p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2.3C43, C8.15C42, (C4×C8)⋊23C4, C8○3(C8⋊C4), C8⋊C4⋊18C4, C8○(C42⋊4C4), (C2×C4).40C42, C4.34(C2×C42), C8○(C2.C42), C42.294(C2×C4), C42⋊4C4.29C2, C22.18(C8○D4), C22.24(C2×C42), C2.3(C8○2M4(2)), C2.C42.26C4, (C2×C42).981C22, (C22×C8).574C22, C23.243(C22×C4), (C22×C4).1596C23, C8○(C2×C8⋊C4), (C2×C4×C8).61C2, C8⋊C4○(C22×C8), (C2×C8)○2(C8⋊C4), (C2×C8).190(C2×C4), (C2×C8⋊C4).42C2, (C2×C8)○(C42⋊4C4), (C2×C4).586(C22×C4), (C22×C4).375(C2×C4), (C22×C8)○(C42⋊4C4), (C2×C8)○(C2×C8⋊C4), SmallGroup(128,458)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2.C43
G = < a,b,c,d | a2=c4=d4=1, b4=a, ab=ba, dcd-1=ac=ca, ad=da, bc=cb, bd=db >
Subgroups: 188 in 164 conjugacy classes, 140 normal (8 characteristic)
C1, C2, C2, C4, C4, C4, C22, C8, C2×C4, C2×C4, C23, C42, C2×C8, C22×C4, C22×C4, C2.C42, C4×C8, C8⋊C4, C2×C42, C22×C8, C22×C8, C42⋊4C4, C2×C4×C8, C2×C8⋊C4, C2.C43
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C2×C42, C8○D4, C43, C8○2M4(2), C2.C43
►Regular action on 128 points
Generators in S128
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)
(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)
(1 125 109 37)(2 126 110 38)(3 127 111 39)(4 128 112 40)(5 121 105 33)(6 122 106 34)(7 123 107 35)(8 124 108 36)(9 119 47 31)(10 120 48 32)(11 113 41 25)(12 114 42 26)(13 115 43 27)(14 116 44 28)(15 117 45 29)(16 118 46 30)(17 93 79 61)(18 94 80 62)(19 95 73 63)(20 96 74 64)(21 89 75 57)(22 90 76 58)(23 91 77 59)(24 92 78 60)(49 97 81 67)(50 98 82 68)(51 99 83 69)(52 100 84 70)(53 101 85 71)(54 102 86 72)(55 103 87 65)(56 104 88 66)
(1 95 87 29)(2 96 88 30)(3 89 81 31)(4 90 82 32)(5 91 83 25)(6 92 84 26)(7 93 85 27)(8 94 86 28)(9 123 75 71)(10 124 76 72)(11 125 77 65)(12 126 78 66)(13 127 79 67)(14 128 80 68)(15 121 73 69)(16 122 74 70)(17 97 43 39)(18 98 44 40)(19 99 45 33)(20 100 46 34)(21 101 47 35)(22 102 48 36)(23 103 41 37)(24 104 42 38)(49 119 111 57)(50 120 112 58)(51 113 105 59)(52 114 106 60)(53 115 107 61)(54 116 108 62)(55 117 109 63)(56 118 110 64)
G:=sub<Sym(128)| (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128), (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), (1,125,109,37)(2,126,110,38)(3,127,111,39)(4,128,112,40)(5,121,105,33)(6,122,106,34)(7,123,107,35)(8,124,108,36)(9,119,47,31)(10,120,48,32)(11,113,41,25)(12,114,42,26)(13,115,43,27)(14,116,44,28)(15,117,45,29)(16,118,46,30)(17,93,79,61)(18,94,80,62)(19,95,73,63)(20,96,74,64)(21,89,75,57)(22,90,76,58)(23,91,77,59)(24,92,78,60)(49,97,81,67)(50,98,82,68)(51,99,83,69)(52,100,84,70)(53,101,85,71)(54,102,86,72)(55,103,87,65)(56,104,88,66), (1,95,87,29)(2,96,88,30)(3,89,81,31)(4,90,82,32)(5,91,83,25)(6,92,84,26)(7,93,85,27)(8,94,86,28)(9,123,75,71)(10,124,76,72)(11,125,77,65)(12,126,78,66)(13,127,79,67)(14,128,80,68)(15,121,73,69)(16,122,74,70)(17,97,43,39)(18,98,44,40)(19,99,45,33)(20,100,46,34)(21,101,47,35)(22,102,48,36)(23,103,41,37)(24,104,42,38)(49,119,111,57)(50,120,112,58)(51,113,105,59)(52,114,106,60)(53,115,107,61)(54,116,108,62)(55,117,109,63)(56,118,110,64)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128), (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), (1,125,109,37)(2,126,110,38)(3,127,111,39)(4,128,112,40)(5,121,105,33)(6,122,106,34)(7,123,107,35)(8,124,108,36)(9,119,47,31)(10,120,48,32)(11,113,41,25)(12,114,42,26)(13,115,43,27)(14,116,44,28)(15,117,45,29)(16,118,46,30)(17,93,79,61)(18,94,80,62)(19,95,73,63)(20,96,74,64)(21,89,75,57)(22,90,76,58)(23,91,77,59)(24,92,78,60)(49,97,81,67)(50,98,82,68)(51,99,83,69)(52,100,84,70)(53,101,85,71)(54,102,86,72)(55,103,87,65)(56,104,88,66), (1,95,87,29)(2,96,88,30)(3,89,81,31)(4,90,82,32)(5,91,83,25)(6,92,84,26)(7,93,85,27)(8,94,86,28)(9,123,75,71)(10,124,76,72)(11,125,77,65)(12,126,78,66)(13,127,79,67)(14,128,80,68)(15,121,73,69)(16,122,74,70)(17,97,43,39)(18,98,44,40)(19,99,45,33)(20,100,46,34)(21,101,47,35)(22,102,48,36)(23,103,41,37)(24,104,42,38)(49,119,111,57)(50,120,112,58)(51,113,105,59)(52,114,106,60)(53,115,107,61)(54,116,108,62)(55,117,109,63)(56,118,110,64) );
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128)], [(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)], [(1,125,109,37),(2,126,110,38),(3,127,111,39),(4,128,112,40),(5,121,105,33),(6,122,106,34),(7,123,107,35),(8,124,108,36),(9,119,47,31),(10,120,48,32),(11,113,41,25),(12,114,42,26),(13,115,43,27),(14,116,44,28),(15,117,45,29),(16,118,46,30),(17,93,79,61),(18,94,80,62),(19,95,73,63),(20,96,74,64),(21,89,75,57),(22,90,76,58),(23,91,77,59),(24,92,78,60),(49,97,81,67),(50,98,82,68),(51,99,83,69),(52,100,84,70),(53,101,85,71),(54,102,86,72),(55,103,87,65),(56,104,88,66)], [(1,95,87,29),(2,96,88,30),(3,89,81,31),(4,90,82,32),(5,91,83,25),(6,92,84,26),(7,93,85,27),(8,94,86,28),(9,123,75,71),(10,124,76,72),(11,125,77,65),(12,126,78,66),(13,127,79,67),(14,128,80,68),(15,121,73,69),(16,122,74,70),(17,97,43,39),(18,98,44,40),(19,99,45,33),(20,100,46,34),(21,101,47,35),(22,102,48,36),(23,103,41,37),(24,104,42,38),(49,119,111,57),(50,120,112,58),(51,113,105,59),(52,114,106,60),(53,115,107,61),(54,116,108,62),(55,117,109,63),(56,118,110,64)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AF | 8A | ··· | 8P | 8Q | ··· | 8AN |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8○D4 |
| kernel | C2.C43 | C42⋊4C4 | C2×C4×C8 | C2×C8⋊C4 | C2.C42 | C4×C8 | C8⋊C4 | C22 |
| # reps | 1 | 1 | 3 | 3 | 8 | 24 | 24 | 16 |
Matrix representation of C2.C43 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 8 | 7 |
| 0 | 0 | 8 | 9 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,0,16,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,4,0,0,0,0,8,8,0,0,7,9],[4,0,0,0,0,1,0,0,0,0,1,0,0,0,15,16] >;
C2.C43 in GAP, Magma, Sage, TeX
C_2.C_4^3
% in TeX
G:=Group("C2.C4^3"); // GroupNames label
G:=SmallGroup(128,458);
// by ID
G=gap.SmallGroup(128,458);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,120,723,184,248]);
// Polycyclic
G:=Group<a,b,c,d|a^2=c^4=d^4=1,b^4=a,a*b=b*a,d*c*d^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b>;
// generators/relations