direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C14, C23⋊2C14, C28⋊4C22, C14.11C23, C4⋊(C2×C14), (C2×C4)⋊2C14, (C2×C28)⋊6C2, C22⋊(C2×C14), (C22×C14)⋊1C2, (C2×C14)⋊2C22, C2.1(C22×C14), SmallGroup(112,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C14
G = < a,b,c | a14=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 70 in 54 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, D4, C23, C14, C14, C14, C2×D4, C28, C2×C14, C2×C14, C2×C14, C2×C28, C7×D4, C22×C14, D4×C14
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C7×D4, C22×C14, D4×C14
►On 56 points
(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 28 36 46)(2 15 37 47)(3 16 38 48)(4 17 39 49)(5 18 40 50)(6 19 41 51)(7 20 42 52)(8 21 29 53)(9 22 30 54)(10 23 31 55)(11 24 32 56)(12 25 33 43)(13 26 34 44)(14 27 35 45)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 43)(41 44)(42 45)
G:=sub<Sym(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), (1,28,36,46)(2,15,37,47)(3,16,38,48)(4,17,39,49)(5,18,40,50)(6,19,41,51)(7,20,42,52)(8,21,29,53)(9,22,30,54)(10,23,31,55)(11,24,32,56)(12,25,33,43)(13,26,34,44)(14,27,35,45), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,43)(41,44)(42,45)>;
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), (1,28,36,46)(2,15,37,47)(3,16,38,48)(4,17,39,49)(5,18,40,50)(6,19,41,51)(7,20,42,52)(8,21,29,53)(9,22,30,54)(10,23,31,55)(11,24,32,56)(12,25,33,43)(13,26,34,44)(14,27,35,45), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,43)(41,44)(42,45) );
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)], [(1,28,36,46),(2,15,37,47),(3,16,38,48),(4,17,39,49),(5,18,40,50),(6,19,41,51),(7,20,42,52),(8,21,29,53),(9,22,30,54),(10,23,31,55),(11,24,32,56),(12,25,33,43),(13,26,34,44),(14,27,35,45)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,15),(10,16),(11,17),(12,18),(13,19),(14,20),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,43),(41,44),(42,45)]])
D4×C14 is a maximal subgroup of
D4⋊Dic7 C28.D4 C23⋊Dic7 D4.D14 C23.18D14 C28.17D4 C23⋊D14 C28⋊2D4 Dic7⋊D4 C28⋊D4 D4⋊6D14
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AP | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | C7×D4 |
| kernel | D4×C14 | C2×C28 | C7×D4 | C22×C14 | C2×D4 | C2×C4 | D4 | C23 | C14 | C2 |
| # reps | 1 | 1 | 4 | 2 | 6 | 6 | 24 | 12 | 2 | 12 |
Matrix representation of D4×C14 ►in GL3(𝔽29) generated by
| 28 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 28 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(29))| [28,0,0,0,4,0,0,0,4],[1,0,0,0,0,28,0,1,0],[1,0,0,0,0,1,0,1,0] >;
D4×C14 in GAP, Magma, Sage, TeX
D_4\times C_{14} % in TeX
G:=Group("D4xC14"); // GroupNames label
G:=SmallGroup(112,38);
// by ID
G=gap.SmallGroup(112,38);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-2,581]);
// Polycyclic
G:=Group<a,b,c|a^14=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations