metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.1D14, C22.2D28, (C2xDic7):C4, (C22xD7):C4, C7:1(C23:C4), C22:C4:1D7, (C2xC14).27D4, C23.D7:1C2, C22.3(C4xD7), C2.4(D14:C4), C14.2(C22:C4), C22.8(C7:D4), (C22xC14).5C22, (C7xC22:C4):1C2, (C2xC14).1(C2xC4), (C2xC7:D4).1C2, SmallGroup(224,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.1D14
G = < a,b,c,d | a2=b2=c28=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >
Quotients: C1, C2, C4, C22, C2xC4, D4, D7, C22:C4, D14, C23:C4, C4xD7, D28, C7:D4, D14:C4, C23.1D14
2C2
2C2
2C2
28C2
4C4
4C22
14C22
14C4
28C4
28C22
2C14
2C14
2C14
4D7
7C23
14C2xC4
14D4
14D4
2Dic7
2D14
4Dic7
4D14
4C28
Smallest permutation representation of C23.1D14
►On 56 points
Generators in S56
(1 29)(3 31)(5 33)(7 35)(9 37)(11 39)(13 41)(15 43)(17 45)(19 47)(21 49)(23 51)(25 53)(27 55)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 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 35 29 7)(2 34)(3 5 31 33)(6 30)(8 28)(9 55 37 27)(10 54)(11 25 39 53)(12 24)(13 51 41 23)(14 50)(15 21 43 49)(16 20)(17 47 45 19)(18 46)(22 42)(26 38)(36 56)(40 52)(44 48)
G:=sub<Sym(56)| (1,29)(3,31)(5,33)(7,35)(9,37)(11,39)(13,41)(15,43)(17,45)(19,47)(21,49)(23,51)(25,53)(27,55), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56), (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), (1,35,29,7)(2,34)(3,5,31,33)(6,30)(8,28)(9,55,37,27)(10,54)(11,25,39,53)(12,24)(13,51,41,23)(14,50)(15,21,43,49)(16,20)(17,47,45,19)(18,46)(22,42)(26,38)(36,56)(40,52)(44,48)>;
G:=Group( (1,29)(3,31)(5,33)(7,35)(9,37)(11,39)(13,41)(15,43)(17,45)(19,47)(21,49)(23,51)(25,53)(27,55), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56), (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), (1,35,29,7)(2,34)(3,5,31,33)(6,30)(8,28)(9,55,37,27)(10,54)(11,25,39,53)(12,24)(13,51,41,23)(14,50)(15,21,43,49)(16,20)(17,47,45,19)(18,46)(22,42)(26,38)(36,56)(40,52)(44,48) );
G=PermutationGroup([[(1,29),(3,31),(5,33),(7,35),(9,37),(11,39),(13,41),(15,43),(17,45),(19,47),(21,49),(23,51),(25,53),(27,55)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56)], [(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)], [(1,35,29,7),(2,34),(3,5,31,33),(6,30),(8,28),(9,55,37,27),(10,54),(11,25,39,53),(12,24),(13,51,41,23),(14,50),(15,21,43,49),(16,20),(17,47,45,19),(18,46),(22,42),(26,38),(36,56),(40,52),(44,48)]])
C23.1D14 is a maximal subgroup of
C23:C4:5D7 C23:D28 C23.5D28 D7xC23:C4 (C2xD28):13C4 C24:D14 C22:C4:D14
C23.1D14 is a maximal quotient of
C14.C4wrC2 C4:Dic7:C4 C23.30D28 (C22xD7):C8 (C2xDic7):C8 C22.2D56 C7:C2wrC4 (C2xC28).D4 C23.D28 C23.2D28 C23.3D28 C23.4D28 (C2xC4).D28 (C2xQ8).D14 C24.2D14
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 28 | 4 | 4 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D7 | D14 | C4xD7 | D28 | C7:D4 | C23:C4 | C23.1D14 |
| kernel | C23.1D14 | C23.D7 | C7xC22:C4 | C2xC7:D4 | C2xDic7 | C22xD7 | C2xC14 | C22:C4 | C23 | C22 | C22 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 1 | 6 |
Matrix representation of C23.1D14 ►in GL4(F29) generated by
| 28 | 0 | 3 | 3 |
| 0 | 28 | 12 | 12 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 12 | 14 | 1 | 22 |
| 19 | 27 | 5 | 12 |
| 19 | 1 | 9 | 9 |
| 18 | 18 | 10 | 10 |
| 13 | 15 | 10 | 22 |
| 8 | 16 | 22 | 12 |
| 0 | 0 | 19 | 10 |
| 0 | 0 | 22 | 10 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,3,12,1,0,3,12,0,1],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[12,19,19,18,14,27,1,18,1,5,9,10,22,12,9,10],[13,8,0,0,15,16,0,0,10,22,19,22,22,12,10,10] >;
C23.1D14 in GAP, Magma, Sage, TeX
C_2^3._1D_{14} % in TeX
G:=Group("C2^3.1D14"); // GroupNames label
G:=SmallGroup(224,12);
// by ID
G=gap.SmallGroup(224,12);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,121,31,362,297,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^28=1,d^2=a,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations
Export