metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.1D14, C22.2D28, (C2×Dic7)⋊C4, (C22×D7)⋊C4, C7⋊1(C23⋊C4), C22⋊C4⋊1D7, (C2×C14).27D4, C23.D7⋊1C2, C22.3(C4×D7), C2.4(D14⋊C4), C14.2(C22⋊C4), C22.8(C7⋊D4), (C22×C14).5C22, (C7×C22⋊C4)⋊1C2, (C2×C14).1(C2×C4), (C2×C7⋊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 >
Smallest permutation representation of C23.1D14
►On 56 points
(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 D7×C23⋊C4 (C2×D28)⋊13C4 C24⋊D14 C22⋊C4⋊D14
C23.1D14 is a maximal quotient of
C14.C4≀C2 C4⋊Dic7⋊C4 C23.30D28 (C22×D7)⋊C8 (C2×Dic7)⋊C8 C22.2D56 C7⋊C2≀C4 (C2×C28).D4 C23.D28 C23.2D28 C23.3D28 C23.4D28 (C2×C4).D28 (C2×Q8).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 | C4×D7 | D28 | C7⋊D4 | C23⋊C4 | C23.1D14 |
| kernel | C23.1D14 | C23.D7 | C7×C22⋊C4 | C2×C7⋊D4 | C2×Dic7 | C22×D7 | C2×C14 | 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