direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2xC7:D4, C23:D7, C14:2D4, C22:2D14, D14:3C22, C14.10C23, Dic7:2C22, C7:3(C2xD4), (C2xC14):3C22, (C22xC14):2C2, (C2xDic7):4C2, (C22xD7):3C2, C2.10(C22xD7), SmallGroup(112,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xC7:D4
G = < a,b,c,d | a2=b7=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 184 in 54 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2xC4, D4, C23, C23, D7, C14, C14, C14, C2xD4, Dic7, D14, D14, C2xC14, C2xC14, C2xC14, C2xDic7, C7:D4, C22xD7, C22xC14, C2xC7:D4
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, D14, C7:D4, C22xD7, C2xC7:D4
►On 56 points
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)
(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 15 8 22)(2 21 9 28)(3 20 10 27)(4 19 11 26)(5 18 12 25)(6 17 13 24)(7 16 14 23)(29 43 36 50)(30 49 37 56)(31 48 38 55)(32 47 39 54)(33 46 40 53)(34 45 41 52)(35 44 42 51)
(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 22)(16 28)(17 27)(18 26)(19 25)(20 24)(21 23)(30 35)(31 34)(32 33)(37 42)(38 41)(39 40)(43 50)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)
G:=sub<Sym(56)| (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49), (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), (1,15,8,22)(2,21,9,28)(3,20,10,27)(4,19,11,26)(5,18,12,25)(6,17,13,24)(7,16,14,23)(29,43,36,50)(30,49,37,56)(31,48,38,55)(32,47,39,54)(33,46,40,53)(34,45,41,52)(35,44,42,51), (2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,22)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(43,50)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)>;
G:=Group( (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49), (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), (1,15,8,22)(2,21,9,28)(3,20,10,27)(4,19,11,26)(5,18,12,25)(6,17,13,24)(7,16,14,23)(29,43,36,50)(30,49,37,56)(31,48,38,55)(32,47,39,54)(33,46,40,53)(34,45,41,52)(35,44,42,51), (2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,22)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(43,50)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51) );
G=PermutationGroup([[(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49)], [(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)], [(1,15,8,22),(2,21,9,28),(3,20,10,27),(4,19,11,26),(5,18,12,25),(6,17,13,24),(7,16,14,23),(29,43,36,50),(30,49,37,56),(31,48,38,55),(32,47,39,54),(33,46,40,53),(34,45,41,52),(35,44,42,51)], [(2,7),(3,6),(4,5),(9,14),(10,13),(11,12),(15,22),(16,28),(17,27),(18,26),(19,25),(20,24),(21,23),(30,35),(31,34),(32,33),(37,42),(38,41),(39,40),(43,50),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51)]])
C2xC7:D4 is a maximal subgroup of
C23.1D14 Dic7:4D4 C22:D28 D14.D4 D14:D4 Dic7.D4 C22.D28 C23.23D14 C28:7D4 C23:D14 C28:2D4 Dic7:D4 C28:D4 C24:D7 C2xD4xD7 D4:6D14
C2xC7:D4 is a maximal quotient of
C28.48D4 C23.23D14 C28:7D4 D4.D14 C23.18D14 C28.17D4 C23:D14 C28:2D4 Dic7:D4 C28:D4 C28.C23 Dic7:Q8 D14:3Q8 C28.23D4 D4:D14 D4.8D14 D4.9D14 C24:D7
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 7A | 7B | 7C | 14A | ··· | 14U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | C7:D4 |
| kernel | C2xC7:D4 | C2xDic7 | C7:D4 | C22xD7 | C22xC14 | C14 | C23 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 3 | 9 | 12 |
Matrix representation of C2xC7:D4 ►in GL3(F29) generated by
| 28 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 3 | 28 |
| 0 | 1 | 0 |
| 28 | 0 | 0 |
| 0 | 5 | 16 |
| 0 | 2 | 24 |
| 28 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 3 | 28 |
G:=sub<GL(3,GF(29))| [28,0,0,0,1,0,0,0,1],[1,0,0,0,3,1,0,28,0],[28,0,0,0,5,2,0,16,24],[28,0,0,0,1,3,0,0,28] >;
C2xC7:D4 in GAP, Magma, Sage, TeX
C_2\times C_7\rtimes D_4
% in TeX
G:=Group("C2xC7:D4"); // GroupNames label
G:=SmallGroup(112,36);
// by ID
G=gap.SmallGroup(112,36);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,182,2404]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^7=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations