p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.42D4, C23.17M4(2), C23⋊C8.9C2, (C2×C42).2C4, (C23×C4).16C4, C24.22(C2×C4), (C2×C4).31M4(2), C42.6C4⋊21C2, C22⋊C8.120C22, C2.8(C24.4C4), (C2×C42).148C22, C23.165(C22×C4), (C22×C4).428C23, C22.17(C2×M4(2)), C22.M4(2)⋊13C2, C2.6(C23.C23), C2.6(M4(2).8C22), (C2×C4⋊C4).33C4, (C2×C4).1125(C2×D4), (C2×C22⋊C4).35C4, (C22×C4).43(C2×C4), (C2×C4⋊C4).736C22, (C2×C42⋊C2).2C2, (C2×C4).312(C22⋊C4), C22.146(C2×C22⋊C4), (C2×C22⋊C4).403C22, SmallGroup(128,196)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.42D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=b-1c3 >
Subgroups: 244 in 126 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C24, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C23⋊C8, C22.M4(2), C42.6C4, C2×C42⋊C2, C42.42D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), C22×C4, C2×D4, C2×C22⋊C4, C2×M4(2), C24.4C4, C23.C23, M4(2).8C22, C42.42D4
►On 32 points
(1 29 12 18)(2 26 13 23)(3 31 14 20)(4 28 15 17)(5 25 16 22)(6 30 9 19)(7 27 10 24)(8 32 11 21)
(1 3 5 7)(2 15 6 11)(4 9 8 13)(10 12 14 16)(17 30 21 26)(18 20 22 24)(19 32 23 28)(25 27 29 31)
(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 9 3 8 5 13 7 4)(2 10 15 12 6 14 11 16)(17 22 30 24 21 18 26 20)(19 27 32 29 23 31 28 25)
G:=sub<Sym(32)| (1,29,12,18)(2,26,13,23)(3,31,14,20)(4,28,15,17)(5,25,16,22)(6,30,9,19)(7,27,10,24)(8,32,11,21), (1,3,5,7)(2,15,6,11)(4,9,8,13)(10,12,14,16)(17,30,21,26)(18,20,22,24)(19,32,23,28)(25,27,29,31), (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), (1,9,3,8,5,13,7,4)(2,10,15,12,6,14,11,16)(17,22,30,24,21,18,26,20)(19,27,32,29,23,31,28,25)>;
G:=Group( (1,29,12,18)(2,26,13,23)(3,31,14,20)(4,28,15,17)(5,25,16,22)(6,30,9,19)(7,27,10,24)(8,32,11,21), (1,3,5,7)(2,15,6,11)(4,9,8,13)(10,12,14,16)(17,30,21,26)(18,20,22,24)(19,32,23,28)(25,27,29,31), (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), (1,9,3,8,5,13,7,4)(2,10,15,12,6,14,11,16)(17,22,30,24,21,18,26,20)(19,27,32,29,23,31,28,25) );
G=PermutationGroup([[(1,29,12,18),(2,26,13,23),(3,31,14,20),(4,28,15,17),(5,25,16,22),(6,30,9,19),(7,27,10,24),(8,32,11,21)], [(1,3,5,7),(2,15,6,11),(4,9,8,13),(10,12,14,16),(17,30,21,26),(18,20,22,24),(19,32,23,28),(25,27,29,31)], [(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)], [(1,9,3,8,5,13,7,4),(2,10,15,12,6,14,11,16),(17,22,30,24,21,18,26,20),(19,27,32,29,23,31,28,25)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | M4(2) | M4(2) | C23.C23 | M4(2).8C22 |
| kernel | C42.42D4 | C23⋊C8 | C22.M4(2) | C42.6C4 | C2×C42⋊C2 | C2×C42 | C2×C22⋊C4 | C2×C4⋊C4 | C23×C4 | C42 | C2×C4 | C23 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of C42.42D4 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 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 | 0 | 0 | 13 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 1 |
| 0 | 0 | 16 | 10 | 1 | 0 |
| 2 | 12 | 0 | 0 | 0 | 0 |
| 5 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 16 | 2 | 0 |
| 0 | 0 | 4 | 3 | 0 | 15 |
| 0 | 0 | 15 | 8 | 5 | 16 |
| 0 | 0 | 4 | 11 | 4 | 14 |
| 12 | 2 | 0 | 0 | 0 | 0 |
| 15 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 16 | 2 | 0 |
| 0 | 0 | 13 | 14 | 0 | 2 |
| 0 | 0 | 11 | 5 | 5 | 1 |
| 0 | 0 | 10 | 2 | 4 | 3 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,7,16,0,0,1,0,1,10,0,0,0,0,0,1,0,0,0,0,1,0],[2,5,0,0,0,0,12,15,0,0,0,0,0,0,12,4,15,4,0,0,16,3,8,11,0,0,2,0,5,4,0,0,0,15,16,14],[12,15,0,0,0,0,2,5,0,0,0,0,0,0,12,13,11,10,0,0,16,14,5,2,0,0,2,0,5,4,0,0,0,2,1,3] >;
C42.42D4 in GAP, Magma, Sage, TeX
C_4^2._{42}D_4 % in TeX
G:=Group("C4^2.42D4"); // GroupNames label
G:=SmallGroup(128,196);
// by ID
G=gap.SmallGroup(128,196);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,184,1123,851,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations