metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D28, D28⋊5C2, C4○Dic14, C4.16D14, Dic14⋊5C2, C14.4C23, C22.2D14, C28.16C22, D14.1C22, Dic7.2C22, (C2×C4)⋊3D7, (C2×C28)⋊4C2, (C4×D7)⋊4C2, C4○(C7⋊D4), C7⋊1(C4○D4), C7⋊D4⋊3C2, C2.5(C22×D7), (C2×C14).11C22, SmallGroup(112,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D28
G = < a,b,c | a4=c2=1, b14=a2, ab=ba, ac=ca, cbc=a2b13 >
Smallest permutation representation of C4○D28
►On 56 points
(1 32 15 46)(2 33 16 47)(3 34 17 48)(4 35 18 49)(5 36 19 50)(6 37 20 51)(7 38 21 52)(8 39 22 53)(9 40 23 54)(10 41 24 55)(11 42 25 56)(12 43 26 29)(13 44 27 30)(14 45 28 31)
(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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)
G:=sub<Sym(56)| (1,32,15,46)(2,33,16,47)(3,34,17,48)(4,35,18,49)(5,36,19,50)(6,37,20,51)(7,38,21,52)(8,39,22,53)(9,40,23,54)(10,41,24,55)(11,42,25,56)(12,43,26,29)(13,44,27,30)(14,45,28,31), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)>;
G:=Group( (1,32,15,46)(2,33,16,47)(3,34,17,48)(4,35,18,49)(5,36,19,50)(6,37,20,51)(7,38,21,52)(8,39,22,53)(9,40,23,54)(10,41,24,55)(11,42,25,56)(12,43,26,29)(13,44,27,30)(14,45,28,31), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50) );
G=PermutationGroup([[(1,32,15,46),(2,33,16,47),(3,34,17,48),(4,35,18,49),(5,36,19,50),(6,37,20,51),(7,38,21,52),(8,39,22,53),(9,40,23,54),(10,41,24,55),(11,42,25,56),(12,43,26,29),(13,44,27,30),(14,45,28,31)], [(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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50)]])
C4○D28 is a maximal subgroup of
Dic14⋊C4 D28⋊4C4 D28.2C4 D56⋊7C2 D28.C4 C8⋊D14 C8.D14 D4.D14 C28.C23 D4.8D14 D4⋊6D14 Q8.10D14 D7×C4○D4 D4⋊8D14 D4.10D14 D28⋊6C6 D28⋊5S3 D84⋊C2 D6.D14 Dic7.D6 D84⋊11C2
C4○D28 is a maximal quotient of
C4×Dic14 C28.6Q8 C42⋊D7 C4×D28 C4.D28 C42⋊2D7 C23.D14 D14.D4 D14⋊D4 Dic7.D4 Dic7.Q8 D14.5D4 D14⋊Q8 C4⋊C4⋊D7 C28.48D4 C23.21D14 C4×C7⋊D4 C23.23D14 C28⋊7D4 D28⋊5S3 D84⋊C2 D6.D14 Dic7.D6 D84⋊11C2
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 14 | 14 | 1 | 1 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | C4○D28 |
| kernel | C4○D28 | Dic14 | C4×D7 | D28 | C7⋊D4 | C2×C28 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 2 | 6 | 3 | 12 |
Matrix representation of C4○D28 ►in GL2(𝔽29) generated by
| 17 | 0 |
| 0 | 17 |
| 17 | 26 |
| 18 | 2 |
| 1 | 0 |
| 5 | 28 |
G:=sub<GL(2,GF(29))| [17,0,0,17],[17,18,26,2],[1,5,0,28] >;
C4○D28 in GAP, Magma, Sage, TeX
C_4\circ D_{28} % in TeX
G:=Group("C4oD28"); // GroupNames label
G:=SmallGroup(112,30);
// by ID
G=gap.SmallGroup(112,30);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,46,182,2404]);
// Polycyclic
G:=Group<a,b,c|a^4=c^2=1,b^14=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^13>;
// generators/relations
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