p-group, metabelian, nilpotent (class 2), monomial
Aliases: C43.3C2, C8⋊C4⋊12C4, (C2×C4).41C42, (C2×C42).31C4, C2.9(C4×M4(2)), C42.243(C2×C4), (C2×C4).58M4(2), C2.4(C42⋊4C4), C22.47(C2×C42), C4.68(C42⋊C2), C2.1(C42.6C4), (C22×C8).374C22, C23.248(C22×C4), (C2×C42).985C22, C22.35(C2×M4(2)), (C22×C4).1601C23, C22.46(C42⋊C2), C22.7C42.38C2, (C2×C8).128(C2×C4), (C2×C8⋊C4).21C2, (C2×C4).911(C4○D4), (C22×C4).432(C2×C4), (C2×C4).591(C22×C4), SmallGroup(128,477)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C43.C2
G = < a,b,c,d | a4=b4=c4=1, d2=c, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=bc2, cd=dc >
Subgroups: 188 in 136 conjugacy classes, 84 normal (8 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×12], C22, C22 [×6], C8 [×8], C2×C4 [×18], C2×C4 [×12], C23, C42 [×4], C42 [×12], C2×C8 [×8], C2×C8 [×8], C22×C4, C22×C4 [×6], C8⋊C4 [×8], C2×C42, C2×C42 [×6], C22×C8 [×4], C22.7C42 [×4], C43, C2×C8⋊C4 [×2], C43.C2
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×8], C22×C4 [×3], C4○D4 [×4], C2×C42, C42⋊C2 [×6], C2×M4(2) [×4], C42⋊4C4, C4×M4(2) [×2], C42.6C4 [×4], C43.C2
►Regular action on 128 points
Generators in S128
(1 119 103 89)(2 64 104 34)(3 113 97 91)(4 58 98 36)(5 115 99 93)(6 60 100 38)(7 117 101 95)(8 62 102 40)(9 32 126 54)(10 81 127 111)(11 26 128 56)(12 83 121 105)(13 28 122 50)(14 85 123 107)(15 30 124 52)(16 87 125 109)(17 63 47 33)(18 120 48 90)(19 57 41 35)(20 114 42 92)(21 59 43 37)(22 116 44 94)(23 61 45 39)(24 118 46 96)(25 69 55 79)(27 71 49 73)(29 65 51 75)(31 67 53 77)(66 108 76 86)(68 110 78 88)(70 112 80 82)(72 106 74 84)
(1 25 17 81)(2 30 18 86)(3 27 19 83)(4 32 20 88)(5 29 21 85)(6 26 22 82)(7 31 23 87)(8 28 24 84)(9 92 78 36)(10 89 79 33)(11 94 80 38)(12 91 73 35)(13 96 74 40)(14 93 75 37)(15 90 76 34)(16 95 77 39)(41 105 97 49)(42 110 98 54)(43 107 99 51)(44 112 100 56)(45 109 101 53)(46 106 102 50)(47 111 103 55)(48 108 104 52)(57 121 113 71)(58 126 114 68)(59 123 115 65)(60 128 116 70)(61 125 117 67)(62 122 118 72)(63 127 119 69)(64 124 120 66)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)
(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)
G:=sub<Sym(128)| (1,119,103,89)(2,64,104,34)(3,113,97,91)(4,58,98,36)(5,115,99,93)(6,60,100,38)(7,117,101,95)(8,62,102,40)(9,32,126,54)(10,81,127,111)(11,26,128,56)(12,83,121,105)(13,28,122,50)(14,85,123,107)(15,30,124,52)(16,87,125,109)(17,63,47,33)(18,120,48,90)(19,57,41,35)(20,114,42,92)(21,59,43,37)(22,116,44,94)(23,61,45,39)(24,118,46,96)(25,69,55,79)(27,71,49,73)(29,65,51,75)(31,67,53,77)(66,108,76,86)(68,110,78,88)(70,112,80,82)(72,106,74,84), (1,25,17,81)(2,30,18,86)(3,27,19,83)(4,32,20,88)(5,29,21,85)(6,26,22,82)(7,31,23,87)(8,28,24,84)(9,92,78,36)(10,89,79,33)(11,94,80,38)(12,91,73,35)(13,96,74,40)(14,93,75,37)(15,90,76,34)(16,95,77,39)(41,105,97,49)(42,110,98,54)(43,107,99,51)(44,112,100,56)(45,109,101,53)(46,106,102,50)(47,111,103,55)(48,108,104,52)(57,121,113,71)(58,126,114,68)(59,123,115,65)(60,128,116,70)(61,125,117,67)(62,122,118,72)(63,127,119,69)(64,124,120,66), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128), (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)>;
G:=Group( (1,119,103,89)(2,64,104,34)(3,113,97,91)(4,58,98,36)(5,115,99,93)(6,60,100,38)(7,117,101,95)(8,62,102,40)(9,32,126,54)(10,81,127,111)(11,26,128,56)(12,83,121,105)(13,28,122,50)(14,85,123,107)(15,30,124,52)(16,87,125,109)(17,63,47,33)(18,120,48,90)(19,57,41,35)(20,114,42,92)(21,59,43,37)(22,116,44,94)(23,61,45,39)(24,118,46,96)(25,69,55,79)(27,71,49,73)(29,65,51,75)(31,67,53,77)(66,108,76,86)(68,110,78,88)(70,112,80,82)(72,106,74,84), (1,25,17,81)(2,30,18,86)(3,27,19,83)(4,32,20,88)(5,29,21,85)(6,26,22,82)(7,31,23,87)(8,28,24,84)(9,92,78,36)(10,89,79,33)(11,94,80,38)(12,91,73,35)(13,96,74,40)(14,93,75,37)(15,90,76,34)(16,95,77,39)(41,105,97,49)(42,110,98,54)(43,107,99,51)(44,112,100,56)(45,109,101,53)(46,106,102,50)(47,111,103,55)(48,108,104,52)(57,121,113,71)(58,126,114,68)(59,123,115,65)(60,128,116,70)(61,125,117,67)(62,122,118,72)(63,127,119,69)(64,124,120,66), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128), (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) );
G=PermutationGroup([(1,119,103,89),(2,64,104,34),(3,113,97,91),(4,58,98,36),(5,115,99,93),(6,60,100,38),(7,117,101,95),(8,62,102,40),(9,32,126,54),(10,81,127,111),(11,26,128,56),(12,83,121,105),(13,28,122,50),(14,85,123,107),(15,30,124,52),(16,87,125,109),(17,63,47,33),(18,120,48,90),(19,57,41,35),(20,114,42,92),(21,59,43,37),(22,116,44,94),(23,61,45,39),(24,118,46,96),(25,69,55,79),(27,71,49,73),(29,65,51,75),(31,67,53,77),(66,108,76,86),(68,110,78,88),(70,112,80,82),(72,106,74,84)], [(1,25,17,81),(2,30,18,86),(3,27,19,83),(4,32,20,88),(5,29,21,85),(6,26,22,82),(7,31,23,87),(8,28,24,84),(9,92,78,36),(10,89,79,33),(11,94,80,38),(12,91,73,35),(13,96,74,40),(14,93,75,37),(15,90,76,34),(16,95,77,39),(41,105,97,49),(42,110,98,54),(43,107,99,51),(44,112,100,56),(45,109,101,53),(46,106,102,50),(47,111,103,55),(48,108,104,52),(57,121,113,71),(58,126,114,68),(59,123,115,65),(60,128,116,70),(61,125,117,67),(62,122,118,72),(63,127,119,69),(64,124,120,66)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120),(121,123,125,127),(122,124,126,128)], [(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)])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AF | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | M4(2) | C4○D4 |
| kernel | C43.C2 | C22.7C42 | C43 | C2×C8⋊C4 | C8⋊C4 | C2×C42 | C2×C4 | C2×C4 |
| # reps | 1 | 4 | 1 | 2 | 16 | 8 | 16 | 8 |
Matrix representation of C43.C2 ►in GL5(𝔽17)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 6 | 11 | 0 | 0 |
| 0 | 11 | 11 | 0 | 0 |
| 0 | 0 | 0 | 8 | 14 |
| 0 | 0 | 0 | 3 | 9 |
G:=sub<GL(5,GF(17))| [4,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,6,11,0,0,0,11,11,0,0,0,0,0,8,3,0,0,0,14,9] >;
C43.C2 in GAP, Magma, Sage, TeX
C_4^3.C_2
% in TeX
G:=Group("C4^3.C2"); // GroupNames label
G:=SmallGroup(128,477);
// by ID
G=gap.SmallGroup(128,477);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,58,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b*c^2,c*d=d*c>;
// generators/relations