direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D28, C4⋊2D14, C14⋊1D4, C28⋊2C22, D14⋊1C22, C14.3C23, C22.10D14, C7⋊1(C2×D4), (C2×C4)⋊2D7, (C2×C28)⋊3C2, (C22×D7)⋊1C2, C2.4(C22×D7), (C2×C14).10C22, SmallGroup(112,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D28
G = < a,b,c | a2=b28=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 232 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, D4, C23, D7, C14, C14, C2×D4, C28, D14, D14, C2×C14, D28, C2×C28, C22×D7, C2×D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, D28, C22×D7, C2×D28
►On 56 points
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 29)(25 30)(26 31)(27 32)(28 33)
(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)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 38)(30 37)(31 36)(32 35)(33 34)(39 56)(40 55)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)
G:=sub<Sym(56)| (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,29)(25,30)(26,31)(27,32)(28,33), (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)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,38)(30,37)(31,36)(32,35)(33,34)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)>;
G:=Group( (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,29)(25,30)(26,31)(27,32)(28,33), (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)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,38)(30,37)(31,36)(32,35)(33,34)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48) );
G=PermutationGroup([[(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,29),(25,30),(26,31),(27,32),(28,33)], [(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),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(29,38),(30,37),(31,36),(32,35),(33,34),(39,56),(40,55),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48)]])
C2×D28 is a maximal subgroup of
C14.D8 C2.D56 C28.46D4 C28⋊4D4 C4.D28 C22⋊D28 D14⋊D4 D28⋊C4 D14.5D4 C4⋊D28 C8⋊D14 C28⋊7D4 C28⋊D4 C28.23D4 D4⋊D14 C2×D4×D7 D4⋊8D14
C2×D28 is a maximal quotient of
C28⋊2Q8 C28⋊4D4 C4.D28 C22⋊D28 C22.D28 C4⋊D28 D14⋊2Q8 D56⋊7C2 C8⋊D14 C8.D14 C28⋊7D4
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D28 |
| kernel | C2×D28 | D28 | C2×C28 | C22×D7 | C14 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 3 | 6 | 3 | 12 |
Matrix representation of C2×D28 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 1 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 22 | 24 |
| 0 | 0 | 25 | 26 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 24 | 17 |
| 0 | 0 | 2 | 5 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[0,28,0,0,1,0,0,0,0,0,22,25,0,0,24,26],[0,1,0,0,1,0,0,0,0,0,24,2,0,0,17,5] >;
C2×D28 in GAP, Magma, Sage, TeX
C_2\times D_{28} % in TeX
G:=Group("C2xD28"); // GroupNames label
G:=SmallGroup(112,29);
// by ID
G=gap.SmallGroup(112,29);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,182,42,2404]);
// Polycyclic
G:=Group<a,b,c|a^2=b^28=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations