direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×D42, C14.1602+ (1+4), C4⋊2(D4×C14), C28⋊14(C2×D4), (D4×C28)⋊43C2, (C4×D4)⋊14C14, C4⋊1D4⋊7C14, C24⋊4(C2×C14), C42⋊8(C2×C14), C22⋊2(D4×C14), C22≀C2⋊5C14, C4⋊D4⋊10C14, (C4×C28)⋊42C22, (C22×D4)⋊8C14, (D4×C14)⋊64C22, (C23×C14)⋊4C22, (C2×C28).674C23, (C2×C14).365C24, (C22×C28)⋊50C22, C14.193(C22×D4), C22.39(C23×C14), C23.15(C22×C14), C2.12(C7×2+ (1+4)), (C22×C14).264C23, (D4×C2×C14)⋊23C2, C2.17(D4×C2×C14), C4⋊C4⋊16(C2×C14), (C2×C14)⋊13(C2×D4), (C2×D4)⋊12(C2×C14), (C7×C4⋊1D4)⋊18C2, (C7×C4⋊D4)⋊37C2, C22⋊C4⋊5(C2×C14), (C7×C4⋊C4)⋊72C22, (C7×C22≀C2)⋊15C2, (C22×C4)⋊10(C2×C14), (C7×C22⋊C4)⋊40C22, (C2×C4).32(C22×C14), SmallGroup(448,1328)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 778 in 428 conjugacy classes, 182 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×4], C4 [×5], C22, C22 [×8], C22 [×36], C7, C2×C4, C2×C4 [×6], C2×C4 [×8], D4 [×8], D4 [×26], C23 [×8], C23 [×20], C14, C14 [×2], C14 [×12], C42, C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×4], C2×D4 [×16], C2×D4 [×16], C24 [×4], C28 [×4], C28 [×5], C2×C14, C2×C14 [×8], C2×C14 [×36], C4×D4 [×2], C22≀C2 [×4], C4⋊D4 [×4], C4⋊1D4, C22×D4 [×4], C2×C28, C2×C28 [×6], C2×C28 [×8], C7×D4 [×8], C7×D4 [×26], C22×C14 [×8], C22×C14 [×20], D42, C4×C28, C7×C22⋊C4 [×8], C7×C4⋊C4 [×2], C22×C28 [×4], D4×C14 [×16], D4×C14 [×16], C23×C14 [×4], D4×C28 [×2], C7×C22≀C2 [×4], C7×C4⋊D4 [×4], C7×C4⋊1D4, D4×C2×C14 [×4], C7×D42
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×8], C23 [×15], C14 [×15], C2×D4 [×12], C24, C2×C14 [×35], C22×D4 [×2], 2+ (1+4), C7×D4 [×8], C22×C14 [×15], D42, D4×C14 [×12], C23×C14, D4×C2×C14 [×2], C7×2+ (1+4), C7×D42
Generators and relations
G = < a,b,c,d,e | a7=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
►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 35 75)(2 51 29 76)(3 52 30 77)(4 53 31 71)(5 54 32 72)(6 55 33 73)(7 56 34 74)(8 84 18 102)(9 78 19 103)(10 79 20 104)(11 80 21 105)(12 81 15 99)(13 82 16 100)(14 83 17 101)(22 92 111 89)(23 93 112 90)(24 94 106 91)(25 95 107 85)(26 96 108 86)(27 97 109 87)(28 98 110 88)(36 70 44 60)(37 64 45 61)(38 65 46 62)(39 66 47 63)(40 67 48 57)(41 68 49 58)(42 69 43 59)
(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(50 75)(51 76)(52 77)(53 71)(54 72)(55 73)(56 74)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)
(1 94 39 78)(2 95 40 79)(3 96 41 80)(4 97 42 81)(5 98 36 82)(6 92 37 83)(7 93 38 84)(8 74 23 62)(9 75 24 63)(10 76 25 57)(11 77 26 58)(12 71 27 59)(13 72 28 60)(14 73 22 61)(15 53 109 69)(16 54 110 70)(17 55 111 64)(18 56 112 65)(19 50 106 66)(20 51 107 67)(21 52 108 68)(29 85 48 104)(30 86 49 105)(31 87 43 99)(32 88 44 100)(33 89 45 101)(34 90 46 102)(35 91 47 103)
(1 94)(2 95)(3 96)(4 97)(5 98)(6 92)(7 93)(8 62)(9 63)(10 57)(11 58)(12 59)(13 60)(14 61)(15 69)(16 70)(17 64)(18 65)(19 66)(20 67)(21 68)(22 73)(23 74)(24 75)(25 76)(26 77)(27 71)(28 72)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 82)(37 83)(38 84)(39 78)(40 79)(41 80)(42 81)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(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), (1,50,35,75)(2,51,29,76)(3,52,30,77)(4,53,31,71)(5,54,32,72)(6,55,33,73)(7,56,34,74)(8,84,18,102)(9,78,19,103)(10,79,20,104)(11,80,21,105)(12,81,15,99)(13,82,16,100)(14,83,17,101)(22,92,111,89)(23,93,112,90)(24,94,106,91)(25,95,107,85)(26,96,108,86)(27,97,109,87)(28,98,110,88)(36,70,44,60)(37,64,45,61)(38,65,46,62)(39,66,47,63)(40,67,48,57)(41,68,49,58)(42,69,43,59), (8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66), (1,94,39,78)(2,95,40,79)(3,96,41,80)(4,97,42,81)(5,98,36,82)(6,92,37,83)(7,93,38,84)(8,74,23,62)(9,75,24,63)(10,76,25,57)(11,77,26,58)(12,71,27,59)(13,72,28,60)(14,73,22,61)(15,53,109,69)(16,54,110,70)(17,55,111,64)(18,56,112,65)(19,50,106,66)(20,51,107,67)(21,52,108,68)(29,85,48,104)(30,86,49,105)(31,87,43,99)(32,88,44,100)(33,89,45,101)(34,90,46,102)(35,91,47,103), (1,94)(2,95)(3,96)(4,97)(5,98)(6,92)(7,93)(8,62)(9,63)(10,57)(11,58)(12,59)(13,60)(14,61)(15,69)(16,70)(17,64)(18,65)(19,66)(20,67)(21,68)(22,73)(23,74)(24,75)(25,76)(26,77)(27,71)(28,72)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,82)(37,83)(38,84)(39,78)(40,79)(41,80)(42,81)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(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), (1,50,35,75)(2,51,29,76)(3,52,30,77)(4,53,31,71)(5,54,32,72)(6,55,33,73)(7,56,34,74)(8,84,18,102)(9,78,19,103)(10,79,20,104)(11,80,21,105)(12,81,15,99)(13,82,16,100)(14,83,17,101)(22,92,111,89)(23,93,112,90)(24,94,106,91)(25,95,107,85)(26,96,108,86)(27,97,109,87)(28,98,110,88)(36,70,44,60)(37,64,45,61)(38,65,46,62)(39,66,47,63)(40,67,48,57)(41,68,49,58)(42,69,43,59), (8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66), (1,94,39,78)(2,95,40,79)(3,96,41,80)(4,97,42,81)(5,98,36,82)(6,92,37,83)(7,93,38,84)(8,74,23,62)(9,75,24,63)(10,76,25,57)(11,77,26,58)(12,71,27,59)(13,72,28,60)(14,73,22,61)(15,53,109,69)(16,54,110,70)(17,55,111,64)(18,56,112,65)(19,50,106,66)(20,51,107,67)(21,52,108,68)(29,85,48,104)(30,86,49,105)(31,87,43,99)(32,88,44,100)(33,89,45,101)(34,90,46,102)(35,91,47,103), (1,94)(2,95)(3,96)(4,97)(5,98)(6,92)(7,93)(8,62)(9,63)(10,57)(11,58)(12,59)(13,60)(14,61)(15,69)(16,70)(17,64)(18,65)(19,66)(20,67)(21,68)(22,73)(23,74)(24,75)(25,76)(26,77)(27,71)(28,72)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,82)(37,83)(38,84)(39,78)(40,79)(41,80)(42,81)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(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)], [(1,50,35,75),(2,51,29,76),(3,52,30,77),(4,53,31,71),(5,54,32,72),(6,55,33,73),(7,56,34,74),(8,84,18,102),(9,78,19,103),(10,79,20,104),(11,80,21,105),(12,81,15,99),(13,82,16,100),(14,83,17,101),(22,92,111,89),(23,93,112,90),(24,94,106,91),(25,95,107,85),(26,96,108,86),(27,97,109,87),(28,98,110,88),(36,70,44,60),(37,64,45,61),(38,65,46,62),(39,66,47,63),(40,67,48,57),(41,68,49,58),(42,69,43,59)], [(8,18),(9,19),(10,20),(11,21),(12,15),(13,16),(14,17),(22,111),(23,112),(24,106),(25,107),(26,108),(27,109),(28,110),(50,75),(51,76),(52,77),(53,71),(54,72),(55,73),(56,74),(57,67),(58,68),(59,69),(60,70),(61,64),(62,65),(63,66)], [(1,94,39,78),(2,95,40,79),(3,96,41,80),(4,97,42,81),(5,98,36,82),(6,92,37,83),(7,93,38,84),(8,74,23,62),(9,75,24,63),(10,76,25,57),(11,77,26,58),(12,71,27,59),(13,72,28,60),(14,73,22,61),(15,53,109,69),(16,54,110,70),(17,55,111,64),(18,56,112,65),(19,50,106,66),(20,51,107,67),(21,52,108,68),(29,85,48,104),(30,86,49,105),(31,87,43,99),(32,88,44,100),(33,89,45,101),(34,90,46,102),(35,91,47,103)], [(1,94),(2,95),(3,96),(4,97),(5,98),(6,92),(7,93),(8,62),(9,63),(10,57),(11,58),(12,59),(13,60),(14,61),(15,69),(16,70),(17,64),(18,65),(19,66),(20,67),(21,68),(22,73),(23,74),(24,75),(25,76),(26,77),(27,71),(28,72),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,82),(37,83),(38,84),(39,78),(40,79),(41,80),(42,81),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 20 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 2 | 0 | 0 |
| 28 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 27 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,16,0,0,0,0,16],[28,0,0,0,0,28,0,0,0,0,0,1,0,0,28,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,28],[28,28,0,0,2,1,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,27,28,0,0,0,0,28,0,0,0,0,28] >;
175 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14BN | 14BO | ··· | 14CL | 28A | ··· | 28X | 28Y | ··· | 28BB |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
175 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | C7×D4 | 2+ (1+4) | C7×2+ (1+4) |
| kernel | C7×D42 | D4×C28 | C7×C22≀C2 | C7×C4⋊D4 | C7×C4⋊1D4 | D4×C2×C14 | D42 | C4×D4 | C22≀C2 | C4⋊D4 | C4⋊1D4 | C22×D4 | C7×D4 | D4 | C14 | C2 |
| # reps | 1 | 2 | 4 | 4 | 1 | 4 | 6 | 12 | 24 | 24 | 6 | 24 | 8 | 48 | 1 | 6 |
In GAP, Magma, Sage, TeX
C_7\times D_4^2
% in TeX
G:=Group("C7xD4^2"); // GroupNames label
G:=SmallGroup(448,1328);
// by ID
G=gap.SmallGroup(448,1328);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,1690]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