direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7xC42:C2, C42:1C14, C4:C4:6C14, (C2xC28):9C4, (C2xC4):4C28, (C4xC28):2C2, C4.9(C2xC28), C28.46(C2xC4), C22:C4.3C14, C22.5(C2xC28), C23.6(C2xC14), C2.3(C22xC28), (C22xC4).4C14, C14.38(C4oD4), (C22xC28).14C2, (C2xC28).79C22, C14.31(C22xC4), (C2xC14).72C23, C22.6(C22xC14), (C22xC14).25C22, (C7xC4:C4):15C2, C2.1(C7xC4oD4), (C2xC14).22(C2xC4), (C2xC4).14(C2xC14), (C7xC22:C4).6C2, SmallGroup(224,152)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7xC42:C2
G = < a,b,c,d | a7=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >
Subgroups: 92 in 76 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2xC4, C2xC4, C23, C14, C14, C14, C42, C22:C4, C4:C4, C22xC4, C28, C28, C2xC14, C2xC14, C2xC14, C42:C2, C2xC28, C2xC28, C22xC14, C4xC28, C7xC22:C4, C7xC4:C4, C22xC28, C7xC42:C2
Quotients: C1, C2, C4, C22, C7, C2xC4, C23, C14, C22xC4, C4oD4, C28, C2xC14, C42:C2, C2xC28, C22xC14, C22xC28, C7xC4oD4, C7xC42:C2
►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 91 35 102)(2 85 29 103)(3 86 30 104)(4 87 31 105)(5 88 32 99)(6 89 33 100)(7 90 34 101)(8 75 18 57)(9 76 19 58)(10 77 20 59)(11 71 21 60)(12 72 15 61)(13 73 16 62)(14 74 17 63)(22 68 111 55)(23 69 112 56)(24 70 106 50)(25 64 107 51)(26 65 108 52)(27 66 109 53)(28 67 110 54)(36 79 48 92)(37 80 49 93)(38 81 43 94)(39 82 44 95)(40 83 45 96)(41 84 46 97)(42 78 47 98)
(1 50 47 74)(2 51 48 75)(3 52 49 76)(4 53 43 77)(5 54 44 71)(6 55 45 72)(7 56 46 73)(8 103 107 79)(9 104 108 80)(10 105 109 81)(11 99 110 82)(12 100 111 83)(13 101 112 84)(14 102 106 78)(15 89 22 96)(16 90 23 97)(17 91 24 98)(18 85 25 92)(19 86 26 93)(20 87 27 94)(21 88 28 95)(29 64 36 57)(30 65 37 58)(31 66 38 59)(32 67 39 60)(33 68 40 61)(34 69 41 62)(35 70 42 63)
(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 25)(9 26)(10 27)(11 28)(12 22)(13 23)(14 24)(15 111)(16 112)(17 106)(18 107)(19 108)(20 109)(21 110)(36 48)(37 49)(38 43)(39 44)(40 45)(41 46)(42 47)(50 70)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 75)(58 76)(59 77)(60 71)(61 72)(62 73)(63 74)(78 91)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(92 103)(93 104)(94 105)(95 99)(96 100)(97 101)(98 102)
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), (1,91,35,102)(2,85,29,103)(3,86,30,104)(4,87,31,105)(5,88,32,99)(6,89,33,100)(7,90,34,101)(8,75,18,57)(9,76,19,58)(10,77,20,59)(11,71,21,60)(12,72,15,61)(13,73,16,62)(14,74,17,63)(22,68,111,55)(23,69,112,56)(24,70,106,50)(25,64,107,51)(26,65,108,52)(27,66,109,53)(28,67,110,54)(36,79,48,92)(37,80,49,93)(38,81,43,94)(39,82,44,95)(40,83,45,96)(41,84,46,97)(42,78,47,98), (1,50,47,74)(2,51,48,75)(3,52,49,76)(4,53,43,77)(5,54,44,71)(6,55,45,72)(7,56,46,73)(8,103,107,79)(9,104,108,80)(10,105,109,81)(11,99,110,82)(12,100,111,83)(13,101,112,84)(14,102,106,78)(15,89,22,96)(16,90,23,97)(17,91,24,98)(18,85,25,92)(19,86,26,93)(20,87,27,94)(21,88,28,95)(29,64,36,57)(30,65,37,58)(31,66,38,59)(32,67,39,60)(33,68,40,61)(34,69,41,62)(35,70,42,63), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,25)(9,26)(10,27)(11,28)(12,22)(13,23)(14,24)(15,111)(16,112)(17,106)(18,107)(19,108)(20,109)(21,110)(36,48)(37,49)(38,43)(39,44)(40,45)(41,46)(42,47)(50,70)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,103)(93,104)(94,105)(95,99)(96,100)(97,101)(98,102)>;
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), (1,91,35,102)(2,85,29,103)(3,86,30,104)(4,87,31,105)(5,88,32,99)(6,89,33,100)(7,90,34,101)(8,75,18,57)(9,76,19,58)(10,77,20,59)(11,71,21,60)(12,72,15,61)(13,73,16,62)(14,74,17,63)(22,68,111,55)(23,69,112,56)(24,70,106,50)(25,64,107,51)(26,65,108,52)(27,66,109,53)(28,67,110,54)(36,79,48,92)(37,80,49,93)(38,81,43,94)(39,82,44,95)(40,83,45,96)(41,84,46,97)(42,78,47,98), (1,50,47,74)(2,51,48,75)(3,52,49,76)(4,53,43,77)(5,54,44,71)(6,55,45,72)(7,56,46,73)(8,103,107,79)(9,104,108,80)(10,105,109,81)(11,99,110,82)(12,100,111,83)(13,101,112,84)(14,102,106,78)(15,89,22,96)(16,90,23,97)(17,91,24,98)(18,85,25,92)(19,86,26,93)(20,87,27,94)(21,88,28,95)(29,64,36,57)(30,65,37,58)(31,66,38,59)(32,67,39,60)(33,68,40,61)(34,69,41,62)(35,70,42,63), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,25)(9,26)(10,27)(11,28)(12,22)(13,23)(14,24)(15,111)(16,112)(17,106)(18,107)(19,108)(20,109)(21,110)(36,48)(37,49)(38,43)(39,44)(40,45)(41,46)(42,47)(50,70)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,103)(93,104)(94,105)(95,99)(96,100)(97,101)(98,102) );
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)], [(1,91,35,102),(2,85,29,103),(3,86,30,104),(4,87,31,105),(5,88,32,99),(6,89,33,100),(7,90,34,101),(8,75,18,57),(9,76,19,58),(10,77,20,59),(11,71,21,60),(12,72,15,61),(13,73,16,62),(14,74,17,63),(22,68,111,55),(23,69,112,56),(24,70,106,50),(25,64,107,51),(26,65,108,52),(27,66,109,53),(28,67,110,54),(36,79,48,92),(37,80,49,93),(38,81,43,94),(39,82,44,95),(40,83,45,96),(41,84,46,97),(42,78,47,98)], [(1,50,47,74),(2,51,48,75),(3,52,49,76),(4,53,43,77),(5,54,44,71),(6,55,45,72),(7,56,46,73),(8,103,107,79),(9,104,108,80),(10,105,109,81),(11,99,110,82),(12,100,111,83),(13,101,112,84),(14,102,106,78),(15,89,22,96),(16,90,23,97),(17,91,24,98),(18,85,25,92),(19,86,26,93),(20,87,27,94),(21,88,28,95),(29,64,36,57),(30,65,37,58),(31,66,38,59),(32,67,39,60),(33,68,40,61),(34,69,41,62),(35,70,42,63)], [(1,35),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,25),(9,26),(10,27),(11,28),(12,22),(13,23),(14,24),(15,111),(16,112),(17,106),(18,107),(19,108),(20,109),(21,110),(36,48),(37,49),(38,43),(39,44),(40,45),(41,46),(42,47),(50,70),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,75),(58,76),(59,77),(60,71),(61,72),(62,73),(63,74),(78,91),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90),(92,103),(93,104),(94,105),(95,99),(96,100),(97,101),(98,102)]])
C7xC42:C2 is a maximal subgroup of
C42:Dic7 C28.2C42 (C2xC28).Q8 C28.(C2xQ8) C4:C4.233D14 C28.5C42 C28.45(C4:C4) C42.43D14 C42.187D14 C4:C4:36D14 C4.(C2xD28) C4:C4.236D14 (C2xC4).47D28 C42:4D14 (C2xD28):13C4 C42.87D14 C42.88D14 C42.89D14 C42.90D14 C42:7D14 C42.188D14 C42.91D14 C42:8D14 C42:9D14 C42.92D14 C42:10D14 C42.93D14 C42.94D14 C42.95D14 C42.96D14 C42.97D14 C42.98D14 C42.99D14 C42.100D14 C4oD4xC28
C7xC42:C2 is a maximal quotient of
C22:C4xC28 C4:C4xC28
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28CF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | C4oD4 | C7xC4oD4 |
| kernel | C7xC42:C2 | C4xC28 | C7xC22:C4 | C7xC4:C4 | C22xC28 | C2xC28 | C42:C2 | C42 | C22:C4 | C4:C4 | C22xC4 | C2xC4 | C14 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 6 | 12 | 12 | 12 | 6 | 48 | 4 | 24 |
Matrix representation of C7xC42:C2 ►in GL3(F29) generated by
| 1 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 17 | 0 | 0 |
| 0 | 1 | 27 |
| 0 | 1 | 28 |
| 1 | 0 | 0 |
| 0 | 17 | 0 |
| 0 | 0 | 17 |
| 1 | 0 | 0 |
| 0 | 28 | 0 |
| 0 | 28 | 1 |
G:=sub<GL(3,GF(29))| [1,0,0,0,16,0,0,0,16],[17,0,0,0,1,1,0,27,28],[1,0,0,0,17,0,0,0,17],[1,0,0,0,28,28,0,0,1] >;
C7xC42:C2 in GAP, Magma, Sage, TeX
C_7\times C_4^2\rtimes C_2
% in TeX
G:=Group("C7xC4^2:C2"); // GroupNames label
G:=SmallGroup(224,152);
// by ID
G=gap.SmallGroup(224,152);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,672,697,266]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,c*d=d*c>;
// generators/relations