direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×C8⋊C4, C8⋊4C42, C2.2C43, C43.1C2, (C4×C8)⋊22C4, C4.33(C2×C42), (C2×C42).28C4, (C2×C4).39C42, C2.2(C4×M4(2)), C42.338(C2×C4), (C2×C4).88M4(2), C22.23(C2×C42), (C22×C8).573C22, C23.242(C22×C4), C22.31(C2×M4(2)), (C22×C4).1595C23, (C2×C42).1143C22, (C2×C4×C8).60C2, C2.2(C2×C8⋊C4), C42○(C2×C8⋊C4), (C2×C8).243(C2×C4), (C2×C8⋊C4).41C2, (C2×C4).585(C22×C4), (C22×C4).503(C2×C4), (C2×C42)○(C2×C8⋊C4), SmallGroup(128,457)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×C8⋊C4
G = < a,b,c | a4=b8=c4=1, ab=ba, ac=ca, cbc-1=b5 >
Subgroups: 204 in 180 conjugacy classes, 156 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C42, C2×C8, C22×C4, C22×C4, C4×C8, C8⋊C4, C2×C42, C2×C42, C22×C8, C43, C2×C4×C8, C2×C8⋊C4, C4×C8⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, C8⋊C4, C2×C42, C2×M4(2), C43, C2×C8⋊C4, C4×M4(2), C4×C8⋊C4
►Regular action on 128 points
Generators in S128
(1 100 121 39)(2 101 122 40)(3 102 123 33)(4 103 124 34)(5 104 125 35)(6 97 126 36)(7 98 127 37)(8 99 128 38)(9 48 69 107)(10 41 70 108)(11 42 71 109)(12 43 72 110)(13 44 65 111)(14 45 66 112)(15 46 67 105)(16 47 68 106)(17 91 81 57)(18 92 82 58)(19 93 83 59)(20 94 84 60)(21 95 85 61)(22 96 86 62)(23 89 87 63)(24 90 88 64)(25 114 75 53)(26 115 76 54)(27 116 77 55)(28 117 78 56)(29 118 79 49)(30 119 80 50)(31 120 73 51)(32 113 74 52)
(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 81 16 31)(2 86 9 28)(3 83 10 25)(4 88 11 30)(5 85 12 27)(6 82 13 32)(7 87 14 29)(8 84 15 26)(17 68 73 121)(18 65 74 126)(19 70 75 123)(20 67 76 128)(21 72 77 125)(22 69 78 122)(23 66 79 127)(24 71 80 124)(33 93 108 53)(34 90 109 50)(35 95 110 55)(36 92 111 52)(37 89 112 49)(38 94 105 54)(39 91 106 51)(40 96 107 56)(41 114 102 59)(42 119 103 64)(43 116 104 61)(44 113 97 58)(45 118 98 63)(46 115 99 60)(47 120 100 57)(48 117 101 62)
G:=sub<Sym(128)| (1,100,121,39)(2,101,122,40)(3,102,123,33)(4,103,124,34)(5,104,125,35)(6,97,126,36)(7,98,127,37)(8,99,128,38)(9,48,69,107)(10,41,70,108)(11,42,71,109)(12,43,72,110)(13,44,65,111)(14,45,66,112)(15,46,67,105)(16,47,68,106)(17,91,81,57)(18,92,82,58)(19,93,83,59)(20,94,84,60)(21,95,85,61)(22,96,86,62)(23,89,87,63)(24,90,88,64)(25,114,75,53)(26,115,76,54)(27,116,77,55)(28,117,78,56)(29,118,79,49)(30,119,80,50)(31,120,73,51)(32,113,74,52), (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), (1,81,16,31)(2,86,9,28)(3,83,10,25)(4,88,11,30)(5,85,12,27)(6,82,13,32)(7,87,14,29)(8,84,15,26)(17,68,73,121)(18,65,74,126)(19,70,75,123)(20,67,76,128)(21,72,77,125)(22,69,78,122)(23,66,79,127)(24,71,80,124)(33,93,108,53)(34,90,109,50)(35,95,110,55)(36,92,111,52)(37,89,112,49)(38,94,105,54)(39,91,106,51)(40,96,107,56)(41,114,102,59)(42,119,103,64)(43,116,104,61)(44,113,97,58)(45,118,98,63)(46,115,99,60)(47,120,100,57)(48,117,101,62)>;
G:=Group( (1,100,121,39)(2,101,122,40)(3,102,123,33)(4,103,124,34)(5,104,125,35)(6,97,126,36)(7,98,127,37)(8,99,128,38)(9,48,69,107)(10,41,70,108)(11,42,71,109)(12,43,72,110)(13,44,65,111)(14,45,66,112)(15,46,67,105)(16,47,68,106)(17,91,81,57)(18,92,82,58)(19,93,83,59)(20,94,84,60)(21,95,85,61)(22,96,86,62)(23,89,87,63)(24,90,88,64)(25,114,75,53)(26,115,76,54)(27,116,77,55)(28,117,78,56)(29,118,79,49)(30,119,80,50)(31,120,73,51)(32,113,74,52), (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), (1,81,16,31)(2,86,9,28)(3,83,10,25)(4,88,11,30)(5,85,12,27)(6,82,13,32)(7,87,14,29)(8,84,15,26)(17,68,73,121)(18,65,74,126)(19,70,75,123)(20,67,76,128)(21,72,77,125)(22,69,78,122)(23,66,79,127)(24,71,80,124)(33,93,108,53)(34,90,109,50)(35,95,110,55)(36,92,111,52)(37,89,112,49)(38,94,105,54)(39,91,106,51)(40,96,107,56)(41,114,102,59)(42,119,103,64)(43,116,104,61)(44,113,97,58)(45,118,98,63)(46,115,99,60)(47,120,100,57)(48,117,101,62) );
G=PermutationGroup([[(1,100,121,39),(2,101,122,40),(3,102,123,33),(4,103,124,34),(5,104,125,35),(6,97,126,36),(7,98,127,37),(8,99,128,38),(9,48,69,107),(10,41,70,108),(11,42,71,109),(12,43,72,110),(13,44,65,111),(14,45,66,112),(15,46,67,105),(16,47,68,106),(17,91,81,57),(18,92,82,58),(19,93,83,59),(20,94,84,60),(21,95,85,61),(22,96,86,62),(23,89,87,63),(24,90,88,64),(25,114,75,53),(26,115,76,54),(27,116,77,55),(28,117,78,56),(29,118,79,49),(30,119,80,50),(31,120,73,51),(32,113,74,52)], [(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)], [(1,81,16,31),(2,86,9,28),(3,83,10,25),(4,88,11,30),(5,85,12,27),(6,82,13,32),(7,87,14,29),(8,84,15,26),(17,68,73,121),(18,65,74,126),(19,70,75,123),(20,67,76,128),(21,72,77,125),(22,69,78,122),(23,66,79,127),(24,71,80,124),(33,93,108,53),(34,90,109,50),(35,95,110,55),(36,92,111,52),(37,89,112,49),(38,94,105,54),(39,91,106,51),(40,96,107,56),(41,114,102,59),(42,119,103,64),(43,116,104,61),(44,113,97,58),(45,118,98,63),(46,115,99,60),(47,120,100,57),(48,117,101,62)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4X | 4Y | ··· | 4AN | 8A | ··· | 8AF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) |
| kernel | C4×C8⋊C4 | C43 | C2×C4×C8 | C2×C8⋊C4 | C4×C8 | C8⋊C4 | C2×C42 | C2×C4 |
| # reps | 1 | 1 | 2 | 4 | 16 | 32 | 8 | 16 |
Matrix representation of C4×C8⋊C4 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 8 | 5 |
| 0 | 0 | 5 | 9 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,13,0,0,0,0,8,5,0,0,5,9],[4,0,0,0,0,4,0,0,0,0,0,4,0,0,13,0] >;
C4×C8⋊C4 in GAP, Magma, Sage, TeX
C_4\times C_8\rtimes C_4
% in TeX
G:=Group("C4xC8:C4"); // GroupNames label
G:=SmallGroup(128,457);
// by ID
G=gap.SmallGroup(128,457);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,925,120,184,248]);
// Polycyclic
G:=Group<a,b,c|a^4=b^8=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations