p-group, metabelian, nilpotent (class 3), monomial
Aliases: C43:7C2, C42:38D4, C4:1C4wrC2, C4:Q8:18C4, C4.15(C4xD4), C4:1D4:14C4, C4.4D4:16C4, C4.50(C4:1D4), C42.268(C2xC4), C23.569(C2xD4), (C22xC4).761D4, C4:M4(2):26C2, C4.90(C4.4D4), C22.16(C4:D4), (C22xC4).1406C23, (C2xC42).1074C22, (C2xM4(2)).206C22, C22.26C24.20C2, C2.14(C24.3C22), (C2xC4wrC2):18C2, C2.43(C2xC4wrC2), (C2xC4).740(C2xD4), (C2xQ8).92(C2xC4), (C2xD4).107(C2xC4), (C2xC4).597(C4oD4), (C2xC4).420(C22xC4), (C2xC4oD4).37C22, (C2xC4).200(C22:C4), C22.284(C2xC22:C4), SmallGroup(128,694)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C43:C2
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=a-1c-1, bc=cb, dbd=b-1, cd=dc >
Subgroups: 356 in 180 conjugacy classes, 60 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C42, C42, C22:C4, C4:C4, C2xC8, M4(2), C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4wrC2, C4:C8, C2xC42, C2xC42, C2xC42, C4xD4, C4:D4, C4.4D4, C4:1D4, C4:Q8, C2xM4(2), C2xC4oD4, C43, C2xC4wrC2, C4:M4(2), C22.26C24, C43:C2
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C4oD4, C4wrC2, C2xC22:C4, C4xD4, C4:D4, C4.4D4, C4:1D4, C24.3C22, C2xC4wrC2, C43:C2
►On 32 points
(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)
(1 15 11 3)(2 16 12 4)(5 9 14 7)(6 10 13 8)(17 23 27 32)(18 24 28 29)(19 21 25 30)(20 22 26 31)
(1 6 12 14)(2 5 11 13)(3 8 16 9)(4 7 15 10)(17 28 19 26)(18 25 20 27)(21 31 23 29)(22 32 24 30)
(1 27)(2 19)(3 32)(4 21)(5 26)(6 18)(7 31)(8 24)(9 22)(10 29)(11 17)(12 25)(13 28)(14 20)(15 23)(16 30)
G:=sub<Sym(32)| (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), (1,15,11,3)(2,16,12,4)(5,9,14,7)(6,10,13,8)(17,23,27,32)(18,24,28,29)(19,21,25,30)(20,22,26,31), (1,6,12,14)(2,5,11,13)(3,8,16,9)(4,7,15,10)(17,28,19,26)(18,25,20,27)(21,31,23,29)(22,32,24,30), (1,27)(2,19)(3,32)(4,21)(5,26)(6,18)(7,31)(8,24)(9,22)(10,29)(11,17)(12,25)(13,28)(14,20)(15,23)(16,30)>;
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), (1,15,11,3)(2,16,12,4)(5,9,14,7)(6,10,13,8)(17,23,27,32)(18,24,28,29)(19,21,25,30)(20,22,26,31), (1,6,12,14)(2,5,11,13)(3,8,16,9)(4,7,15,10)(17,28,19,26)(18,25,20,27)(21,31,23,29)(22,32,24,30), (1,27)(2,19)(3,32)(4,21)(5,26)(6,18)(7,31)(8,24)(9,22)(10,29)(11,17)(12,25)(13,28)(14,20)(15,23)(16,30) );
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)], [(1,15,11,3),(2,16,12,4),(5,9,14,7),(6,10,13,8),(17,23,27,32),(18,24,28,29),(19,21,25,30),(20,22,26,31)], [(1,6,12,14),(2,5,11,13),(3,8,16,9),(4,7,15,10),(17,28,19,26),(18,25,20,27),(21,31,23,29),(22,32,24,30)], [(1,27),(2,19),(3,32),(4,21),(5,26),(6,18),(7,31),(8,24),(9,22),(10,29),(11,17),(12,25),(13,28),(14,20),(15,23),(16,30)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4AD | 4AE | 4AF | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | C4oD4 | C4wrC2 |
| kernel | C43:C2 | C43 | C2xC4wrC2 | C4:M4(2) | C22.26C24 | C4.4D4 | C4:1D4 | C4:Q8 | C42 | C22xC4 | C2xC4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 2 | 2 | 6 | 2 | 4 | 16 |
Matrix representation of C43:C2 ►in GL4(F17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,13,0,0,0,0,1,0,0,0,0,4],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,13],[13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[0,16,0,0,16,0,0,0,0,0,0,1,0,0,1,0] >;
C43:C2 in GAP, Magma, Sage, TeX
C_4^3\rtimes C_2
% in TeX
G:=Group("C4^3:C2"); // GroupNames label
G:=SmallGroup(128,694);
// by ID
G=gap.SmallGroup(128,694);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,248,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1*c^-1,b*c=c*b,d*b*d=b^-1,c*d=d*c>;
// generators/relations