metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C4.Dic7, C28.1C4, C7⋊2M4(2), C4.15D14, C22.Dic7, C28.15C22, C7⋊C8⋊5C2, (C2×C4).2D7, C14.7(C2×C4), (C2×C28).5C2, (C2×C14).3C4, C2.3(C2×Dic7), SmallGroup(112,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.Dic7
G = < a,b,c | a4=1, b14=a2, c2=a2b7, ab=ba, cac-1=a-1, cbc-1=b13 >
Smallest permutation representation of C4.Dic7
►On 56 points
(1 22 15 8)(2 23 16 9)(3 24 17 10)(4 25 18 11)(5 26 19 12)(6 27 20 13)(7 28 21 14)(29 36 43 50)(30 37 44 51)(31 38 45 52)(32 39 46 53)(33 40 47 54)(34 41 48 55)(35 42 49 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 43 22 36 15 29 8 50)(2 56 23 49 16 42 9 35)(3 41 24 34 17 55 10 48)(4 54 25 47 18 40 11 33)(5 39 26 32 19 53 12 46)(6 52 27 45 20 38 13 31)(7 37 28 30 21 51 14 44)
G:=sub<Sym(56)| (1,22,15,8)(2,23,16,9)(3,24,17,10)(4,25,18,11)(5,26,19,12)(6,27,20,13)(7,28,21,14)(29,36,43,50)(30,37,44,51)(31,38,45,52)(32,39,46,53)(33,40,47,54)(34,41,48,55)(35,42,49,56), (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), (1,43,22,36,15,29,8,50)(2,56,23,49,16,42,9,35)(3,41,24,34,17,55,10,48)(4,54,25,47,18,40,11,33)(5,39,26,32,19,53,12,46)(6,52,27,45,20,38,13,31)(7,37,28,30,21,51,14,44)>;
G:=Group( (1,22,15,8)(2,23,16,9)(3,24,17,10)(4,25,18,11)(5,26,19,12)(6,27,20,13)(7,28,21,14)(29,36,43,50)(30,37,44,51)(31,38,45,52)(32,39,46,53)(33,40,47,54)(34,41,48,55)(35,42,49,56), (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), (1,43,22,36,15,29,8,50)(2,56,23,49,16,42,9,35)(3,41,24,34,17,55,10,48)(4,54,25,47,18,40,11,33)(5,39,26,32,19,53,12,46)(6,52,27,45,20,38,13,31)(7,37,28,30,21,51,14,44) );
G=PermutationGroup([[(1,22,15,8),(2,23,16,9),(3,24,17,10),(4,25,18,11),(5,26,19,12),(6,27,20,13),(7,28,21,14),(29,36,43,50),(30,37,44,51),(31,38,45,52),(32,39,46,53),(33,40,47,54),(34,41,48,55),(35,42,49,56)], [(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)], [(1,43,22,36,15,29,8,50),(2,56,23,49,16,42,9,35),(3,41,24,34,17,55,10,48),(4,54,25,47,18,40,11,33),(5,39,26,32,19,53,12,46),(6,52,27,45,20,38,13,31),(7,37,28,30,21,51,14,44)]])
C4.Dic7 is a maximal subgroup of
Dic14⋊C4 C56.C4 C28.53D4 C28.46D4 C4.12D28 C28.D4 C28.10D4 D4⋊2Dic7 D28.2C4 D7×M4(2) D4.D14 C28.C23 Q8.Dic7 D4⋊D14 D4.9D14 C28.C12 D6.Dic7 C84.C4
C4.Dic7 is a maximal quotient of
C42.D7 C28⋊C8 C28.55D4 D6.Dic7 C84.C4
34 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | D7 | M4(2) | Dic7 | D14 | Dic7 | C4.Dic7 |
| kernel | C4.Dic7 | C7⋊C8 | C2×C28 | C28 | C2×C14 | C2×C4 | C7 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 3 | 2 | 3 | 3 | 3 | 12 |
Matrix representation of C4.Dic7 ►in GL2(𝔽29) generated by
| 17 | 0 |
| 0 | 12 |
| 26 | 0 |
| 0 | 10 |
| 0 | 12 |
| 1 | 0 |
G:=sub<GL(2,GF(29))| [17,0,0,12],[26,0,0,10],[0,1,12,0] >;
C4.Dic7 in GAP, Magma, Sage, TeX
C_4.{\rm Dic}_7 % in TeX
G:=Group("C4.Dic7"); // GroupNames label
G:=SmallGroup(112,9);
// by ID
G=gap.SmallGroup(112,9);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,20,101,42,2404]);
// Polycyclic
G:=Group<a,b,c|a^4=1,b^14=a^2,c^2=a^2*b^7,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^13>;
// generators/relations
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