direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C4⋊1D4, C28⋊6D4, C42⋊6C14, C4⋊1(C7×D4), (C4×C28)⋊13C2, (C2×D4)⋊3C14, C2.9(D4×C14), (D4×C14)⋊12C2, C14.72(C2×D4), C23.4(C2×C14), (C2×C14).82C23, (C2×C28).125C22, (C22×C14).4C22, C22.17(C22×C14), (C2×C4).23(C2×C14), SmallGroup(224,162)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4⋊1D4
G = < a,b,c,d | a7=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 180 in 108 conjugacy classes, 52 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, D4, C23, C14, C14, C42, C2×D4, C28, C2×C14, C2×C14, C4⋊1D4, C2×C28, C7×D4, C22×C14, C4×C28, D4×C14, C7×C4⋊1D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C4⋊1D4, C7×D4, C22×C14, D4×C14, C7×C4⋊1D4
►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)
(1 106 43 9)(2 107 44 10)(3 108 45 11)(4 109 46 12)(5 110 47 13)(6 111 48 14)(7 112 49 8)(15 35 24 39)(16 29 25 40)(17 30 26 41)(18 31 27 42)(19 32 28 36)(20 33 22 37)(21 34 23 38)(50 91 63 78)(51 85 57 79)(52 86 58 80)(53 87 59 81)(54 88 60 82)(55 89 61 83)(56 90 62 84)(64 104 76 92)(65 105 77 93)(66 99 71 94)(67 100 72 95)(68 101 73 96)(69 102 74 97)(70 103 75 98)
(1 63 35 66)(2 57 29 67)(3 58 30 68)(4 59 31 69)(5 60 32 70)(6 61 33 64)(7 62 34 65)(8 90 21 93)(9 91 15 94)(10 85 16 95)(11 86 17 96)(12 87 18 97)(13 88 19 98)(14 89 20 92)(22 104 111 83)(23 105 112 84)(24 99 106 78)(25 100 107 79)(26 101 108 80)(27 102 109 81)(28 103 110 82)(36 75 47 54)(37 76 48 55)(38 77 49 56)(39 71 43 50)(40 72 44 51)(41 73 45 52)(42 74 46 53)
(1 91)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 62)(9 63)(10 57)(11 58)(12 59)(13 60)(14 61)(15 66)(16 67)(17 68)(18 69)(19 70)(20 64)(21 65)(22 76)(23 77)(24 71)(25 72)(26 73)(27 74)(28 75)(29 95)(30 96)(31 97)(32 98)(33 92)(34 93)(35 94)(36 103)(37 104)(38 105)(39 99)(40 100)(41 101)(42 102)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)
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), (1,106,43,9)(2,107,44,10)(3,108,45,11)(4,109,46,12)(5,110,47,13)(6,111,48,14)(7,112,49,8)(15,35,24,39)(16,29,25,40)(17,30,26,41)(18,31,27,42)(19,32,28,36)(20,33,22,37)(21,34,23,38)(50,91,63,78)(51,85,57,79)(52,86,58,80)(53,87,59,81)(54,88,60,82)(55,89,61,83)(56,90,62,84)(64,104,76,92)(65,105,77,93)(66,99,71,94)(67,100,72,95)(68,101,73,96)(69,102,74,97)(70,103,75,98), (1,63,35,66)(2,57,29,67)(3,58,30,68)(4,59,31,69)(5,60,32,70)(6,61,33,64)(7,62,34,65)(8,90,21,93)(9,91,15,94)(10,85,16,95)(11,86,17,96)(12,87,18,97)(13,88,19,98)(14,89,20,92)(22,104,111,83)(23,105,112,84)(24,99,106,78)(25,100,107,79)(26,101,108,80)(27,102,109,81)(28,103,110,82)(36,75,47,54)(37,76,48,55)(38,77,49,56)(39,71,43,50)(40,72,44,51)(41,73,45,52)(42,74,46,53), (1,91)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,62)(9,63)(10,57)(11,58)(12,59)(13,60)(14,61)(15,66)(16,67)(17,68)(18,69)(19,70)(20,64)(21,65)(22,76)(23,77)(24,71)(25,72)(26,73)(27,74)(28,75)(29,95)(30,96)(31,97)(32,98)(33,92)(34,93)(35,94)(36,103)(37,104)(38,105)(39,99)(40,100)(41,101)(42,102)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)>;
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), (1,106,43,9)(2,107,44,10)(3,108,45,11)(4,109,46,12)(5,110,47,13)(6,111,48,14)(7,112,49,8)(15,35,24,39)(16,29,25,40)(17,30,26,41)(18,31,27,42)(19,32,28,36)(20,33,22,37)(21,34,23,38)(50,91,63,78)(51,85,57,79)(52,86,58,80)(53,87,59,81)(54,88,60,82)(55,89,61,83)(56,90,62,84)(64,104,76,92)(65,105,77,93)(66,99,71,94)(67,100,72,95)(68,101,73,96)(69,102,74,97)(70,103,75,98), (1,63,35,66)(2,57,29,67)(3,58,30,68)(4,59,31,69)(5,60,32,70)(6,61,33,64)(7,62,34,65)(8,90,21,93)(9,91,15,94)(10,85,16,95)(11,86,17,96)(12,87,18,97)(13,88,19,98)(14,89,20,92)(22,104,111,83)(23,105,112,84)(24,99,106,78)(25,100,107,79)(26,101,108,80)(27,102,109,81)(28,103,110,82)(36,75,47,54)(37,76,48,55)(38,77,49,56)(39,71,43,50)(40,72,44,51)(41,73,45,52)(42,74,46,53), (1,91)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,62)(9,63)(10,57)(11,58)(12,59)(13,60)(14,61)(15,66)(16,67)(17,68)(18,69)(19,70)(20,64)(21,65)(22,76)(23,77)(24,71)(25,72)(26,73)(27,74)(28,75)(29,95)(30,96)(31,97)(32,98)(33,92)(34,93)(35,94)(36,103)(37,104)(38,105)(39,99)(40,100)(41,101)(42,102)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112) );
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)], [(1,106,43,9),(2,107,44,10),(3,108,45,11),(4,109,46,12),(5,110,47,13),(6,111,48,14),(7,112,49,8),(15,35,24,39),(16,29,25,40),(17,30,26,41),(18,31,27,42),(19,32,28,36),(20,33,22,37),(21,34,23,38),(50,91,63,78),(51,85,57,79),(52,86,58,80),(53,87,59,81),(54,88,60,82),(55,89,61,83),(56,90,62,84),(64,104,76,92),(65,105,77,93),(66,99,71,94),(67,100,72,95),(68,101,73,96),(69,102,74,97),(70,103,75,98)], [(1,63,35,66),(2,57,29,67),(3,58,30,68),(4,59,31,69),(5,60,32,70),(6,61,33,64),(7,62,34,65),(8,90,21,93),(9,91,15,94),(10,85,16,95),(11,86,17,96),(12,87,18,97),(13,88,19,98),(14,89,20,92),(22,104,111,83),(23,105,112,84),(24,99,106,78),(25,100,107,79),(26,101,108,80),(27,102,109,81),(28,103,110,82),(36,75,47,54),(37,76,48,55),(38,77,49,56),(39,71,43,50),(40,72,44,51),(41,73,45,52),(42,74,46,53)], [(1,91),(2,85),(3,86),(4,87),(5,88),(6,89),(7,90),(8,62),(9,63),(10,57),(11,58),(12,59),(13,60),(14,61),(15,66),(16,67),(17,68),(18,69),(19,70),(20,64),(21,65),(22,76),(23,77),(24,71),(25,72),(26,73),(27,74),(28,75),(29,95),(30,96),(31,97),(32,98),(33,92),(34,93),(35,94),(36,103),(37,104),(38,105),(39,99),(40,100),(41,101),(42,102),(43,78),(44,79),(45,80),(46,81),(47,82),(48,83),(49,84),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112)]])
C7×C4⋊1D4 is a maximal subgroup of
C28.9D8 C42⋊3Dic7 C28.16D8 C42.72D14 C28⋊2D8 C28⋊D8 C42.74D14 Dic14⋊9D4 C28⋊4SD16 D28⋊5D4 C42.166D14 C42⋊26D14 C42.238D14 D28⋊11D4 Dic14⋊11D4 C42.168D14 C42⋊28D14 C7×D42
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AP | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D4 | C7×D4 |
| kernel | C7×C4⋊1D4 | C4×C28 | D4×C14 | C4⋊1D4 | C42 | C2×D4 | C28 | C4 |
| # reps | 1 | 1 | 6 | 6 | 6 | 36 | 6 | 36 |
Matrix representation of C7×C4⋊1D4 ►in GL4(𝔽29) generated by
| 24 | 0 | 0 | 0 |
| 0 | 24 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 |
| 0 | 1 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 28 |
G:=sub<GL(4,GF(29))| [24,0,0,0,0,24,0,0,0,0,25,0,0,0,0,25],[28,0,0,0,0,28,0,0,0,0,0,28,0,0,1,0],[0,28,0,0,1,0,0,0,0,0,0,28,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,28] >;
C7×C4⋊1D4 in GAP, Magma, Sage, TeX
C_7\times C_4\rtimes_1D_4
% in TeX
G:=Group("C7xC4:1D4"); // GroupNames label
G:=SmallGroup(224,162);
// by ID
G=gap.SmallGroup(224,162);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,343,2090,518]);
// 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^-1,d*c*d=c^-1>;
// generators/relations