direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7xC8:C22, C56:C23, D8:3D14, SD16:1D14, D56:1C22, D28:3C23, M4(2):7D14, C28.19C24, Dic14:3C23, (D7xD8):1C2, C7:C8:3C23, C4oD4:9D14, C8:1(C22xD7), (C2xD4):29D14, D8:D7:1C2, C8:D14:1C2, D56:C2:1C2, D4:D7:5C22, (D7xSD16):1C2, (C4xD7).42D4, C4.189(D4xD7), D4:3(C22xD7), (D4xD7):8C22, (C7xD8):1C22, (C7xD4):3C23, (C8xD7):1C22, Q8:D7:4C22, D4:D14:9C2, (Q8xD7):9C22, Q8:3(C22xD7), (C7xQ8):3C23, D14.67(C2xD4), C28.240(C2xD4), C4oD28:7C22, C56:C2:1C22, C8:D7:1C22, (D7xM4(2)):1C2, D4.D7:4C22, C4.19(C23xD7), C22.46(D4xD7), D4.D14:9C2, (D4xC14):21C22, D4:2D7:9C22, (C2xD28):35C22, Dic7.59(C2xD4), (C2xDic7).81D4, Q8:2D7:9C22, (C7xSD16):1C22, (C4xD7).12C23, (C2xC28).110C23, (C22xD7).101D4, C14.120(C22xD4), (C7xM4(2)):1C22, C4.Dic7:12C22, (C2xD4xD7):24C2, C7:4(C2xC8:C22), C2.93(C2xD4xD7), (D7xC4oD4):3C2, (C7xC8:C22):1C2, (C2xC14).65(C2xD4), (C7xC4oD4):5C22, (C2xC4xD7).160C22, (C2xC4).94(C22xD7), SmallGroup(448,1225)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7xC8:C22
G = < a,b,c,d,e | a7=b2=c8=d2=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, de=ed >
Subgroups: 1772 in 298 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2xC4, C2xC4, D4, D4, D4, Q8, Q8, C23, D7, D7, C14, C14, C2xC8, M4(2), M4(2), D8, D8, SD16, SD16, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C24, Dic7, Dic7, C28, C28, D14, D14, C2xC14, C2xC14, C2xM4(2), C2xD8, C2xSD16, C8:C22, C8:C22, C22xD4, C2xC4oD4, C7:C8, C56, Dic14, Dic14, C4xD7, C4xD7, D28, D28, D28, C2xDic7, C2xDic7, C7:D4, C2xC28, C2xC28, C7xD4, C7xD4, C7xD4, C7xQ8, C22xD7, C22xD7, C22xC14, C2xC8:C22, C8xD7, C8:D7, C56:C2, D56, C4.Dic7, D4:D7, D4.D7, Q8:D7, C7xM4(2), C7xD8, C7xSD16, C2xC4xD7, C2xC4xD7, C2xD28, C4oD28, C4oD28, D4xD7, D4xD7, D4xD7, D4:2D7, D4:2D7, Q8xD7, Q8:2D7, C2xC7:D4, D4xC14, C7xC4oD4, C23xD7, D7xM4(2), C8:D14, D7xD8, D8:D7, D7xSD16, D56:C2, D4.D14, D4:D14, C7xC8:C22, C2xD4xD7, D7xC4oD4, D7xC8:C22
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, C24, D14, C8:C22, C22xD4, C22xD7, C2xC8:C22, D4xD7, C23xD7, C2xD4xD7, D7xC8:C22
(1 37 55 14 17 45 27)(2 38 56 15 18 46 28)(3 39 49 16 19 47 29)(4 40 50 9 20 48 30)(5 33 51 10 21 41 31)(6 34 52 11 22 42 32)(7 35 53 12 23 43 25)(8 36 54 13 24 44 26)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(17 55)(18 56)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(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)(3 5)(4 8)(9 13)(10 16)(12 14)(17 23)(19 21)(20 24)(25 27)(26 30)(29 31)(33 39)(35 37)(36 40)(41 47)(43 45)(44 48)(49 51)(50 54)(53 55)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)
G:=sub<Sym(56)| (1,37,55,14,17,45,27)(2,38,56,15,18,46,28)(3,39,49,16,19,47,29)(4,40,50,9,20,48,30)(5,33,51,10,21,41,31)(6,34,52,11,22,42,32)(7,35,53,12,23,43,25)(8,36,54,13,24,44,26), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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)(3,5)(4,8)(9,13)(10,16)(12,14)(17,23)(19,21)(20,24)(25,27)(26,30)(29,31)(33,39)(35,37)(36,40)(41,47)(43,45)(44,48)(49,51)(50,54)(53,55), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)>;
G:=Group( (1,37,55,14,17,45,27)(2,38,56,15,18,46,28)(3,39,49,16,19,47,29)(4,40,50,9,20,48,30)(5,33,51,10,21,41,31)(6,34,52,11,22,42,32)(7,35,53,12,23,43,25)(8,36,54,13,24,44,26), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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)(3,5)(4,8)(9,13)(10,16)(12,14)(17,23)(19,21)(20,24)(25,27)(26,30)(29,31)(33,39)(35,37)(36,40)(41,47)(43,45)(44,48)(49,51)(50,54)(53,55), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56) );
