p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.395D4, C23⋊C8⋊16C2, C24.2(C2×C4), (C22×D4).5C4, (C22×Q8).4C4, C4.4(C4.D4), C42.6C4⋊23C2, C42.12C4⋊3C2, C22.10(C8○D4), C22⋊C8.125C22, C23.170(C22×C4), (C2×C42).151C22, (C22×C4).433C23, C2.9(C23.C23), (C2×C22⋊C4).8C4, C2.7(C2×C4.D4), (C2×C4).1130(C2×D4), (C22×C4).10(C2×C4), (C2×C4.4D4).1C2, (C2×C4).163(C22⋊C4), C2.7((C22×C8)⋊C2), (C2×C22⋊C4).87C22, C22.151(C2×C22⋊C4), SmallGroup(128,201)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.395D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=ab2, ad=da, cbc-1=a2b, bd=db, dcd-1=b-1c3 >
Subgroups: 308 in 128 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, C23⋊C8, C42.12C4, C42.6C4, C2×C4.4D4, C42.395D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.D4, C2×C22⋊C4, C8○D4, (C22×C8)⋊C2, C23.C23, C2×C4.D4, C42.395D4
►On 32 points
(1 15 32 17)(2 12 25 22)(3 9 26 19)(4 14 27 24)(5 11 28 21)(6 16 29 18)(7 13 30 23)(8 10 31 20)
(1 3 5 7)(2 27 6 31)(4 29 8 25)(9 11 13 15)(10 22 14 18)(12 24 16 20)(17 19 21 23)(26 28 30 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 14 3 18 5 10 7 22)(2 15 27 9 6 11 31 13)(4 19 29 21 8 23 25 17)(12 32 24 26 16 28 20 30)
G:=sub<Sym(32)| (1,15,32,17)(2,12,25,22)(3,9,26,19)(4,14,27,24)(5,11,28,21)(6,16,29,18)(7,13,30,23)(8,10,31,20), (1,3,5,7)(2,27,6,31)(4,29,8,25)(9,11,13,15)(10,22,14,18)(12,24,16,20)(17,19,21,23)(26,28,30,32), (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), (1,14,3,18,5,10,7,22)(2,15,27,9,6,11,31,13)(4,19,29,21,8,23,25,17)(12,32,24,26,16,28,20,30)>;
G:=Group( (1,15,32,17)(2,12,25,22)(3,9,26,19)(4,14,27,24)(5,11,28,21)(6,16,29,18)(7,13,30,23)(8,10,31,20), (1,3,5,7)(2,27,6,31)(4,29,8,25)(9,11,13,15)(10,22,14,18)(12,24,16,20)(17,19,21,23)(26,28,30,32), (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), (1,14,3,18,5,10,7,22)(2,15,27,9,6,11,31,13)(4,19,29,21,8,23,25,17)(12,32,24,26,16,28,20,30) );
G=PermutationGroup([[(1,15,32,17),(2,12,25,22),(3,9,26,19),(4,14,27,24),(5,11,28,21),(6,16,29,18),(7,13,30,23),(8,10,31,20)], [(1,3,5,7),(2,27,6,31),(4,29,8,25),(9,11,13,15),(10,22,14,18),(12,24,16,20),(17,19,21,23),(26,28,30,32)], [(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)], [(1,14,3,18,5,10,7,22),(2,15,27,9,6,11,31,13),(4,19,29,21,8,23,25,17),(12,32,24,26,16,28,20,30)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | C8○D4 | C4.D4 | C23.C23 |
| kernel | C42.395D4 | C23⋊C8 | C42.12C4 | C42.6C4 | C2×C4.4D4 | C2×C22⋊C4 | C22×D4 | C22×Q8 | C42 | C22 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 8 | 2 | 2 |
Matrix representation of C42.395D4 ►in GL6(𝔽17)
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 2 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 16 |
| 0 | 0 | 0 | 9 | 1 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 15 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 1 |
| 0 | 0 | 0 | 8 | 16 | 0 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 15 | 0 |
| 0 | 0 | 9 | 0 | 13 | 16 |
| 0 | 0 | 0 | 1 | 2 | 0 |
| 0 | 0 | 16 | 0 | 8 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 13 | 1 |
| 0 | 0 | 0 | 1 | 2 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,13,0,0,0,0,0,0,0,13,0,0,0,0,0,2,4,15,9,0,0,0,0,0,1,0,0,0,0,16,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,15,13,2,8,0,0,0,0,0,16,0,0,0,0,1,0],[15,0,0,0,0,0,0,2,0,0,0,0,0,0,15,9,0,16,0,0,0,0,1,0,0,0,15,13,2,8,0,0,0,16,0,0],[0,15,0,0,0,0,2,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,1,0,0,0,15,13,2,8,0,0,0,1,0,0] >;
C42.395D4 in GAP, Magma, Sage, TeX
C_4^2._{395}D_4 % in TeX
G:=Group("C4^2.395D4"); // GroupNames label
G:=SmallGroup(128,201);
// by ID
G=gap.SmallGroup(128,201);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,723,184,1123,851,172]);
// 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=a*b^2,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations