direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4×D7, C28⋊C23, C23⋊3D14, D28⋊7C22, D14⋊2C23, C14.5C24, Dic7⋊1C23, (C2×C4)⋊6D14, C14⋊2(C2×D4), (C2×C14)⋊C23, C7⋊2(C22×D4), (D4×C14)⋊5C2, C4⋊1(C22×D7), (C2×D28)⋊11C2, (C2×C28)⋊2C22, (C23×D7)⋊4C2, (C4×D7)⋊3C22, (C7×D4)⋊5C22, C7⋊D4⋊1C22, C2.6(C23×D7), C22⋊1(C22×D7), (C22×C14)⋊4C22, (C2×Dic7)⋊8C22, (C22×D7)⋊6C22, (C2×C4×D7)⋊3C2, (C2×C7⋊D4)⋊9C2, SmallGroup(224,178)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4×D7
G = < a,b,c,d,e | a2=b4=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1054 in 236 conjugacy classes, 97 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, D4, C23, C23, D7, D7, C14, C14, C14, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C22×D4, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C2×C4×D7, C2×D28, D4×D7, C2×C7⋊D4, D4×C14, C23×D7, C2×D4×D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D4×D7, C23×D7, C2×D4×D7
►On 56 points
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)
(1 34 13 41)(2 35 14 42)(3 29 8 36)(4 30 9 37)(5 31 10 38)(6 32 11 39)(7 33 12 40)(15 43 22 50)(16 44 23 51)(17 45 24 52)(18 46 25 53)(19 47 26 54)(20 48 27 55)(21 49 28 56)
(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)
(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)
(1 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 17)(9 16)(10 15)(11 21)(12 20)(13 19)(14 18)(29 52)(30 51)(31 50)(32 56)(33 55)(34 54)(35 53)(36 45)(37 44)(38 43)(39 49)(40 48)(41 47)(42 46)
G:=sub<Sym(56)| (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56), (1,34,13,41)(2,35,14,42)(3,29,8,36)(4,30,9,37)(5,31,10,38)(6,32,11,39)(7,33,12,40)(15,43,22,50)(16,44,23,51)(17,45,24,52)(18,46,25,53)(19,47,26,54)(20,48,27,55)(21,49,28,56), (29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,17)(9,16)(10,15)(11,21)(12,20)(13,19)(14,18)(29,52)(30,51)(31,50)(32,56)(33,55)(34,54)(35,53)(36,45)(37,44)(38,43)(39,49)(40,48)(41,47)(42,46)>;
G:=Group( (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56), (1,34,13,41)(2,35,14,42)(3,29,8,36)(4,30,9,37)(5,31,10,38)(6,32,11,39)(7,33,12,40)(15,43,22,50)(16,44,23,51)(17,45,24,52)(18,46,25,53)(19,47,26,54)(20,48,27,55)(21,49,28,56), (29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,17)(9,16)(10,15)(11,21)(12,20)(13,19)(14,18)(29,52)(30,51)(31,50)(32,56)(33,55)(34,54)(35,53)(36,45)(37,44)(38,43)(39,49)(40,48)(41,47)(42,46) );
G=PermutationGroup([[(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56)], [(1,34,13,41),(2,35,14,42),(3,29,8,36),(4,30,9,37),(5,31,10,38),(6,32,11,39),(7,33,12,40),(15,43,22,50),(16,44,23,51),(17,45,24,52),(18,46,25,53),(19,47,26,54),(20,48,27,55),(21,49,28,56)], [(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56)], [(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)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,28),(7,27),(8,17),(9,16),(10,15),(11,21),(12,20),(13,19),(14,18),(29,52),(30,51),(31,50),(32,56),(33,55),(34,54),(35,53),(36,45),(37,44),(38,43),(39,49),(40,48),(41,47),(42,46)]])
C2×D4×D7 is a maximal subgroup of
(D4×D7)⋊C4 D4⋊D28 D4.6D28 D28⋊D4 D14⋊6SD16 C42⋊11D14 D4⋊5D28 C24⋊2D14 C24⋊3D14 C14.372+ 1+4 C14.382+ 1+4 D28⋊19D4 C14.402+ 1+4 D28⋊20D4 C14.1202+ 1+4 C14.1212+ 1+4 C42⋊18D14 D28⋊10D4 C42⋊26D14 D28⋊11D4 C14.1452+ 1+4
C2×D4×D7 is a maximal quotient of
C24.27D14 C14.2- 1+4 C42⋊12D14 C42.228D14 D28⋊23D4 D28⋊24D4 Dic14⋊23D4 Dic14⋊24D4 C24.56D14 C24⋊2D14 C24⋊3D14 C24.33D14 C24.34D14 C28⋊(C4○D4) C14.682- 1+4 Dic14⋊19D4 Dic14⋊20D4 C14.372+ 1+4 C4⋊C4⋊21D14 C14.382+ 1+4 C14.722- 1+4 D28⋊19D4 C14.402+ 1+4 C14.732- 1+4 D28⋊20D4 C4⋊C4⋊26D14 C14.162- 1+4 C14.172- 1+4 D28⋊21D4 D28⋊22D4 Dic14⋊21D4 Dic14⋊22D4 C14.792- 1+4 C14.1202+ 1+4 C14.1212+ 1+4 C14.822- 1+4 C4⋊C4⋊28D14 C42.233D14 C42⋊18D14 C42.141D14 D28⋊10D4 Dic14⋊10D4 C42⋊26D14 C42.238D14 D28⋊11D4 Dic14⋊11D4 C42.171D14 C42.240D14 D28⋊12D4 D28⋊8Q8 D8⋊13D14 D28.29D4 D28.30D4 D8⋊10D14 D8⋊15D14 D8⋊11D14 D8.10D14 SD16⋊D14 D8⋊5D14 D8⋊6D14 D56⋊C22 C56.C23 D28.44D4
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 7 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | D4×D7 |
| kernel | C2×D4×D7 | C2×C4×D7 | C2×D28 | D4×D7 | C2×C7⋊D4 | D4×C14 | C23×D7 | D14 | C2×D4 | C2×C4 | D4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 4 | 3 | 3 | 12 | 6 | 6 |
Matrix representation of C2×D4×D7 ►in GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 13 |
| 0 | 0 | 11 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 11 | 28 |
| 19 | 1 | 0 | 0 |
| 9 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 22 | 4 | 0 | 0 |
| 17 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,1,11,0,0,13,28],[1,0,0,0,0,1,0,0,0,0,1,11,0,0,0,28],[19,9,0,0,1,28,0,0,0,0,1,0,0,0,0,1],[22,17,0,0,4,7,0,0,0,0,1,0,0,0,0,1] >;
C2×D4×D7 in GAP, Magma, Sage, TeX
C_2\times D_4\times D_7
% in TeX
G:=Group("C2xD4xD7"); // GroupNames label
G:=SmallGroup(224,178);
// by ID
G=gap.SmallGroup(224,178);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,159,6917]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^7=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