direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C42⋊C22, C4≀C2⋊5C14, C4○D4⋊4C28, (C2×Q8)⋊8C28, (D4×C14)⋊22C4, (C2×D4)⋊10C28, C42⋊1(C2×C14), (Q8×C14)⋊18C4, D4.7(C2×C28), C4.73(D4×C14), Q8.7(C2×C28), (C4×C28)⋊34C22, (C2×C28).520D4, C28.478(C2×D4), C4.8(C22×C28), C23.12(C7×D4), C42⋊C2⋊4C14, C22.13(D4×C14), (C22×C14).30D4, C28.84(C22⋊C4), (C14×M4(2))⋊31C2, M4(2)⋊10(C2×C14), (C2×M4(2))⋊13C14, (C2×C28).897C23, C28.153(C22×C4), (C7×M4(2))⋊39C22, (C22×C28).413C22, (C7×C4≀C2)⋊13C2, (C7×C4○D4)⋊10C4, (C2×C4).25(C7×D4), (C2×C4).23(C2×C28), (C2×C4○D4).7C14, C4○D4.7(C2×C14), (C7×D4).29(C2×C4), C4.16(C7×C22⋊C4), (C7×Q8).31(C2×C4), (C2×C28).196(C2×C4), (C14×C4○D4).21C2, (C2×C14).408(C2×D4), C2.24(C14×C22⋊C4), (C7×C42⋊C2)⋊25C2, C14.112(C2×C22⋊C4), (C2×C4).72(C22×C14), (C22×C4).32(C2×C14), (C7×C4○D4).52C22, C22.22(C7×C22⋊C4), (C2×C14).83(C22⋊C4), SmallGroup(448,829)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C42⋊C22
G = < a,b,c,d,e | a7=b4=c4=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1c, ebe=bc2, cd=dc, ce=ec, de=ed >
Subgroups: 258 in 154 conjugacy classes, 78 normal (46 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C28, C28, C2×C14, C2×C14, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, C42⋊C22, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C7×M4(2), C7×M4(2), C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C7×C4≀C2, C7×C42⋊C2, C14×M4(2), C14×C4○D4, C7×C42⋊C22
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C28, C2×C14, C2×C22⋊C4, C2×C28, C7×D4, C22×C14, C42⋊C22, C7×C22⋊C4, C22×C28, D4×C14, C14×C22⋊C4, C7×C42⋊C22
►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 74 46 67)(2 75 47 68)(3 76 48 69)(4 77 49 70)(5 71 43 64)(6 72 44 65)(7 73 45 66)(8 81)(9 82)(10 83)(11 84)(12 78)(13 79)(14 80)(15 90)(16 91)(17 85)(18 86)(19 87)(20 88)(21 89)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 92)(29 57 36 55)(30 58 37 56)(31 59 38 50)(32 60 39 51)(33 61 40 52)(34 62 41 53)(35 63 42 54)(99 110)(100 111)(101 112)(102 106)(103 107)(104 108)(105 109)
(1 31 46 38)(2 32 47 39)(3 33 48 40)(4 34 49 41)(5 35 43 42)(6 29 44 36)(7 30 45 37)(8 109 15 27)(9 110 16 28)(10 111 17 22)(11 112 18 23)(12 106 19 24)(13 107 20 25)(14 108 21 26)(50 74 59 67)(51 75 60 68)(52 76 61 69)(53 77 62 70)(54 71 63 64)(55 72 57 65)(56 73 58 66)(78 102 87 95)(79 103 88 96)(80 104 89 97)(81 105 90 98)(82 99 91 92)(83 100 85 93)(84 101 86 94)
(1 102)(2 103)(3 104)(4 105)(5 99)(6 100)(7 101)(8 70)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 77)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 83)(37 84)(38 78)(39 79)(40 80)(41 81)(42 82)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)
(1 67)(2 68)(3 69)(4 70)(5 64)(6 65)(7 66)(8 105)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 98)(16 92)(17 93)(18 94)(19 95)(20 96)(21 97)(22 83)(23 84)(24 78)(25 79)(26 80)(27 81)(28 82)(29 55)(30 56)(31 50)(32 51)(33 52)(34 53)(35 54)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(85 111)(86 112)(87 106)(88 107)(89 108)(90 109)(91 110)
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), (1,74,46,67)(2,75,47,68)(3,76,48,69)(4,77,49,70)(5,71,43,64)(6,72,44,65)(7,73,45,66)(8,81)(9,82)(10,83)(11,84)(12,78)(13,79)(14,80)(15,90)(16,91)(17,85)(18,86)(19,87)(20,88)(21,89)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,92)(29,57,36,55)(30,58,37,56)(31,59,38,50)(32,60,39,51)(33,61,40,52)(34,62,41,53)(35,63,42,54)(99,110)(100,111)(101,112)(102,106)(103,107)(104,108)(105,109), (1,31,46,38)(2,32,47,39)(3,33,48,40)(4,34,49,41)(5,35,43,42)(6,29,44,36)(7,30,45,37)(8,109,15,27)(9,110,16,28)(10,111,17,22)(11,112,18,23)(12,106,19,24)(13,107,20,25)(14,108,21,26)(50,74,59,67)(51,75,60,68)(52,76,61,69)(53,77,62,70)(54,71,63,64)(55,72,57,65)(56,73,58,66)(78,102,87,95)(79,103,88,96)(80,104,89,97)(81,105,90,98)(82,99,91,92)(83,100,85,93)(84,101,86,94), (1,102)(2,103)(3,104)(4,105)(5,99)(6,100)(7,101)(8,70)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,77)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,83)(37,84)(38,78)(39,79)(40,80)(41,81)(42,82)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112), (1,67)(2,68)(3,69)(4,70)(5,64)(6,65)(7,66)(8,105)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,98)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,83)(23,84)(24,78)(25,79)(26,80)(27,81)(28,82)(29,55)(30,56)(31,50)(32,51)(33,52)(34,53)(35,54)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(85,111)(86,112)(87,106)(88,107)(89,108)(90,109)(91,110)>;
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), (1,74,46,67)(2,75,47,68)(3,76,48,69)(4,77,49,70)(5,71,43,64)(6,72,44,65)(7,73,45,66)(8,81)(9,82)(10,83)(11,84)(12,78)(13,79)(14,80)(15,90)(16,91)(17,85)(18,86)(19,87)(20,88)(21,89)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,92)(29,57,36,55)(30,58,37,56)(31,59,38,50)(32,60,39,51)(33,61,40,52)(34,62,41,53)(35,63,42,54)(99,110)(100,111)(101,112)(102,106)(103,107)(104,108)(105,109), (1,31,46,38)(2,32,47,39)(3,33,48,40)(4,34,49,41)(5,35,43,42)(6,29,44,36)(7,30,45,37)(8,109,15,27)(9,110,16,28)(10,111,17,22)(11,112,18,23)(12,106,19,24)(13,107,20,25)(14,108,21,26)(50,74,59,67)(51,75,60,68)(52,76,61,69)(53,77,62,70)(54,71,63,64)(55,72,57,65)(56,73,58,66)(78,102,87,95)(79,103,88,96)(80,104,89,97)(81,105,90,98)(82,99,91,92)(83,100,85,93)(84,101,86,94), (1,102)(2,103)(3,104)(4,105)(5,99)(6,100)(7,101)(8,70)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,77)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,83)(37,84)(38,78)(39,79)(40,80)(41,81)(42,82)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112), (1,67)(2,68)(3,69)(4,70)(5,64)(6,65)(7,66)(8,105)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,98)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,83)(23,84)(24,78)(25,79)(26,80)(27,81)(28,82)(29,55)(30,56)(31,50)(32,51)(33,52)(34,53)(35,54)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(85,111)(86,112)(87,106)(88,107)(89,108)(90,109)(91,110) );
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)], [(1,74,46,67),(2,75,47,68),(3,76,48,69),(4,77,49,70),(5,71,43,64),(6,72,44,65),(7,73,45,66),(8,81),(9,82),(10,83),(11,84),(12,78),(13,79),(14,80),(15,90),(16,91),(17,85),(18,86),(19,87),(20,88),(21,89),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,92),(29,57,36,55),(30,58,37,56),(31,59,38,50),(32,60,39,51),(33,61,40,52),(34,62,41,53),(35,63,42,54),(99,110),(100,111),(101,112),(102,106),(103,107),(104,108),(105,109)], [(1,31,46,38),(2,32,47,39),(3,33,48,40),(4,34,49,41),(5,35,43,42),(6,29,44,36),(7,30,45,37),(8,109,15,27),(9,110,16,28),(10,111,17,22),(11,112,18,23),(12,106,19,24),(13,107,20,25),(14,108,21,26),(50,74,59,67),(51,75,60,68),(52,76,61,69),(53,77,62,70),(54,71,63,64),(55,72,57,65),(56,73,58,66),(78,102,87,95),(79,103,88,96),(80,104,89,97),(81,105,90,98),(82,99,91,92),(83,100,85,93),(84,101,86,94)], [(1,102),(2,103),(3,104),(4,105),(5,99),(6,100),(7,101),(8,70),(9,64),(10,65),(11,66),(12,67),(13,68),(14,69),(15,77),(16,71),(17,72),(18,73),(19,74),(20,75),(21,76),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,83),(37,84),(38,78),(39,79),(40,80),(41,81),(42,82),(43,92),(44,93),(45,94),(46,95),(47,96),(48,97),(49,98),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112)], [(1,67),(2,68),(3,69),(4,70),(5,64),(6,65),(7,66),(8,105),(9,99),(10,100),(11,101),(12,102),(13,103),(14,104),(15,98),(16,92),(17,93),(18,94),(19,95),(20,96),(21,97),(22,83),(23,84),(24,78),(25,79),(26,80),(27,81),(28,82),(29,55),(30,56),(31,50),(32,51),(33,52),(34,53),(35,54),(36,57),(37,58),(38,59),(39,60),(40,61),(41,62),(42,63),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(85,111),(86,112),(87,106),(88,107),(89,108),(90,109),(91,110)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4K | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14F | 14G | ··· | 14X | 14Y | ··· | 14AJ | 28A | ··· | 28L | 28M | ··· | 28AD | 28AE | ··· | 28BN | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | C28 | C28 | D4 | D4 | C7×D4 | C7×D4 | C42⋊C22 | C7×C42⋊C22 |
| kernel | C7×C42⋊C22 | C7×C4≀C2 | C7×C42⋊C2 | C14×M4(2) | C14×C4○D4 | D4×C14 | Q8×C14 | C7×C4○D4 | C42⋊C22 | C4≀C2 | C42⋊C2 | C2×M4(2) | C2×C4○D4 | C2×D4 | C2×Q8 | C4○D4 | C2×C28 | C22×C14 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 24 | 6 | 6 | 6 | 12 | 12 | 24 | 3 | 1 | 18 | 6 | 2 | 12 |
Matrix representation of C7×C42⋊C22 ►in GL4(𝔽113) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 98 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 0 | 112 | 0 |
G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[0,1,0,0,112,0,0,0,0,0,0,98,0,0,15,0],[15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,112,0,0,112,0,0,0,0,0,0,112,0,0,112,0] >;
C7×C42⋊C22 in GAP, Magma, Sage, TeX
C_7\times C_4^2\rtimes C_2^2
% in TeX
G:=Group("C7xC4^2:C2^2"); // GroupNames label
G:=SmallGroup(448,829);
// by ID
G=gap.SmallGroup(448,829);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,2403,9804,4911,172,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^4=d^2=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d=b^-1*c,e*b*e=b*c^2,c*d=d*c,c*e=e*c,d*e=e*d>;
// generators/relations