direct product, metabelian, supersoluble, monomial
Aliases: Q8xF7, Dic14:3C6, D7:(C3xQ8), C7:2(C6xQ8), (Q8xD7):3C3, (C7xQ8):4C6, C4.F7:3C2, C4.6(C2xF7), C28.6(C2xC6), (C4xD7).1C6, (C4xF7).1C2, D14.5(C2xC6), C7:C12.4C22, C2.8(C22xF7), C14.7(C22xC6), Dic7.4(C2xC6), (C2xF7).5C22, C7:C3:2(C2xQ8), (Q8xC7:C3):2C2, (C4xC7:C3).6C22, (C2xC7:C3).7C23, SmallGroup(336,127)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8xF7
G = < a,b,c,d | a4=c7=d6=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 268 in 76 conjugacy classes, 44 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C7, C2xC4, Q8, Q8, C12, C2xC6, D7, C14, C2xQ8, C7:C3, C2xC12, C3xQ8, Dic7, C28, D14, F7, C2xC7:C3, C6xQ8, Dic14, C4xD7, C7xQ8, C7:C12, C4xC7:C3, C2xF7, Q8xD7, C4.F7, C4xF7, Q8xC7:C3, Q8xF7
Quotients: C1, C2, C3, C22, C6, Q8, C23, C2xC6, C2xQ8, C3xQ8, C22xC6, F7, C6xQ8, C2xF7, C22xF7, Q8xF7
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(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)
(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 8)(2 11 3 14 5 13)(4 10 7 12 6 9)(15 22)(16 25 17 28 19 27)(18 24 21 26 20 23)(29 36)(30 39 31 42 33 41)(32 38 35 40 34 37)(43 50)(44 53 45 56 47 55)(46 52 49 54 48 51)
G:=sub<Sym(56)| (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(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), (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), (1,8)(2,11,3,14,5,13)(4,10,7,12,6,9)(15,22)(16,25,17,28,19,27)(18,24,21,26,20,23)(29,36)(30,39,31,42,33,41)(32,38,35,40,34,37)(43,50)(44,53,45,56,47,55)(46,52,49,54,48,51)>;
G:=Group( (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(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), (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), (1,8)(2,11,3,14,5,13)(4,10,7,12,6,9)(15,22)(16,25,17,28,19,27)(18,24,21,26,20,23)(29,36)(30,39,31,42,33,41)(32,38,35,40,34,37)(43,50)(44,53,45,56,47,55)(46,52,49,54,48,51) );
G=PermutationGroup([[(1,22,8,15),(2,23,9,16),(3,24,10,17),(4,25,11,18),(5,26,12,19),(6,27,13,20),(7,28,14,21),(29,43,36,50),(30,44,37,51),(31,45,38,52),(32,46,39,53),(33,47,40,54),(34,48,41,55),(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)], [(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)], [(1,8),(2,11,3,14,5,13),(4,10,7,12,6,9),(15,22),(16,25,17,28,19,27),(18,24,21,26,20,23),(29,36),(30,39,31,42,33,41),(32,38,35,40,34,37),(43,50),(44,53,45,56,47,55),(46,52,49,54,48,51)]])
35 conjugacy classes
class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 7 | 12A | ··· | 12L | 14 | 28A | 28B | 28C |
order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 7 | 12 | ··· | 12 | 14 | 28 | 28 | 28 |
size | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 14 | 14 | 14 | 7 | ··· | 7 | 6 | 14 | ··· | 14 | 6 | 12 | 12 | 12 |
35 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 6 | 6 |
type | + | + | + | + | - | - | + | + | |||||
image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | Q8xF7 | Q8 | C3xQ8 | F7 | C2xF7 |
kernel | Q8xF7 | C4.F7 | C4xF7 | Q8xC7:C3 | Q8xD7 | Dic14 | C4xD7 | C7xQ8 | C1 | F7 | D7 | Q8 | C4 |
# reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 1 | 2 | 4 | 1 | 3 |
Matrix representation of Q8xF7 ►in GL8(F337)
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
336 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
138 | 294 | 0 | 0 | 0 | 0 | 0 | 0 |
294 | 199 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 336 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 336 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 336 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 336 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 336 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 336 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 336 | 336 | 336 | 336 | 336 | 336 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
209 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 209 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 336 | 336 | 336 | 336 | 336 | 336 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(337))| [0,336,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[138,294,0,0,0,0,0,0,294,199,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,336,1,0,0,0,0,0,0,336,0,1,0,0,0,0,0,336,0,0,1,0,0,0,0,336,0,0,0,1,0,0,0,336,0,0,0,0,1,0,0,336,0,0,0,0,0],[209,0,0,0,0,0,0,0,0,209,0,0,0,0,0,0,0,0,1,0,0,0,336,0,0,0,0,0,0,1,336,0,0,0,0,0,0,0,336,0,0,0,0,0,1,0,336,0,0,0,0,0,0,0,336,1,0,0,0,1,0,0,336,0] >;
Q8xF7 in GAP, Magma, Sage, TeX
Q_8\times F_7
% in TeX
G:=Group("Q8xF7");
// GroupNames label
G:=SmallGroup(336,127);
// by ID
G=gap.SmallGroup(336,127);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-7,151,260,122,10373,887]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^7=d^6=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations