direct product, metabelian, soluble, monomial, A-group
Aliases: C7×C32⋊C4, C32⋊C28, C3⋊S3.C14, (C3×C21)⋊1C4, (C7×C3⋊S3).1C2, SmallGroup(252,31)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C7×C32⋊C4 |
Generators and relations for C7×C32⋊C4
G = < a,b,c,d | a7=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >
Smallest permutation representation of C7×C32⋊C4
►On 42 points
Generators in S42
(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)
(8 42 16)(9 36 17)(10 37 18)(11 38 19)(12 39 20)(13 40 21)(14 41 15)
(1 30 23)(2 31 24)(3 32 25)(4 33 26)(5 34 27)(6 35 28)(7 29 22)(8 42 16)(9 36 17)(10 37 18)(11 38 19)(12 39 20)(13 40 21)(14 41 15)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 36)(8 35 16 28)(9 29 17 22)(10 30 18 23)(11 31 19 24)(12 32 20 25)(13 33 21 26)(14 34 15 27)
G:=sub<Sym(42)| (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), (8,42,16)(9,36,17)(10,37,18)(11,38,19)(12,39,20)(13,40,21)(14,41,15), (1,30,23)(2,31,24)(3,32,25)(4,33,26)(5,34,27)(6,35,28)(7,29,22)(8,42,16)(9,36,17)(10,37,18)(11,38,19)(12,39,20)(13,40,21)(14,41,15), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,36)(8,35,16,28)(9,29,17,22)(10,30,18,23)(11,31,19,24)(12,32,20,25)(13,33,21,26)(14,34,15,27)>;
G:=Group( (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), (8,42,16)(9,36,17)(10,37,18)(11,38,19)(12,39,20)(13,40,21)(14,41,15), (1,30,23)(2,31,24)(3,32,25)(4,33,26)(5,34,27)(6,35,28)(7,29,22)(8,42,16)(9,36,17)(10,37,18)(11,38,19)(12,39,20)(13,40,21)(14,41,15), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,36)(8,35,16,28)(9,29,17,22)(10,30,18,23)(11,31,19,24)(12,32,20,25)(13,33,21,26)(14,34,15,27) );
G=PermutationGroup([[(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)], [(8,42,16),(9,36,17),(10,37,18),(11,38,19),(12,39,20),(13,40,21),(14,41,15)], [(1,30,23),(2,31,24),(3,32,25),(4,33,26),(5,34,27),(6,35,28),(7,29,22),(8,42,16),(9,36,17),(10,37,18),(11,38,19),(12,39,20),(13,40,21),(14,41,15)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,36),(8,35,16,28),(9,29,17,22),(10,30,18,23),(11,31,19,24),(12,32,20,25),(13,33,21,26),(14,34,15,27)]])
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 7A | ··· | 7F | 14A | ··· | 14F | 21A | ··· | 21L | 28A | ··· | 28L |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 |
| size | 1 | 9 | 4 | 4 | 9 | 9 | 1 | ··· | 1 | 9 | ··· | 9 | 4 | ··· | 4 | 9 | ··· | 9 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | |||||
| image | C1 | C2 | C4 | C7 | C14 | C28 | C32⋊C4 | C7×C32⋊C4 |
| kernel | C7×C32⋊C4 | C7×C3⋊S3 | C3×C21 | C32⋊C4 | C3⋊S3 | C32 | C7 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 2 | 12 |
Matrix representation of C7×C32⋊C4 ►in GL4(𝔽337) generated by
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 336 | 336 | 336 | 336 |
| 0 | 336 | 0 | 0 |
| 1 | 336 | 0 | 0 |
| 0 | 1 | 0 | 1 |
| 336 | 0 | 336 | 336 |
| 0 | 0 | 336 | 1 |
| 336 | 336 | 335 | 336 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 |
G:=sub<GL(4,GF(337))| [52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[1,0,0,336,0,1,0,336,0,0,0,336,0,0,1,336],[0,1,0,336,336,336,1,0,0,0,0,336,0,0,1,336],[0,336,0,0,0,336,0,1,336,335,1,1,1,336,0,0] >;
C7×C32⋊C4 in GAP, Magma, Sage, TeX
C_7\times C_3^2\rtimes C_4
% in TeX
G:=Group("C7xC3^2:C4"); // GroupNames label
G:=SmallGroup(252,31);
// by ID
G=gap.SmallGroup(252,31);
# by ID
G:=PCGroup([5,-2,-7,-2,-3,3,70,3923,93,5604,314]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations
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