p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.444D4, C42.323C23, C4○D4⋊9D4, D4⋊2(C2×D4), Q8⋊2(C2×D4), C4⋊C8⋊3C22, C4⋊D8⋊18C2, C4⋊5(C8⋊C22), C4⋊SD16⋊2C2, (C4×D4)⋊83C22, (C2×D8)⋊16C22, (C2×C8).13C23, (C4×Q8)⋊79C22, C4.69(C22×D4), C4⋊C4.379C23, C4⋊M4(2)⋊4C2, (C2×C4).242C24, (C2×SD16)⋊7C22, (C2×D4).51C23, C23.654(C2×D4), (C22×C4).422D4, C4.104(C4⋊D4), D4⋊C4⋊16C22, C23.37D4⋊6C2, (C2×Q8).359C23, C4⋊1D4.138C22, C22.77(C4⋊D4), (C22×C4).972C23, (C2×C42).811C22, C22.502(C22×D4), (C22×D4).338C22, (C2×M4(2)).49C22, C42⋊C2.311C22, (C4×C4○D4)⋊7C2, (C2×C4⋊1D4)⋊16C2, (C2×C8⋊C22)⋊15C2, C4.152(C2×C4○D4), C2.60(C2×C4⋊D4), C2.16(C2×C8⋊C22), (C2×C4).1421(C2×D4), (C2×C4).273(C4○D4), (C2×C4○D4).298C22, SmallGroup(128,1770)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.444D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=b2c3 >
Subgroups: 676 in 292 conjugacy classes, 104 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, D4⋊C4, C4⋊C8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊1D4, C4⋊1D4, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C22×D4, C2×C4○D4, C23.37D4, C4⋊M4(2), C4⋊D8, C4⋊SD16, C4×C4○D4, C2×C4⋊1D4, C2×C8⋊C22, C42.444D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C8⋊C22, C22×D4, C2×C4○D4, C2×C4⋊D4, C2×C8⋊C22, C42.444D4
►On 32 points
(1 18 30 10)(2 11 31 19)(3 20 32 12)(4 13 25 21)(5 22 26 14)(6 15 27 23)(7 24 28 16)(8 9 29 17)
(1 32 5 28)(2 29 6 25)(3 26 7 30)(4 31 8 27)(9 23 13 19)(10 20 14 24)(11 17 15 21)(12 22 16 18)
(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 10)(2 9)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 32)
G:=sub<Sym(32)| (1,18,30,10)(2,11,31,19)(3,20,32,12)(4,13,25,21)(5,22,26,14)(6,15,27,23)(7,24,28,16)(8,9,29,17), (1,32,5,28)(2,29,6,25)(3,26,7,30)(4,31,8,27)(9,23,13,19)(10,20,14,24)(11,17,15,21)(12,22,16,18), (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), (1,10)(2,9)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,32)>;
G:=Group( (1,18,30,10)(2,11,31,19)(3,20,32,12)(4,13,25,21)(5,22,26,14)(6,15,27,23)(7,24,28,16)(8,9,29,17), (1,32,5,28)(2,29,6,25)(3,26,7,30)(4,31,8,27)(9,23,13,19)(10,20,14,24)(11,17,15,21)(12,22,16,18), (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), (1,10)(2,9)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,32) );
G=PermutationGroup([[(1,18,30,10),(2,11,31,19),(3,20,32,12),(4,13,25,21),(5,22,26,14),(6,15,27,23),(7,24,28,16),(8,9,29,17)], [(1,32,5,28),(2,29,6,25),(3,26,7,30),(4,31,8,27),(9,23,13,19),(10,20,14,24),(11,17,15,21),(12,22,16,18)], [(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)], [(1,10),(2,9),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(24,32)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C8⋊C22 |
| kernel | C42.444D4 | C23.37D4 | C4⋊M4(2) | C4⋊D8 | C4⋊SD16 | C4×C4○D4 | C2×C4⋊1D4 | C2×C8⋊C22 | C42 | C22×C4 | C4○D4 | C2×C4 | C4 |
| # reps | 1 | 2 | 1 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C42.444D4 ►in GL6(𝔽17)
| 13 | 15 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 |
| 1 | 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 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 1 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 13 | 15 | 0 | 0 | 0 | 0 |
| 16 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 16 | 0 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,15,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[4,1,0,0,0,0,2,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,16,0,0,0],[13,16,0,0,0,0,15,4,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,16,0] >;
C42.444D4 in GAP, Magma, Sage, TeX
C_4^2._{444}D_4 % in TeX
G:=Group("C4^2.444D4"); // GroupNames label
G:=SmallGroup(128,1770);
// by ID
G=gap.SmallGroup(128,1770);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,2019,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=b^2*c^3>;
// generators/relations