direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C4.9C42, C42⋊1C28, C28.32C42, (C2×C56)⋊3C4, (C2×C8)⋊1C28, (C4×C28)⋊4C4, C4.9(C4×C28), C28.50(C4⋊C4), (C2×C28).36Q8, C23.8(C7×D4), (C2×C28).277D4, (C22×C14).27D4, C42⋊C2.1C14, (C2×M4(2)).4C14, C28.101(C22⋊C4), (C14×M4(2)).16C2, (C22×C28).386C22, C14.21(C2.C42), C4.1(C7×C4⋊C4), (C2×C4).8(C7×D4), (C2×C4).1(C7×Q8), C22.1(C7×C4⋊C4), (C2×C4).63(C2×C28), C4.17(C7×C22⋊C4), (C2×C14).18(C4⋊C4), (C2×C28).324(C2×C4), C22.7(C7×C22⋊C4), (C22×C4).16(C2×C14), C2.2(C7×C2.C42), (C2×C14).70(C22⋊C4), (C7×C42⋊C2).15C2, SmallGroup(448,141)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4.9C42
G = < a,b,c,d | a7=b4=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >
Subgroups: 154 in 94 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C42⋊C2, C2×M4(2), C56, C2×C28, C2×C28, C2×C28, C22×C14, C4.9C42, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C7×M4(2), C22×C28, C7×C42⋊C2, C14×M4(2), C7×C4.9C42
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C14, C42, C22⋊C4, C4⋊C4, C28, C2×C14, C2.C42, C2×C28, C7×D4, C7×Q8, C4.9C42, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C2.C42, C7×C4.9C42
►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 50 78 59)(2 51 79 60)(3 52 80 61)(4 53 81 62)(5 54 82 63)(6 55 83 57)(7 56 84 58)(8 100 25 44)(9 101 26 45)(10 102 27 46)(11 103 28 47)(12 104 22 48)(13 105 23 49)(14 99 24 43)(15 35 109 42)(16 29 110 36)(17 30 111 37)(18 31 112 38)(19 32 106 39)(20 33 107 40)(21 34 108 41)(64 90 71 98)(65 91 72 92)(66 85 73 93)(67 86 74 94)(68 87 75 95)(69 88 76 96)(70 89 77 97)
(1 22 68 32)(2 23 69 33)(3 24 70 34)(4 25 64 35)(5 26 65 29)(6 27 66 30)(7 28 67 31)(8 71 42 81)(9 72 36 82)(10 73 37 83)(11 74 38 84)(12 75 39 78)(13 76 40 79)(14 77 41 80)(15 62 100 98)(16 63 101 92)(17 57 102 93)(18 58 103 94)(19 59 104 95)(20 60 105 96)(21 61 99 97)(43 89 108 52)(44 90 109 53)(45 91 110 54)(46 85 111 55)(47 86 112 56)(48 87 106 50)(49 88 107 51)
(8 44 25 100)(9 45 26 101)(10 46 27 102)(11 47 28 103)(12 48 22 104)(13 49 23 105)(14 43 24 99)(15 35 109 42)(16 29 110 36)(17 30 111 37)(18 31 112 38)(19 32 106 39)(20 33 107 40)(21 34 108 41)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(91 92)
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), (1,50,78,59)(2,51,79,60)(3,52,80,61)(4,53,81,62)(5,54,82,63)(6,55,83,57)(7,56,84,58)(8,100,25,44)(9,101,26,45)(10,102,27,46)(11,103,28,47)(12,104,22,48)(13,105,23,49)(14,99,24,43)(15,35,109,42)(16,29,110,36)(17,30,111,37)(18,31,112,38)(19,32,106,39)(20,33,107,40)(21,34,108,41)(64,90,71,98)(65,91,72,92)(66,85,73,93)(67,86,74,94)(68,87,75,95)(69,88,76,96)(70,89,77,97), (1,22,68,32)(2,23,69,33)(3,24,70,34)(4,25,64,35)(5,26,65,29)(6,27,66,30)(7,28,67,31)(8,71,42,81)(9,72,36,82)(10,73,37,83)(11,74,38,84)(12,75,39,78)(13,76,40,79)(14,77,41,80)(15,62,100,98)(16,63,101,92)(17,57,102,93)(18,58,103,94)(19,59,104,95)(20,60,105,96)(21,61,99,97)(43,89,108,52)(44,90,109,53)(45,91,110,54)(46,85,111,55)(47,86,112,56)(48,87,106,50)(49,88,107,51), (8,44,25,100)(9,45,26,101)(10,46,27,102)(11,47,28,103)(12,48,22,104)(13,49,23,105)(14,43,24,99)(15,35,109,42)(16,29,110,36)(17,30,111,37)(18,31,112,38)(19,32,106,39)(20,33,107,40)(21,34,108,41)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92)>;
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), (1,50,78,59)(2,51,79,60)(3,52,80,61)(4,53,81,62)(5,54,82,63)(6,55,83,57)(7,56,84,58)(8,100,25,44)(9,101,26,45)(10,102,27,46)(11,103,28,47)(12,104,22,48)(13,105,23,49)(14,99,24,43)(15,35,109,42)(16,29,110,36)(17,30,111,37)(18,31,112,38)(19,32,106,39)(20,33,107,40)(21,34,108,41)(64,90,71,98)(65,91,72,92)(66,85,73,93)(67,86,74,94)(68,87,75,95)(69,88,76,96)(70,89,77,97), (1,22,68,32)(2,23,69,33)(3,24,70,34)(4,25,64,35)(5,26,65,29)(6,27,66,30)(7,28,67,31)(8,71,42,81)(9,72,36,82)(10,73,37,83)(11,74,38,84)(12,75,39,78)(13,76,40,79)(14,77,41,80)(15,62,100,98)(16,63,101,92)(17,57,102,93)(18,58,103,94)(19,59,104,95)(20,60,105,96)(21,61,99,97)(43,89,108,52)(44,90,109,53)(45,91,110,54)(46,85,111,55)(47,86,112,56)(48,87,106,50)(49,88,107,51), (8,44,25,100)(9,45,26,101)(10,46,27,102)(11,47,28,103)(12,48,22,104)(13,49,23,105)(14,43,24,99)(15,35,109,42)(16,29,110,36)(17,30,111,37)(18,31,112,38)(19,32,106,39)(20,33,107,40)(21,34,108,41)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92) );
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)], [(1,50,78,59),(2,51,79,60),(3,52,80,61),(4,53,81,62),(5,54,82,63),(6,55,83,57),(7,56,84,58),(8,100,25,44),(9,101,26,45),(10,102,27,46),(11,103,28,47),(12,104,22,48),(13,105,23,49),(14,99,24,43),(15,35,109,42),(16,29,110,36),(17,30,111,37),(18,31,112,38),(19,32,106,39),(20,33,107,40),(21,34,108,41),(64,90,71,98),(65,91,72,92),(66,85,73,93),(67,86,74,94),(68,87,75,95),(69,88,76,96),(70,89,77,97)], [(1,22,68,32),(2,23,69,33),(3,24,70,34),(4,25,64,35),(5,26,65,29),(6,27,66,30),(7,28,67,31),(8,71,42,81),(9,72,36,82),(10,73,37,83),(11,74,38,84),(12,75,39,78),(13,76,40,79),(14,77,41,80),(15,62,100,98),(16,63,101,92),(17,57,102,93),(18,58,103,94),(19,59,104,95),(20,60,105,96),(21,61,99,97),(43,89,108,52),(44,90,109,53),(45,91,110,54),(46,85,111,55),(47,86,112,56),(48,87,106,50),(49,88,107,51)], [(8,44,25,100),(9,45,26,101),(10,46,27,102),(11,47,28,103),(12,48,22,104),(13,49,23,105),(14,43,24,99),(15,35,109,42),(16,29,110,36),(17,30,111,37),(18,31,112,38),(19,32,106,39),(20,33,107,40),(21,34,108,41),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77),(85,93),(86,94),(87,95),(88,96),(89,97),(90,98),(91,92)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14F | 14G | ··· | 14X | 28A | ··· | 28L | 28M | ··· | 28AD | 28AE | ··· | 28BZ | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C28 | C28 | D4 | Q8 | D4 | C7×D4 | C7×Q8 | C7×D4 | C4.9C42 | C7×C4.9C42 |
| kernel | C7×C4.9C42 | C7×C42⋊C2 | C14×M4(2) | C4×C28 | C2×C56 | C4.9C42 | C42⋊C2 | C2×M4(2) | C42 | C2×C8 | C2×C28 | C2×C28 | C22×C14 | C2×C4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 2 | 1 | 8 | 4 | 6 | 12 | 6 | 48 | 24 | 2 | 1 | 1 | 12 | 6 | 6 | 2 | 12 |
Matrix representation of C7×C4.9C42 ►in GL4(𝔽113) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 111 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 112 | 112 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 15 | 15 |
G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[0,0,112,0,0,0,111,1,1,0,0,0,0,1,0,0],[1,112,0,0,0,112,0,0,0,0,98,15,0,0,0,15] >;
C7×C4.9C42 in GAP, Magma, Sage, TeX
C_7\times C_4._9C_4^2
% in TeX
G:=Group("C7xC4.9C4^2"); // GroupNames label
G:=SmallGroup(448,141);
// by ID
G=gap.SmallGroup(448,141);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,792,248,4911,14117]);
// 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*c*d^-1=b*c=c*b,b*d=d*b>;
// generators/relations