p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.102D4, C4⋊3C4≀C2, D4⋊2(C4⋊C4), (C4×D4)⋊15C4, Q8⋊2(C4⋊C4), (C4×Q8)⋊15C4, C4○D4.3Q8, C4○D4.22D4, C42⋊6C4⋊13C2, C42.140(C2×C4), (C22×C4).270D4, C23.548(C2×D4), C4.181(C4⋊D4), C4⋊M4(2)⋊24C2, C4.110(C22⋊Q8), C4.35(C42⋊C2), C22.38(C4⋊D4), (C2×C42).251C22, C22.48(C22⋊Q8), (C22×C4).1328C23, C2.15(C23.7Q8), C42⋊C2.267C22, C2.39(C42⋊C22), (C2×M4(2)).153C22, (C4×C4⋊C4)⋊5C2, C4.5(C2×C4⋊C4), (C2×C4≀C2).7C2, C2.39(C2×C4≀C2), (C4×C4○D4).7C2, C4⋊C4.193(C2×C4), (C2×C4).978(C2×D4), (C2×C4).263(C2×Q8), (C2×D4).207(C2×C4), (C2×Q8).190(C2×C4), (C2×C4).864(C4○D4), (C2×C4).366(C22×C4), (C2×C4).360(C22⋊C4), (C2×C4○D4).255C22, C22.251(C2×C22⋊C4), SmallGroup(128,538)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.102D4
G = < a,b,c,d | a4=b4=c4=1, d2=b, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 292 in 156 conjugacy classes, 62 normal (44 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C2.C42, C4≀C2, C4⋊C8, C2×C42, C2×C42, C2×C42, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C2×M4(2), C2×C4○D4, C42⋊6C4, C4×C4⋊C4, C2×C4≀C2, C4⋊M4(2), C4×C4○D4, C42.102D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4≀C2, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C23.7Q8, C2×C4≀C2, C42⋊C22, C42.102D4
►On 32 points
(1 24 26 15)(2 16 27 17)(3 18 28 9)(4 10 29 19)(5 20 30 11)(6 12 31 21)(7 22 32 13)(8 14 25 23)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)
(1 9)(2 12 6 16)(3 11)(4 14 8 10)(5 13)(7 15)(17 27 21 31)(18 26)(19 29 23 25)(20 28)(22 30)(24 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)
G:=sub<Sym(32)| (1,24,26,15)(2,16,27,17)(3,18,28,9)(4,10,29,19)(5,20,30,11)(6,12,31,21)(7,22,32,13)(8,14,25,23), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (1,9)(2,12,6,16)(3,11)(4,14,8,10)(5,13)(7,15)(17,27,21,31)(18,26)(19,29,23,25)(20,28)(22,30)(24,32), (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)>;
G:=Group( (1,24,26,15)(2,16,27,17)(3,18,28,9)(4,10,29,19)(5,20,30,11)(6,12,31,21)(7,22,32,13)(8,14,25,23), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (1,9)(2,12,6,16)(3,11)(4,14,8,10)(5,13)(7,15)(17,27,21,31)(18,26)(19,29,23,25)(20,28)(22,30)(24,32), (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) );
G=PermutationGroup([[(1,24,26,15),(2,16,27,17),(3,18,28,9),(4,10,29,19),(5,20,30,11),(6,12,31,21),(7,22,32,13),(8,14,25,23)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32)], [(1,9),(2,12,6,16),(3,11),(4,14,8,10),(5,13),(7,15),(17,27,21,31),(18,26),(19,29,23,25),(20,28),(22,30),(24,32)], [(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)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4Z | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D4 | Q8 | C4○D4 | C4≀C2 | C42⋊C22 |
| kernel | C42.102D4 | C42⋊6C4 | C4×C4⋊C4 | C2×C4≀C2 | C4⋊M4(2) | C4×C4○D4 | C4×D4 | C4×Q8 | C42 | C22×C4 | C4○D4 | C4○D4 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 8 | 2 |
Matrix representation of C42.102D4 ►in GL4(𝔽17) generated by
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 13 |
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [0,4,0,0,4,0,0,0,0,0,13,0,0,0,0,4],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[0,4,0,0,13,0,0,0,0,0,16,0,0,0,0,13],[4,0,0,0,0,13,0,0,0,0,0,16,0,0,13,0] >;
C42.102D4 in GAP, Magma, Sage, TeX
C_4^2._{102}D_4 % in TeX
G:=Group("C4^2.102D4"); // GroupNames label
G:=SmallGroup(128,538);
// by ID
G=gap.SmallGroup(128,538);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2804,718,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations