metabelian, supersoluble, monomial
Aliases: C72⋊2Q8, Dic7.D7, C7⋊1Dic14, C14.5D14, C2.5D72, C7⋊Dic7.2C2, (C7×C14).5C22, (C7×Dic7).1C2, SmallGroup(392,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C72⋊2Q8
G = < a,b,c,d | a7=b7=c4=1, d2=c2, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Smallest permutation representation of C72⋊2Q8
►On 56 points
Generators in S56
(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 6 5 4 3 2)(8 14 13 12 11 10 9)(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 49 48 47 46 45 44)(50 56 55 54 53 52 51)
(1 27 8 20)(2 26 9 19)(3 25 10 18)(4 24 11 17)(5 23 12 16)(6 22 13 15)(7 28 14 21)(29 48 36 55)(30 47 37 54)(31 46 38 53)(32 45 39 52)(33 44 40 51)(34 43 41 50)(35 49 42 56)
(1 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)
G:=sub<Sym(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), (1,7,6,5,4,3,2)(8,14,13,12,11,10,9)(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)(43,49,48,47,46,45,44)(50,56,55,54,53,52,51), (1,27,8,20)(2,26,9,19)(3,25,10,18)(4,24,11,17)(5,23,12,16)(6,22,13,15)(7,28,14,21)(29,48,36,55)(30,47,37,54)(31,46,38,53)(32,45,39,52)(33,44,40,51)(34,43,41,50)(35,49,42,56), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49)>;
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)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56), (1,7,6,5,4,3,2)(8,14,13,12,11,10,9)(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)(43,49,48,47,46,45,44)(50,56,55,54,53,52,51), (1,27,8,20)(2,26,9,19)(3,25,10,18)(4,24,11,17)(5,23,12,16)(6,22,13,15)(7,28,14,21)(29,48,36,55)(30,47,37,54)(31,46,38,53)(32,45,39,52)(33,44,40,51)(34,43,41,50)(35,49,42,56), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49) );
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),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56)], [(1,7,6,5,4,3,2),(8,14,13,12,11,10,9),(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),(43,49,48,47,46,45,44),(50,56,55,54,53,52,51)], [(1,27,8,20),(2,26,9,19),(3,25,10,18),(4,24,11,17),(5,23,12,16),(6,22,13,15),(7,28,14,21),(29,48,36,55),(30,47,37,54),(31,46,38,53),(32,45,39,52),(33,44,40,51),(34,43,41,50),(35,49,42,56)], [(1,36,8,29),(2,37,9,30),(3,38,10,31),(4,39,11,32),(5,40,12,33),(6,41,13,34),(7,42,14,35),(15,50,22,43),(16,51,23,44),(17,52,24,45),(18,53,25,46),(19,54,26,47),(20,55,27,48),(21,56,28,49)]])
47 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 7A | ··· | 7F | 7G | ··· | 7O | 14A | ··· | 14F | 14G | ··· | 14O | 28A | ··· | 28L |
| order | 1 | 2 | 4 | 4 | 4 | 7 | ··· | 7 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 14 | 14 | 98 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 14 | ··· | 14 |
47 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | + | + | - | + | - |
| image | C1 | C2 | C2 | Q8 | D7 | D14 | Dic14 | D72 | C72⋊2Q8 |
| kernel | C72⋊2Q8 | C7×Dic7 | C7⋊Dic7 | C72 | Dic7 | C14 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 6 | 6 | 12 | 9 | 9 |
Matrix representation of C72⋊2Q8 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 14 |
| 0 | 0 | 0 | 0 | 18 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 26 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 11 | 1 |
| 13 | 27 | 0 | 0 | 0 | 0 |
| 27 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 19 | 0 | 0 |
| 0 | 0 | 24 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,18,0,0,0,0,14,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,26,0,0,0,0,10,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,11,0,0,0,0,0,1],[13,27,0,0,0,0,27,16,0,0,0,0,0,0,3,24,0,0,0,0,19,26,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C72⋊2Q8 in GAP, Magma, Sage, TeX
C_7^2\rtimes_2Q_8
% in TeX
G:=Group("C7^2:2Q8"); // GroupNames label
G:=SmallGroup(392,22);
// by ID
G=gap.SmallGroup(392,22);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-7,20,61,26,488,8404]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^7=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
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