6 | 2 | 16 | 4 | 4 | 4 | 4 | 8 |
Matrix representation of C42.327D4 ►in GL5(𝔽17)
| 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 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 8 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 14 | 0 | 0 |
| 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(5,GF(17))| [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],[13,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,4],[9,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,0,8,0,0,0,9,0],[4,0,0,0,0,0,3,14,0,0,0,14,14,0,0,0,0,0,1,0,0,0,0,0,16] >;
C42.327D4 in GAP, Magma, Sage, TeX
C_4^2._{327}D_4 % in TeX
G:=Group("C4^2.327D4"); // GroupNames label
G:=SmallGroup(128,716);
// by ID
G=gap.SmallGroup(128,716);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,100,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations