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, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22×C8, C22×Q8, C22.7C42, C2×C4×C8, C2×C4⋊C8, C2×C4×Q8, C42.327D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C22⋊C8, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C8○D4, C23.67C23, C2×C22⋊C8, (C22×C8)⋊C2, C8×Q8, C8⋊4Q8, C42.327D4
►Regular action on 128 points
Generators in S128
(1 80 27 71)(2 73 28 72)(3 74 29 65)(4 75 30 66)(5 76 31 67)(6 77 32 68)(7 78 25 69)(8 79 26 70)(9 86 109 122)(10 87 110 123)(11 88 111 124)(12 81 112 125)(13 82 105 126)(14 83 106 127)(15 84 107 128)(16 85 108 121)(17 104 63 54)(18 97 64 55)(19 98 57 56)(20 99 58 49)(21 100 59 50)(22 101 60 51)(23 102 61 52)(24 103 62 53)(33 48 117 94)(34 41 118 95)(35 42 119 96)(36 43 120 89)(37 44 113 90)(38 45 114 91)(39 46 115 92)(40 47 116 93)
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 90 13 94)(10 91 14 95)(11 92 15 96)(12 93 16 89)(25 63 29 59)(26 64 30 60)(27 57 31 61)(28 58 32 62)(33 86 37 82)(34 87 38 83)(35 88 39 84)(36 81 40 85)(41 110 45 106)(42 111 46 107)(43 112 47 108)(44 105 48 109)(49 68 53 72)(50 69 54 65)(51 70 55 66)(52 71 56 67)(73 99 77 103)(74 100 78 104)(75 101 79 97)(76 102 80 98)(113 126 117 122)(114 127 118 123)(115 128 119 124)(116 121 120 125)
(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 86 31 126)(2 120 32 40)(3 88 25 128)(4 114 26 34)(5 82 27 122)(6 116 28 36)(7 84 29 124)(8 118 30 38)(9 67 105 80)(10 101 106 55)(11 69 107 74)(12 103 108 49)(13 71 109 76)(14 97 110 51)(15 65 111 78)(16 99 112 53)(17 35 59 115)(18 123 60 83)(19 37 61 117)(20 125 62 85)(21 39 63 119)(22 127 64 87)(23 33 57 113)(24 121 58 81)(41 66 91 79)(42 100 92 54)(43 68 93 73)(44 102 94 56)(45 70 95 75)(46 104 96 50)(47 72 89 77)(48 98 90 52)
G:=sub<Sym(128)| (1,80,27,71)(2,73,28,72)(3,74,29,65)(4,75,30,66)(5,76,31,67)(6,77,32,68)(7,78,25,69)(8,79,26,70)(9,86,109,122)(10,87,110,123)(11,88,111,124)(12,81,112,125)(13,82,105,126)(14,83,106,127)(15,84,107,128)(16,85,108,121)(17,104,63,54)(18,97,64,55)(19,98,57,56)(20,99,58,49)(21,100,59,50)(22,101,60,51)(23,102,61,52)(24,103,62,53)(33,48,117,94)(34,41,118,95)(35,42,119,96)(36,43,120,89)(37,44,113,90)(38,45,114,91)(39,46,115,92)(40,47,116,93), (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,90,13,94)(10,91,14,95)(11,92,15,96)(12,93,16,89)(25,63,29,59)(26,64,30,60)(27,57,31,61)(28,58,32,62)(33,86,37,82)(34,87,38,83)(35,88,39,84)(36,81,40,85)(41,110,45,106)(42,111,46,107)(43,112,47,108)(44,105,48,109)(49,68,53,72)(50,69,54,65)(51,70,55,66)(52,71,56,67)(73,99,77,103)(74,100,78,104)(75,101,79,97)(76,102,80,98)(113,126,117,122)(114,127,118,123)(115,128,119,124)(116,121,120,125), (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), (1,86,31,126)(2,120,32,40)(3,88,25,128)(4,114,26,34)(5,82,27,122)(6,116,28,36)(7,84,29,124)(8,118,30,38)(9,67,105,80)(10,101,106,55)(11,69,107,74)(12,103,108,49)(13,71,109,76)(14,97,110,51)(15,65,111,78)(16,99,112,53)(17,35,59,115)(18,123,60,83)(19,37,61,117)(20,125,62,85)(21,39,63,119)(22,127,64,87)(23,33,57,113)(24,121,58,81)(41,66,91,79)(42,100,92,54)(43,68,93,73)(44,102,94,56)(45,70,95,75)(46,104,96,50)(47,72,89,77)(48,98,90,52)>;
G:=Group( (1,80,27,71)(2,73,28,72)(3,74,29,65)(4,75,30,66)(5,76,31,67)(6,77,32,68)(7,78,25,69)(8,79,26,70)(9,86,109,122)(10,87,110,123)(11,88,111,124)(12,81,112,125)(13,82,105,126)(14,83,106,127)(15,84,107,128)(16,85,108,121)(17,104,63,54)(18,97,64,55)(19,98,57,56)(20,99,58,49)(21,100,59,50)(22,101,60,51)(23,102,61,52)(24,103,62,53)(33,48,117,94)(34,41,118,95)(35,42,119,96)(36,43,120,89)(37,44,113,90)(38,45,114,91)(39,46,115,92)(40,47,116,93), (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,90,13,94)(10,91,14,95)(11,92,15,96)(12,93,16,89)(25,63,29,59)(26,64,30,60)(27,57,31,61)(28,58,32,62)(33,86,37,82)(34,87,38,83)(35,88,39,84)(36,81,40,85)(41,110,45,106)(42,111,46,107)(43,112,47,108)(44,105,48,109)(49,68,53,72)(50,69,54,65)(51,70,55,66)(52,71,56,67)(73,99,77,103)(74,100,78,104)(75,101,79,97)(76,102,80,98)(113,126,117,122)(114,127,118,123)(115,128,119,124)(116,121,120,125), (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), (1,86,31,126)(2,120,32,40)(3,88,25,128)(4,114,26,34)(5,82,27,122)(6,116,28,36)(7,84,29,124)(8,118,30,38)(9,67,105,80)(10,101,106,55)(11,69,107,74)(12,103,108,49)(13,71,109,76)(14,97,110,51)(15,65,111,78)(16,99,112,53)(17,35,59,115)(18,123,60,83)(19,37,61,117)(20,125,62,85)(21,39,63,119)(22,127,64,87)(23,33,57,113)(24,121,58,81)(41,66,91,79)(42,100,92,54)(43,68,93,73)(44,102,94,56)(45,70,95,75)(46,104,96,50)(47,72,89,77)(48,98,90,52) );
G=PermutationGroup([[(1,80,27,71),(2,73,28,72),(3,74,29,65),(4,75,30,66),(5,76,31,67),(6,77,32,68),(7,78,25,69),(8,79,26,70),(9,86,109,122),(10,87,110,123),(11,88,111,124),(12,81,112,125),(13,82,105,126),(14,83,106,127),(15,84,107,128),(16,85,108,121),(17,104,63,54),(18,97,64,55),(19,98,57,56),(20,99,58,49),(21,100,59,50),(22,101,60,51),(23,102,61,52),(24,103,62,53),(33,48,117,94),(34,41,118,95),(35,42,119,96),(36,43,120,89),(37,44,113,90),(38,45,114,91),(39,46,115,92),(40,47,116,93)], [(1,19,5,23),(2,20,6,24),(3,21,7,17),(4,22,8,18),(9,90,13,94),(10,91,14,95),(11,92,15,96),(12,93,16,89),(25,63,29,59),(26,64,30,60),(27,57,31,61),(28,58,32,62),(33,86,37,82),(34,87,38,83),(35,88,39,84),(36,81,40,85),(41,110,45,106),(42,111,46,107),(43,112,47,108),(44,105,48,109),(49,68,53,72),(50,69,54,65),(51,70,55,66),(52,71,56,67),(73,99,77,103),(74,100,78,104),(75,101,79,97),(76,102,80,98),(113,126,117,122),(114,127,118,123),(115,128,119,124),(116,121,120,125)], [(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)], [(1,86,31,126),(2,120,32,40),(3,88,25,128),(4,114,26,34),(5,82,27,122),(6,116,28,36),(7,84,29,124),(8,118,30,38),(9,67,105,80),(10,101,106,55),(11,69,107,74),(12,103,108,49),(13,71,109,76),(14,97,110,51),(15,65,111,78),(16,99,112,53),(17,35,59,115),(18,123,60,83),(19,37,61,117),(20,125,62,85),(21,39,63,119),(22,127,64,87),(23,33,57,113),(24,121,58,81),(41,66,91,79),(42,100,92,54),(43,68,93,73),(44,102,94,56),(45,70,95,75),(46,104,96,50),(47,72,89,77),(48,98,90,52)]])
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