metabelian, supersoluble, monomial
Aliases: C32.F7, D7⋊33- 1+2, C7⋊C9⋊3C6, C7⋊C18⋊3C3, C3.5(C3×F7), C21.5(C3×C6), (C3×C21).4C6, C21.C32⋊C2, (C32×D7).3C3, (C3×D7).5C32, C7⋊3(C2×3- 1+2), SmallGroup(378,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32.F7
G = < a,b,c,d | a3=b3=c7=1, d6=b-1, ab=ba, ac=ca, dad-1=ab-1, bc=cb, bd=db, dcd-1=c5 >
Smallest permutation representation of C32.F7
►On 63 points
Generators in S63
(2 8 5)(3 6 9)(11 17 23)(12 24 18)(14 20 26)(15 27 21)(28 40 34)(30 36 42)(31 43 37)(33 39 45)(47 53 59)(48 60 54)(50 56 62)(51 63 57)
(1 4 7)(2 5 8)(3 6 9)(10 22 16)(11 23 17)(12 24 18)(13 25 19)(14 26 20)(15 27 21)(28 40 34)(29 41 35)(30 42 36)(31 43 37)(32 44 38)(33 45 39)(46 58 52)(47 59 53)(48 60 54)(49 61 55)(50 62 56)(51 63 57)
(1 32 10 49 58 19 41)(2 20 50 33 42 59 11)(3 60 34 21 12 43 51)(4 44 22 61 52 13 35)(5 14 62 45 36 53 23)(6 54 28 15 24 37 63)(7 38 16 55 46 25 29)(8 26 56 39 30 47 17)(9 48 40 27 18 31 57)
(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 57 58 59 60 61 62 63)
G:=sub<Sym(63)| (2,8,5)(3,6,9)(11,17,23)(12,24,18)(14,20,26)(15,27,21)(28,40,34)(30,36,42)(31,43,37)(33,39,45)(47,53,59)(48,60,54)(50,56,62)(51,63,57), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(46,58,52)(47,59,53)(48,60,54)(49,61,55)(50,62,56)(51,63,57), (1,32,10,49,58,19,41)(2,20,50,33,42,59,11)(3,60,34,21,12,43,51)(4,44,22,61,52,13,35)(5,14,62,45,36,53,23)(6,54,28,15,24,37,63)(7,38,16,55,46,25,29)(8,26,56,39,30,47,17)(9,48,40,27,18,31,57), (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)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)>;
G:=Group( (2,8,5)(3,6,9)(11,17,23)(12,24,18)(14,20,26)(15,27,21)(28,40,34)(30,36,42)(31,43,37)(33,39,45)(47,53,59)(48,60,54)(50,56,62)(51,63,57), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(46,58,52)(47,59,53)(48,60,54)(49,61,55)(50,62,56)(51,63,57), (1,32,10,49,58,19,41)(2,20,50,33,42,59,11)(3,60,34,21,12,43,51)(4,44,22,61,52,13,35)(5,14,62,45,36,53,23)(6,54,28,15,24,37,63)(7,38,16,55,46,25,29)(8,26,56,39,30,47,17)(9,48,40,27,18,31,57), (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)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63) );
G=PermutationGroup([[(2,8,5),(3,6,9),(11,17,23),(12,24,18),(14,20,26),(15,27,21),(28,40,34),(30,36,42),(31,43,37),(33,39,45),(47,53,59),(48,60,54),(50,56,62),(51,63,57)], [(1,4,7),(2,5,8),(3,6,9),(10,22,16),(11,23,17),(12,24,18),(13,25,19),(14,26,20),(15,27,21),(28,40,34),(29,41,35),(30,42,36),(31,43,37),(32,44,38),(33,45,39),(46,58,52),(47,59,53),(48,60,54),(49,61,55),(50,62,56),(51,63,57)], [(1,32,10,49,58,19,41),(2,20,50,33,42,59,11),(3,60,34,21,12,43,51),(4,44,22,61,52,13,35),(5,14,62,45,36,53,23),(6,54,28,15,24,37,63),(7,38,16,55,46,25,29),(8,26,56,39,30,47,17),(9,48,40,27,18,31,57)], [(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),(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)]])
31 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 7 | 9A | ··· | 9F | 18A | ··· | 18F | 21A | ··· | 21H |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 7 | 9 | ··· | 9 | 18 | ··· | 18 | 21 | ··· | 21 |
| size | 1 | 7 | 1 | 1 | 3 | 3 | 7 | 7 | 21 | 21 | 6 | 21 | ··· | 21 | 21 | ··· | 21 | 6 | ··· | 6 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 6 | 6 | 6 |
| type | + | + | + | ||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | 3- 1+2 | C2×3- 1+2 | F7 | C3×F7 | C32.F7 |
| kernel | C32.F7 | C21.C32 | C7⋊C18 | C32×D7 | C7⋊C9 | C3×C21 | D7 | C7 | C32 | C3 | C1 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 2 | 2 | 1 | 2 | 6 |
Matrix representation of C32.F7 ►in GL6(𝔽127)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 0 | 0 | 0 |
| 0 | 0 | 0 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 107 | 0 |
| 0 | 0 | 0 | 0 | 0 | 107 |
| 19 | 0 | 0 | 0 | 0 | 0 |
| 0 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 0 | 0 | 0 |
| 0 | 0 | 0 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 0 |
| 0 | 0 | 0 | 0 | 0 | 19 |
| 37 | 126 | 0 | 0 | 0 | 0 |
| 38 | 126 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 25 | 0 | 0 |
| 0 | 0 | 66 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 | 91 |
| 0 | 0 | 0 | 0 | 98 | 90 |
| 0 | 0 | 90 | 1 | 0 | 0 |
| 0 | 0 | 29 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 90 | 1 |
| 0 | 0 | 0 | 0 | 29 | 37 |
| 59 | 19 | 0 | 0 | 0 | 0 |
| 43 | 68 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(127))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,0,0,0,0,0,0,19,0,0,0,0,0,0,107,0,0,0,0,0,0,107],[19,0,0,0,0,0,0,19,0,0,0,0,0,0,19,0,0,0,0,0,0,19,0,0,0,0,0,0,19,0,0,0,0,0,0,19],[37,38,0,0,0,0,126,126,0,0,0,0,0,0,66,66,0,0,0,0,25,0,0,0,0,0,0,0,61,98,0,0,0,0,91,90],[0,0,0,0,59,43,0,0,0,0,19,68,90,29,0,0,0,0,1,37,0,0,0,0,0,0,90,29,0,0,0,0,1,37,0,0] >;
C32.F7 in GAP, Magma, Sage, TeX
C_3^2.F_7
% in TeX
G:=Group("C3^2.F7"); // GroupNames label
G:=SmallGroup(378,11);
// by ID
G=gap.SmallGroup(378,11);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,96,187,8104,2709]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^7=1,d^6=b^-1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations
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