p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.375D4, (C4×D4)⋊1C4, (C4×Q8)⋊1C4, C4.45C4≀C2, C4.4D4⋊7C4, C42.C2⋊3C4, C42⋊4C4⋊4C2, C42.69(C2×C4), C4○(C22.SD16), C23.490(C2×D4), (C22×C4).660D4, C4○(C23.31D4), C22.16(C4○D8), C22.SD16.7C2, C42.12C4⋊15C2, C4⋊D4.129C22, C23.31D4⋊23C2, C22⋊C8.161C22, (C22×C4).622C23, (C2×C42).172C22, C22⋊Q8.134C22, C2.8(C23.24D4), C23.36C23.5C2, C2.C42.499C22, C2.12(C23.C23), C4⋊C4.1(C2×C4), C2.17(C2×C4≀C2), (C2×D4).3(C2×C4), (C2×Q8).3(C2×C4), (C2×C4).1146(C2×D4), (C2×C4).112(C22×C4), (C2×C4)○(C22.SD16), (C2×C4).170(C22⋊C4), (C2×C4)○(C23.31D4), C22.176(C2×C22⋊C4), SmallGroup(128,232)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.375D4
G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=b-1c-1 >
Subgroups: 244 in 119 conjugacy classes, 46 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2.C42, C2.C42, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22.SD16, C23.31D4, C42⋊4C4, C42.12C4, C23.36C23, C42.375D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, C4○D8, C23.C23, C23.24D4, C2×C4≀C2, C42.375D4
►On 32 points
(1 10 31 21)(2 11 32 22)(3 12 25 23)(4 13 26 24)(5 14 27 17)(6 15 28 18)(7 16 29 19)(8 9 30 20)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)
(1 17 31 14)(2 20 6 24)(3 16 25 19)(4 11 8 15)(5 21 27 10)(7 12 29 23)(9 28 13 32)(18 26 22 30)
(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)| (1,10,31,21)(2,11,32,22)(3,12,25,23)(4,13,26,24)(5,14,27,17)(6,15,28,18)(7,16,29,19)(8,9,30,20), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (1,17,31,14)(2,20,6,24)(3,16,25,19)(4,11,8,15)(5,21,27,10)(7,12,29,23)(9,28,13,32)(18,26,22,30), (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( (1,10,31,21)(2,11,32,22)(3,12,25,23)(4,13,26,24)(5,14,27,17)(6,15,28,18)(7,16,29,19)(8,9,30,20), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (1,17,31,14)(2,20,6,24)(3,16,25,19)(4,11,8,15)(5,21,27,10)(7,12,29,23)(9,28,13,32)(18,26,22,30), (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([[(1,10,31,21),(2,11,32,22),(3,12,25,23),(4,13,26,24),(5,14,27,17),(6,15,28,18),(7,16,29,19),(8,9,30,20)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28)], [(1,17,31,14),(2,20,6,24),(3,16,25,19),(4,11,8,15),(5,21,27,10),(7,12,29,23),(9,28,13,32),(18,26,22,30)], [(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 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4T | 4U | 4V | 4W | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 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 | C4 | D4 | D4 | C4≀C2 | C4○D8 | C23.C23 |
| kernel | C42.375D4 | C22.SD16 | C23.31D4 | C42⋊4C4 | C42.12C4 | C23.36C23 | C4×D4 | C4×Q8 | C4.4D4 | C42.C2 | C42 | C22×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 2 |
Matrix representation of C42.375D4 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 13 |
| 12 | 5 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 9 | 9 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[0,1,0,0,16,0,0,0,0,0,4,0,0,0,0,4],[13,0,0,0,0,4,0,0,0,0,1,12,0,0,0,13],[12,12,0,0,5,12,0,0,0,0,9,0,0,0,9,8] >;
C42.375D4 in GAP, Magma, Sage, TeX
C_4^2._{375}D_4 % in TeX
G:=Group("C4^2.375D4"); // GroupNames label
G:=SmallGroup(128,232);
// by ID
G=gap.SmallGroup(128,232);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,184,1123,1018,248,1971]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations