direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7xC42:6C4, C42:6C28, M4(2):2C28, C28.27C42, C4:C4:3C28, (C4xC28):16C4, C4.1(C4xC28), C14.26C4wrC2, C28.51(C4:C4), (C2xC28).69Q8, (C2xC28).506D4, (C7xM4(2)):8C4, (C2xC42).7C14, C23.31(C7xD4), C42:C2.2C14, (C22xC14).151D4, (C2xM4(2)).6C14, C28.110(C22:C4), (C14xM4(2)).18C2, (C22xC28).570C22, C14.23(C2.C42), C4.2(C7xC4:C4), (C7xC4:C4):10C4, C2.3(C7xC4wrC2), (C2xC4xC28).30C2, C22.3(C7xC4:C4), (C2xC4).12(C7xQ8), (C2xC4).65(C2xC28), (C2xC4).142(C7xD4), C4.25(C7xC22:C4), (C2xC14).20(C4:C4), (C2xC28).260(C2xC4), C22.28(C7xC22:C4), C2.4(C7xC2.C42), (C2xC14).71(C22:C4), (C7xC42:C2).16C2, (C22xC4).103(C2xC14), SmallGroup(448,143)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7xC42:6C4
G = < a,b,c,d | a7=b4=c4=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=c-1 >
Subgroups: 170 in 110 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2xC4, C2xC4, C23, C14, C14, C14, C42, C42, C22:C4, C4:C4, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C28, C28, C2xC14, C2xC14, C2xC42, C42:C2, C2xM4(2), C56, C2xC28, C2xC28, C22xC14, C42:6C4, C4xC28, C4xC28, C7xC22:C4, C7xC4:C4, C2xC56, C7xM4(2), C7xM4(2), C22xC28, C22xC28, C2xC4xC28, C7xC42:C2, C14xM4(2), C7xC42:6C4
Quotients: C1, C2, C4, C22, C7, C2xC4, D4, Q8, C14, C42, C22:C4, C4:C4, C28, C2xC14, C2.C42, C4wrC2, C2xC28, C7xD4, C7xQ8, C42:6C4, C4xC28, C7xC22:C4, C7xC4:C4, C7xC2.C42, C7xC4wrC2, C7xC42:6C4
►On 112 points
Generators in S112
(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)
(1 83)(2 84)(3 78)(4 79)(5 80)(6 81)(7 82)(8 104 39 48)(9 105 40 49)(10 99 41 43)(11 100 42 44)(12 101 36 45)(13 102 37 46)(14 103 38 47)(15 28 30 107)(16 22 31 108)(17 23 32 109)(18 24 33 110)(19 25 34 111)(20 26 35 112)(21 27 29 106)(50 95)(51 96)(52 97)(53 98)(54 92)(55 93)(56 94)(57 68)(58 69)(59 70)(60 64)(61 65)(62 66)(63 67)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)(77 85)
(1 50 77 59)(2 51 71 60)(3 52 72 61)(4 53 73 62)(5 54 74 63)(6 55 75 57)(7 56 76 58)(8 27 39 106)(9 28 40 107)(10 22 41 108)(11 23 42 109)(12 24 36 110)(13 25 37 111)(14 26 38 112)(15 105 30 49)(16 99 31 43)(17 100 32 44)(18 101 33 45)(19 102 34 46)(20 103 35 47)(21 104 29 48)(64 84 96 86)(65 78 97 87)(66 79 98 88)(67 80 92 89)(68 81 93 90)(69 82 94 91)(70 83 95 85)
(1 21 85 8)(2 15 86 9)(3 16 87 10)(4 17 88 11)(5 18 89 12)(6 19 90 13)(7 20 91 14)(22 61 99 97)(23 62 100 98)(24 63 101 92)(25 57 102 93)(26 58 103 94)(27 59 104 95)(28 60 105 96)(29 83 39 77)(30 84 40 71)(31 78 41 72)(32 79 42 73)(33 80 36 74)(34 81 37 75)(35 82 38 76)(43 65 108 52)(44 66 109 53)(45 67 110 54)(46 68 111 55)(47 69 112 56)(48 70 106 50)(49 64 107 51)
G:=sub<Sym(112)| (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), (1,83)(2,84)(3,78)(4,79)(5,80)(6,81)(7,82)(8,104,39,48)(9,105,40,49)(10,99,41,43)(11,100,42,44)(12,101,36,45)(13,102,37,46)(14,103,38,47)(15,28,30,107)(16,22,31,108)(17,23,32,109)(18,24,33,110)(19,25,34,111)(20,26,35,112)(21,27,29,106)(50,95)(51,96)(52,97)(53,98)(54,92)(55,93)(56,94)(57,68)(58,69)(59,70)(60,64)(61,65)(62,66)(63,67)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,85), (1,50,77,59)(2,51,71,60)(3,52,72,61)(4,53,73,62)(5,54,74,63)(6,55,75,57)(7,56,76,58)(8,27,39,106)(9,28,40,107)(10,22,41,108)(11,23,42,109)(12,24,36,110)(13,25,37,111)(14,26,38,112)(15,105,30,49)(16,99,31,43)(17,100,32,44)(18,101,33,45)(19,102,34,46)(20,103,35,47)(21,104,29,48)(64,84,96,86)(65,78,97,87)(66,79,98,88)(67,80,92,89)(68,81,93,90)(69,82,94,91)(70,83,95,85), (1,21,85,8)(2,15,86,9)(3,16,87,10)(4,17,88,11)(5,18,89,12)(6,19,90,13)(7,20,91,14)(22,61,99,97)(23,62,100,98)(24,63,101,92)(25,57,102,93)(26,58,103,94)(27,59,104,95)(28,60,105,96)(29,83,39,77)(30,84,40,71)(31,78,41,72)(32,79,42,73)(33,80,36,74)(34,81,37,75)(35,82,38,76)(43,65,108,52)(44,66,109,53)(45,67,110,54)(46,68,111,55)(47,69,112,56)(48,70,106,50)(49,64,107,51)>;
G:=Group( (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), (1,83)(2,84)(3,78)(4,79)(5,80)(6,81)(7,82)(8,104,39,48)(9,105,40,49)(10,99,41,43)(11,100,42,44)(12,101,36,45)(13,102,37,46)(14,103,38,47)(15,28,30,107)(16,22,31,108)(17,23,32,109)(18,24,33,110)(19,25,34,111)(20,26,35,112)(21,27,29,106)(50,95)(51,96)(52,97)(53,98)(54,92)(55,93)(56,94)(57,68)(58,69)(59,70)(60,64)(61,65)(62,66)(63,67)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,85), (1,50,77,59)(2,51,71,60)(3,52,72,61)(4,53,73,62)(5,54,74,63)(6,55,75,57)(7,56,76,58)(8,27,39,106)(9,28,40,107)(10,22,41,108)(11,23,42,109)(12,24,36,110)(13,25,37,111)(14,26,38,112)(15,105,30,49)(16,99,31,43)(17,100,32,44)(18,101,33,45)(19,102,34,46)(20,103,35,47)(21,104,29,48)(64,84,96,86)(65,78,97,87)(66,79,98,88)(67,80,92,89)(68,81,93,90)(69,82,94,91)(70,83,95,85), (1,21,85,8)(2,15,86,9)(3,16,87,10)(4,17,88,11)(5,18,89,12)(6,19,90,13)(7,20,91,14)(22,61,99,97)(23,62,100,98)(24,63,101,92)(25,57,102,93)(26,58,103,94)(27,59,104,95)(28,60,105,96)(29,83,39,77)(30,84,40,71)(31,78,41,72)(32,79,42,73)(33,80,36,74)(34,81,37,75)(35,82,38,76)(43,65,108,52)(44,66,109,53)(45,67,110,54)(46,68,111,55)(47,69,112,56)(48,70,106,50)(49,64,107,51) );
G=PermutationGroup([[(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)], [(1,83),(2,84),(3,78),(4,79),(5,80),(6,81),(7,82),(8,104,39,48),(9,105,40,49),(10,99,41,43),(11,100,42,44),(12,101,36,45),(13,102,37,46),(14,103,38,47),(15,28,30,107),(16,22,31,108),(17,23,32,109),(18,24,33,110),(19,25,34,111),(20,26,35,112),(21,27,29,106),(50,95),(51,96),(52,97),(53,98),(54,92),(55,93),(56,94),(57,68),(58,69),(59,70),(60,64),(61,65),(62,66),(63,67),(71,86),(72,87),(73,88),(74,89),(75,90),(76,91),(77,85)], [(1,50,77,59),(2,51,71,60),(3,52,72,61),(4,53,73,62),(5,54,74,63),(6,55,75,57),(7,56,76,58),(8,27,39,106),(9,28,40,107),(10,22,41,108),(11,23,42,109),(12,24,36,110),(13,25,37,111),(14,26,38,112),(15,105,30,49),(16,99,31,43),(17,100,32,44),(18,101,33,45),(19,102,34,46),(20,103,35,47),(21,104,29,48),(64,84,96,86),(65,78,97,87),(66,79,98,88),(67,80,92,89),(68,81,93,90),(69,82,94,91),(70,83,95,85)], [(1,21,85,8),(2,15,86,9),(3,16,87,10),(4,17,88,11),(5,18,89,12),(6,19,90,13),(7,20,91,14),(22,61,99,97),(23,62,100,98),(24,63,101,92),(25,57,102,93),(26,58,103,94),(27,59,104,95),(28,60,105,96),(29,83,39,77),(30,84,40,71),(31,78,41,72),(32,79,42,73),(33,80,36,74),(34,81,37,75),(35,82,38,76),(43,65,108,52),(44,66,109,53),(45,67,110,54),(46,68,111,55),(47,69,112,56),(48,70,106,50),(49,64,107,51)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28CF | 28CG | ··· | 28DD | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C7 | C14 | C14 | C14 | C28 | C28 | C28 | D4 | Q8 | D4 | C4wrC2 | C7xD4 | C7xQ8 | C7xD4 | C7xC4wrC2 |
| kernel | C7xC42:6C4 | C2xC4xC28 | C7xC42:C2 | C14xM4(2) | C4xC28 | C7xC4:C4 | C7xM4(2) | C42:6C4 | C2xC42 | C42:C2 | C2xM4(2) | C42 | C4:C4 | M4(2) | C2xC28 | C2xC28 | C22xC14 | C14 | C2xC4 | C2xC4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 24 | 24 | 24 | 2 | 1 | 1 | 8 | 12 | 6 | 6 | 48 |
Matrix representation of C7xC42:6C4 ►in GL3(F113) generated by
| 1 | 0 | 0 |
| 0 | 49 | 0 |
| 0 | 0 | 49 |
| 1 | 0 | 0 |
| 0 | 112 | 0 |
| 0 | 0 | 98 |
| 1 | 0 | 0 |
| 0 | 15 | 0 |
| 0 | 0 | 98 |
| 98 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(113))| [1,0,0,0,49,0,0,0,49],[1,0,0,0,112,0,0,0,98],[1,0,0,0,15,0,0,0,98],[98,0,0,0,0,1,0,1,0] >;
C7xC42:6C4 in GAP, Magma, Sage, TeX
C_7\times C_4^2\rtimes_6C_4
% in TeX
G:=Group("C7xC4^2:6C4"); // GroupNames label
G:=SmallGroup(448,143);
// by ID
G=gap.SmallGroup(448,143);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,792,7059,248,14117]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=c^-1>;
// generators/relations