p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2xD4).301D4, C2.15(D4oD8), (C2xQ8).236D4, C4.Q8:11C22, C2.D8:22C22, C4:C4.394C23, C22:C8:12C22, (C2xC4).294C24, (C2xC8).144C23, (C2xD4).82C23, C23.245(C2xD4), C2.23(D4oSD16), C4:D4.23C22, C22.D8:13C2, C23.46D4:3C2, C23.25D4:7C2, M4(2):C4:26C2, C42:C2:15C22, C23.37D4:11C2, C23.19D4:14C2, (C22xC8).184C22, C22.554(C22xD4), D4:C4.180C22, C22.29C24.12C2, C23.33C23:9C2, (C22xC4).1010C23, C4.82(C22.D4), (C22xD4).360C22, (C2xM4(2)).76C22, C22.18(C22.D4), (C2xC4:C4):49C22, C4.104(C2xC4oD4), (C2xC4).489(C2xD4), (C2xD4:C4):30C2, (C22xC8):C2:9C2, (C2xC4).296(C4oD4), (C2xC4oD4).139C22, C2.59(C2xC22.D4), SmallGroup(128,1828)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2xD4).301D4
G = < a,b,c,d,e | a2=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=ebe=ab-1, dcd-1=ece=ab2c, ede=ad3 >
Subgroups: 452 in 209 conjugacy classes, 92 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C23, C42, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C22:C8, D4:C4, C4.Q8, C4.Q8, C2.D8, C2.D8, C2xC4:C4, C2xC4:C4, C42:C2, C42:C2, C4xD4, C4xQ8, C22wrC2, C4:D4, C4.4D4, C4:1D4, C22xC8, C2xM4(2), C22xD4, C2xC4oD4, (C22xC8):C2, C2xD4:C4, C23.37D4, C23.25D4, M4(2):C4, C22.D8, C23.46D4, C23.19D4, C23.33C23, C22.29C24, (C2xD4).301D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C22.D4, C22xD4, C2xC4oD4, C2xC22.D4, D4oD8, D4oSD16, (C2xD4).301D4
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)
(1 28 5 32)(2 15 6 11)(3 30 7 26)(4 9 8 13)(10 23 14 19)(12 17 16 21)(18 31 22 27)(20 25 24 29)
(1 32)(2 11)(3 26)(4 13)(5 28)(6 15)(7 30)(8 9)(10 23)(12 17)(14 19)(16 21)(18 27)(20 29)(22 31)(24 25)
(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)
(2 22)(3 7)(4 20)(6 18)(8 24)(9 11)(10 32)(12 30)(13 15)(14 28)(16 26)(17 21)(25 31)(27 29)
G:=sub<Sym(32)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26), (1,28,5,32)(2,15,6,11)(3,30,7,26)(4,9,8,13)(10,23,14,19)(12,17,16,21)(18,31,22,27)(20,25,24,29), (1,32)(2,11)(3,26)(4,13)(5,28)(6,15)(7,30)(8,9)(10,23)(12,17)(14,19)(16,21)(18,27)(20,29)(22,31)(24,25), (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), (2,22)(3,7)(4,20)(6,18)(8,24)(9,11)(10,32)(12,30)(13,15)(14,28)(16,26)(17,21)(25,31)(27,29)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26), (1,28,5,32)(2,15,6,11)(3,30,7,26)(4,9,8,13)(10,23,14,19)(12,17,16,21)(18,31,22,27)(20,25,24,29), (1,32)(2,11)(3,26)(4,13)(5,28)(6,15)(7,30)(8,9)(10,23)(12,17)(14,19)(16,21)(18,27)(20,29)(22,31)(24,25), (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), (2,22)(3,7)(4,20)(6,18)(8,24)(9,11)(10,32)(12,30)(13,15)(14,28)(16,26)(17,21)(25,31)(27,29) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,25),(16,26)], [(1,28,5,32),(2,15,6,11),(3,30,7,26),(4,9,8,13),(10,23,14,19),(12,17,16,21),(18,31,22,27),(20,25,24,29)], [(1,32),(2,11),(3,26),(4,13),(5,28),(6,15),(7,30),(8,9),(10,23),(12,17),(14,19),(16,21),(18,27),(20,29),(22,31),(24,25)], [(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)], [(2,22),(3,7),(4,20),(6,18),(8,24),(9,11),(10,32),(12,30),(13,15),(14,28),(16,26),(17,21),(25,31),(27,29)]])
32 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 8A | 8B | 8C | 8D | 8E | 8F |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4oD4 | D4oD8 | D4oSD16 |
kernel | (C2xD4).301D4 | (C22xC8):C2 | C2xD4:C4 | C23.37D4 | C23.25D4 | M4(2):C4 | C22.D8 | C23.46D4 | C23.19D4 | C23.33C23 | C22.29C24 | C2xD4 | C2xQ8 | C2xC4 | C2 | C2 |
# reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 3 | 1 | 8 | 2 | 2 |
Matrix representation of (C2xD4).301D4 ►in GL6(F17)
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
0 | 16 | 0 | 0 | 0 | 0 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 15 | 0 |
0 | 0 | 0 | 0 | 1 | 1 |
0 | 0 | 1 | 0 | 16 | 0 |
0 | 0 | 16 | 16 | 1 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 15 | 0 |
0 | 0 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 1 | 1 | 0 |
13 | 0 | 0 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 | 0 |
0 | 0 | 6 | 6 | 0 | 0 |
0 | 0 | 14 | 0 | 0 | 0 |
0 | 0 | 0 | 3 | 3 | 14 |
0 | 0 | 14 | 14 | 3 | 3 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 16 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 16 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,0,1,16,0,0,0,0,0,16,0,0,15,1,16,1,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,15,1,16,1,0,0,0,1,0,0],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,6,14,0,14,0,0,6,0,3,14,0,0,0,0,3,3,0,0,0,0,14,3],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,16,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
(C2xD4).301D4 in GAP, Magma, Sage, TeX
(C_2\times D_4)._{301}D_4
% in TeX
G:=Group("(C2xD4).301D4");
// GroupNames label
G:=SmallGroup(128,1828);
// by ID
G=gap.SmallGroup(128,1828);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,100,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=e^2=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e=a*b^-1,d*c*d^-1=e*c*e=a*b^2*c,e*d*e=a*d^3>;
// generators/relations