G=PermutationGroup([[(1,37,55,14,17,45,27),(2,38,56,15,18,46,28),(3,39,49,16,19,47,29),(4,40,50,9,20,48,30),(5,33,51,10,21,41,31),(6,34,52,11,22,42,32),(7,35,53,12,23,43,25),(8,36,54,13,24,44,26)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(17,55),(18,56),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(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),(3,5),(4,8),(9,13),(10,16),(12,14),(17,23),(19,21),(20,24),(25,27),(26,30),(29,31),(33,39),(35,37),(36,40),(41,47),(43,45),(44,48),(49,51),(50,54),(53,55)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(26,30),(28,32),(34,38),(36,40),(42,46),(44,48),(50,54),(52,56)]])
55 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14O | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56F |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
size | 1 | 1 | 2 | 4 | 4 | 4 | 7 | 7 | 14 | 28 | 28 | 28 | 2 | 2 | 4 | 14 | 14 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
55 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | C8:C22 | D4xD7 | D4xD7 | D7xC8:C22 |
kernel | D7xC8:C22 | D7xM4(2) | C8:D14 | D7xD8 | D8:D7 | D7xSD16 | D56:C2 | D4.D14 | D4:D14 | C7xC8:C22 | C2xD4xD7 | D7xC4oD4 | C4xD7 | C2xDic7 | C22xD7 | C8:C22 | M4(2) | D8 | SD16 | C2xD4 | C4oD4 | D7 | C4 | C22 | C1 |
# reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 6 | 6 | 3 | 3 | 2 | 3 | 3 | 3 |
Matrix representation of D7xC8:C22 ►in GL8(F113)
79 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 79 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 112 | 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 |
88 | 25 | 0 | 0 | 0 | 0 | 0 | 0 |
79 | 25 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 88 | 25 | 0 | 0 | 0 | 0 |
0 | 0 | 79 | 25 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 112 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 112 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 112 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 112 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 112 | 22 | 91 | 2 |
0 | 0 | 0 | 0 | 0 | 1 | 111 | 72 |
0 | 0 | 0 | 0 | 0 | 1 | 112 | 72 |
0 | 0 | 0 | 0 | 112 | 22 | 0 | 1 |
112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 112 | 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 | 112 | 22 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 41 | 1 | 1 | 0 |
0 | 0 | 0 | 0 | 112 | 0 | 22 | 112 |
112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 72 | 0 | 112 | 0 |
0 | 0 | 0 | 0 | 2 | 91 | 0 | 112 |
G:=sub<GL(8,GF(113))| [79,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,79,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[88,79,0,0,0,0,0,0,25,25,0,0,0,0,0,0,0,0,88,79,0,0,0,0,0,0,25,25,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112],[0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,112,0,0,112,0,0,0,0,22,1,1,22,0,0,0,0,91,111,112,0,0,0,0,0,2,72,72,1],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,41,112,0,0,0,0,22,1,1,0,0,0,0,0,0,0,1,22,0,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,72,2,0,0,0,0,0,1,0,91,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112] >;
D7xC8:C22 in GAP, Magma, Sage, TeX
D_7\times C_8\rtimes C_2^2
% in TeX
G:=Group("D7xC8:C2^2");
// GroupNames label
G:=SmallGroup(448,1225);
// by ID
G=gap.SmallGroup(448,1225);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,570,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^8=d^2=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^3,e*c*e=c^5,d*e=e*d>;
// generators/relations