p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.232D4, C42.348C23, D8⋊C4⋊7C2, C4⋊C8⋊13C22, (C4×D4)⋊8C22, D4.Q8⋊18C2, (C4×Q8)⋊8C22, C8⋊C4⋊4C22, D4.7(C4○D4), C22⋊D8.3C2, C22⋊SD16⋊5C2, C4⋊C4.67C23, (C2×C8).41C23, C4.Q8⋊13C22, C2.D8⋊24C22, SD16⋊C4⋊8C2, D4.2D4⋊19C2, C42.6C4⋊5C2, (C2×C4).312C24, (C2×D8).59C22, (C22×C4).452D4, C23.676(C2×D4), (C2×Q8).78C23, D4⋊C4⋊22C22, Q8⋊C4⋊24C22, (C2×D4).405C23, C22.D8⋊16C2, C23.47D4⋊5C2, C4.4D4⋊55C22, C22⋊C8.25C22, C42.C2⋊32C22, C4⋊D4.167C22, C22.30(C8⋊C22), (C2×C42).839C22, (C2×SD16).13C22, C22.572(C22×D4), C22⋊Q8.172C22, C2.31(D8⋊C22), C23.36C23⋊4C2, (C22×C4).1028C23, (C22×D4).577C22, C2.113(C22.19C24), (C2×C4×D4)⋊65C2, C4.197(C2×C4○D4), C2.35(C2×C8⋊C22), (C2×C4).1220(C2×D4), (C2×C4⋊C4).940C22, SmallGroup(128,1846)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.232D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=dbd=a2b, dcd=a2c3 >
Subgroups: 460 in 225 conjugacy classes, 90 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C24, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C2×D8, C2×SD16, C23×C4, C22×D4, C42.6C4, SD16⋊C4, D8⋊C4, C22⋊D8, C22⋊SD16, D4.2D4, D4.Q8, C22.D8, C23.47D4, C2×C4×D4, C23.36C23, C42.232D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8⋊C22, C22×D4, C2×C4○D4, C22.19C24, C2×C8⋊C22, D8⋊C22, C42.232D4
►On 32 points
(1 25 5 29)(2 8 6 4)(3 27 7 31)(9 15 13 11)(10 21 14 17)(12 23 16 19)(18 24 22 20)(26 32 30 28)
(1 16 27 21)(2 13 28 18)(3 10 29 23)(4 15 30 20)(5 12 31 17)(6 9 32 22)(7 14 25 19)(8 11 26 24)
(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 26)(2 25)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 19)(10 18)(11 17)(12 24)(13 23)(14 22)(15 21)(16 20)
G:=sub<Sym(32)| (1,25,5,29)(2,8,6,4)(3,27,7,31)(9,15,13,11)(10,21,14,17)(12,23,16,19)(18,24,22,20)(26,32,30,28), (1,16,27,21)(2,13,28,18)(3,10,29,23)(4,15,30,20)(5,12,31,17)(6,9,32,22)(7,14,25,19)(8,11,26,24), (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), (1,26)(2,25)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,19)(10,18)(11,17)(12,24)(13,23)(14,22)(15,21)(16,20)>;
G:=Group( (1,25,5,29)(2,8,6,4)(3,27,7,31)(9,15,13,11)(10,21,14,17)(12,23,16,19)(18,24,22,20)(26,32,30,28), (1,16,27,21)(2,13,28,18)(3,10,29,23)(4,15,30,20)(5,12,31,17)(6,9,32,22)(7,14,25,19)(8,11,26,24), (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), (1,26)(2,25)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,19)(10,18)(11,17)(12,24)(13,23)(14,22)(15,21)(16,20) );
G=PermutationGroup([[(1,25,5,29),(2,8,6,4),(3,27,7,31),(9,15,13,11),(10,21,14,17),(12,23,16,19),(18,24,22,20),(26,32,30,28)], [(1,16,27,21),(2,13,28,18),(3,10,29,23),(4,15,30,20),(5,12,31,17),(6,9,32,22),(7,14,25,19),(8,11,26,24)], [(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)], [(1,26),(2,25),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,19),(10,18),(11,17),(12,24),(13,23),(14,22),(15,21),(16,20)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | ··· | 4H | 4I | ··· | 4N | 4O | 4P | 4Q | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 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 | C2 | D4 | D4 | C4○D4 | C8⋊C22 | D8⋊C22 |
| kernel | C42.232D4 | C42.6C4 | SD16⋊C4 | D8⋊C4 | C22⋊D8 | C22⋊SD16 | D4.2D4 | D4.Q8 | C22.D8 | C23.47D4 | C2×C4×D4 | C23.36C23 | C42 | C22×C4 | D4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.232D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 16 |
| 0 | 0 | 14 | 3 | 1 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 5 | 12 | 4 | 0 |
| 0 | 0 | 12 | 12 | 0 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 3 | 2 | 0 |
| 0 | 0 | 14 | 14 | 0 | 15 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 15 | 0 |
| 0 | 0 | 14 | 14 | 0 | 15 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 3 | 3 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,14,14,0,0,16,0,14,3,0,0,0,0,0,1,0,0,0,0,16,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,5,12,0,0,0,13,12,12,0,0,0,0,4,0,0,0,0,0,0,4],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,2,0,3,14,0,0,0,15,3,3],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,0,15,0,14,3,0,0,0,15,3,3] >;
C42.232D4 in GAP, Magma, Sage, TeX
C_4^2._{232}D_4 % in TeX
G:=Group("C4^2.232D4"); // GroupNames label
G:=SmallGroup(128,1846);
// by ID
G=gap.SmallGroup(128,1846);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,521,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1*b^2,c*b*c^-1=d*b*d=a^2*b,d*c*d=a^2*c^3>;
// generators/relations