p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.116D4, (C2×D8)⋊11C4, C4.24(C4×D4), (C2×Q16)⋊11C4, (C2×SD16)⋊6C4, (C2×C8).112D4, C8.3(C22⋊C4), C4.53(C4⋊1D4), C22.188(C4×D4), C2.19(C8.26D4), C4.206(C4⋊D4), C23.212(C4○D4), C22.20(C4⋊D4), (C2×C42).325C22, (C22×C8).409C22, C22.2(C4.4D4), (C22×C4).1415C23, (C2×M4(2)).212C22, C2.27(C24.3C22), (C2×C4≀C2)⋊21C2, (C2×C8⋊C4)⋊3C2, (C2×C8).74(C2×C4), (C2×C4○D8).7C2, (C2×C4).743(C2×D4), C4.43(C2×C22⋊C4), (C2×Q8).99(C2×C4), (C2×C8.C4)⋊11C2, (C2×D4).114(C2×C4), (C22×C8)⋊C2⋊25C2, (C2×C4).606(C4○D4), (C2×C4).429(C22×C4), (C2×C4○D4).40C22, SmallGroup(128,707)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.116D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1, ab=ba, cac-1=ab2, dad-1=a-1b, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 292 in 146 conjugacy classes, 56 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C2×C8, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, C4≀C2, C8.C4, C2×C42, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C8⋊C4, (C22×C8)⋊C2, C2×C4≀C2, C2×C8.C4, C2×C4○D8, C42.116D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C24.3C22, C8.26D4, C42.116D4
►On 32 points
(1 7 5 3)(2 4 6 8)(9 28)(10 25)(11 30)(12 27)(13 32)(14 29)(15 26)(16 31)(17 23 21 19)(18 20 22 24)
(1 23 5 19)(2 24 6 20)(3 17 7 21)(4 18 8 22)(9 26 13 30)(10 27 14 31)(11 28 15 32)(12 29 16 25)
(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 19 31 5 14 23 27)(2 13 20 26 6 9 24 30)(3 16 21 29 7 12 17 25)(4 11 22 32 8 15 18 28)
G:=sub<Sym(32)| (1,7,5,3)(2,4,6,8)(9,28)(10,25)(11,30)(12,27)(13,32)(14,29)(15,26)(16,31)(17,23,21,19)(18,20,22,24), (1,23,5,19)(2,24,6,20)(3,17,7,21)(4,18,8,22)(9,26,13,30)(10,27,14,31)(11,28,15,32)(12,29,16,25), (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), (1,10,19,31,5,14,23,27)(2,13,20,26,6,9,24,30)(3,16,21,29,7,12,17,25)(4,11,22,32,8,15,18,28)>;
G:=Group( (1,7,5,3)(2,4,6,8)(9,28)(10,25)(11,30)(12,27)(13,32)(14,29)(15,26)(16,31)(17,23,21,19)(18,20,22,24), (1,23,5,19)(2,24,6,20)(3,17,7,21)(4,18,8,22)(9,26,13,30)(10,27,14,31)(11,28,15,32)(12,29,16,25), (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), (1,10,19,31,5,14,23,27)(2,13,20,26,6,9,24,30)(3,16,21,29,7,12,17,25)(4,11,22,32,8,15,18,28) );
G=PermutationGroup([[(1,7,5,3),(2,4,6,8),(9,28),(10,25),(11,30),(12,27),(13,32),(14,29),(15,26),(16,31),(17,23,21,19),(18,20,22,24)], [(1,23,5,19),(2,24,6,20),(3,17,7,21),(4,18,8,22),(9,26,13,30),(10,27,14,31),(11,28,15,32),(12,29,16,25)], [(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)], [(1,10,19,31,5,14,23,27),(2,13,20,26,6,9,24,30),(3,16,21,29,7,12,17,25),(4,11,22,32,8,15,18,28)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | C4○D4 | C4○D4 | C8.26D4 |
| kernel | C42.116D4 | C2×C8⋊C4 | (C22×C8)⋊C2 | C2×C4≀C2 | C2×C8.C4 | C2×C4○D8 | C2×D8 | C2×SD16 | C2×Q16 | C42 | C2×C8 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 6 | 2 | 2 | 4 |
Matrix representation of C42.116D4 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 16 | 15 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 9 | 3 | 0 | 0 | 0 | 0 |
| 1 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,4,0],[9,1,0,0,0,0,3,8,0,0,0,0,0,0,0,0,0,16,0,0,0,0,4,0,0,0,0,16,0,0,0,0,4,0,0,0] >;
C42.116D4 in GAP, Magma, Sage, TeX
C_4^2._{116}D_4 % in TeX
G:=Group("C4^2.116D4"); // GroupNames label
G:=SmallGroup(128,707);
// by ID
G=gap.SmallGroup(128,707);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,100,2019,248,2804,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b^-1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations