metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14⋊C4, C14.6D4, C2.2D28, C22.6D14, (C2×C4)⋊1D7, (C2×C28)⋊1C2, C2.5(C4×D7), C7⋊1(C22⋊C4), C14.5(C2×C4), (C22×D7).C2, (C2×Dic7)⋊1C2, C2.2(C7⋊D4), (C2×C14).6C22, SmallGroup(112,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D14⋊C4
G = < a,b,c | a14=b2=c4=1, bab=a-1, ac=ca, cbc-1=a7b >
Smallest permutation representation of D14⋊C4
►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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 28)(11 27)(12 26)(13 25)(14 24)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 56)(39 55)(40 54)(41 53)(42 52)
(1 55 24 33)(2 56 25 34)(3 43 26 35)(4 44 27 36)(5 45 28 37)(6 46 15 38)(7 47 16 39)(8 48 17 40)(9 49 18 41)(10 50 19 42)(11 51 20 29)(12 52 21 30)(13 53 22 31)(14 54 23 32)
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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,28)(11,27)(12,26)(13,25)(14,24)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,56)(39,55)(40,54)(41,53)(42,52), (1,55,24,33)(2,56,25,34)(3,43,26,35)(4,44,27,36)(5,45,28,37)(6,46,15,38)(7,47,16,39)(8,48,17,40)(9,49,18,41)(10,50,19,42)(11,51,20,29)(12,52,21,30)(13,53,22,31)(14,54,23,32)>;
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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,28)(11,27)(12,26)(13,25)(14,24)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,56)(39,55)(40,54)(41,53)(42,52), (1,55,24,33)(2,56,25,34)(3,43,26,35)(4,44,27,36)(5,45,28,37)(6,46,15,38)(7,47,16,39)(8,48,17,40)(9,49,18,41)(10,50,19,42)(11,51,20,29)(12,52,21,30)(13,53,22,31)(14,54,23,32) );
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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,28),(11,27),(12,26),(13,25),(14,24),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,56),(39,55),(40,54),(41,53),(42,52)], [(1,55,24,33),(2,56,25,34),(3,43,26,35),(4,44,27,36),(5,45,28,37),(6,46,15,38),(7,47,16,39),(8,48,17,40),(9,49,18,41),(10,50,19,42),(11,51,20,29),(12,52,21,30),(13,53,22,31),(14,54,23,32)]])
D14⋊C4 is a maximal subgroup of
C42⋊D7 C4×D28 C4.D28 C42⋊2D7 D7×C22⋊C4 Dic7⋊4D4 C22⋊D28 D14.D4 D14⋊D4 Dic7.D4 C22.D28 C4⋊C4⋊7D7 D28⋊C4 D14.5D4 C4⋊D28 D14⋊Q8 D14⋊2Q8 C4⋊C4⋊D7 C4×C7⋊D4 C23.23D14 C28⋊7D4 C23⋊D14 Dic7⋊D4 D14⋊3Q8 C28.23D4 D14⋊C12 D14⋊Dic3 D42⋊C4 C2.D84
D14⋊C4 is a maximal quotient of
Dic14⋊C4 C23.1D14 C14.D8 C14.Q16 C28.44D4 D14⋊C8 C2.D56 C28.46D4 C4.12D28 D28⋊4C4 C14.C42 D14⋊Dic3 D42⋊C4 C2.D84
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D7 | D14 | C4×D7 | D28 | C7⋊D4 |
| kernel | D14⋊C4 | C2×Dic7 | C2×C28 | C22×D7 | D14 | C14 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 3 | 6 | 6 | 6 |
Matrix representation of D14⋊C4 ►in GL4(𝔽29) generated by
| 25 | 25 | 0 | 0 |
| 4 | 11 | 0 | 0 |
| 0 | 0 | 11 | 8 |
| 0 | 0 | 18 | 0 |
| 4 | 4 | 0 | 0 |
| 18 | 25 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 24 | 28 |
| 17 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 16 | 18 |
| 0 | 0 | 26 | 13 |
G:=sub<GL(4,GF(29))| [25,4,0,0,25,11,0,0,0,0,11,18,0,0,8,0],[4,18,0,0,4,25,0,0,0,0,1,24,0,0,0,28],[17,0,0,0,0,17,0,0,0,0,16,26,0,0,18,13] >;
D14⋊C4 in GAP, Magma, Sage, TeX
D_{14}\rtimes C_4 % in TeX
G:=Group("D14:C4"); // GroupNames label
G:=SmallGroup(112,13);
// by ID
G=gap.SmallGroup(112,13);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,101,26,2404]);
// Polycyclic
G:=Group<a,b,c|a^14=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;
// generators/relations
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