p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.222D4, C42.338C23, C4⋊C8⋊75C22, (C4×C8)⋊40C22, (C2×C4)⋊14SD16, C4⋊Q8⋊58C22, (C4×Q8)⋊5C22, C4○2(D4⋊2Q8), D4⋊2Q8⋊47C2, D4.1(C4○D4), (C4×SD16)⋊27C2, C4⋊C4.54C23, C4.70(C2×SD16), C4.Q8⋊65C22, C4○2(C22⋊SD16), C4○2(D4.D4), D4.D4⋊49C2, (C2×C8).314C23, (C2×C4).299C24, C22⋊SD16.5C2, C23.667(C2×D4), (C22×C4).802D4, (C2×Q8).73C23, Q8⋊C4⋊83C22, (C4×D4).319C22, (C2×D4).401C23, C22.24(C2×SD16), C2.14(C22×SD16), C42.12C4⋊40C2, C4○2(C23.47D4), C23.47D4⋊36C2, C22⋊C8.214C22, (C2×C42).826C22, C22.559(C22×D4), C22⋊Q8.163C22, D4⋊C4.182C22, C2.26(D8⋊C22), C23.37C23⋊4C2, (C22×C4).1015C23, (C2×SD16).136C22, (C22×D4).573C22, C2.100(C22.19C24), (C2×C4×D4).84C2, (C2×C4)○(D4⋊2Q8), C4.184(C2×C4○D4), (C2×C4)○(D4.D4), (C2×C4).1580(C2×D4), (C2×C4⋊C4).931C22, SmallGroup(128,1833)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.222D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c3 >
Subgroups: 436 in 224 conjugacy classes, 98 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×SD16, C23×C4, C22×D4, C42.12C4, C4×SD16, C22⋊SD16, D4.D4, D4⋊2Q8, C23.47D4, C2×C4×D4, C23.37C23, C42.222D4
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C24, C2×SD16, C22×D4, C2×C4○D4, C22.19C24, C22×SD16, D8⋊C22, C42.222D4
►On 32 points
(1 14 5 10)(2 4 6 8)(3 16 7 12)(9 11 13 15)(17 19 21 23)(18 31 22 27)(20 25 24 29)(26 28 30 32)
(1 31 16 24)(2 32 9 17)(3 25 10 18)(4 26 11 19)(5 27 12 20)(6 28 13 21)(7 29 14 22)(8 30 15 23)
(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 15 5 11)(2 10 6 14)(3 13 7 9)(4 16 8 12)(17 29 21 25)(18 32 22 28)(19 27 23 31)(20 30 24 26)
G:=sub<Sym(32)| (1,14,5,10)(2,4,6,8)(3,16,7,12)(9,11,13,15)(17,19,21,23)(18,31,22,27)(20,25,24,29)(26,28,30,32), (1,31,16,24)(2,32,9,17)(3,25,10,18)(4,26,11,19)(5,27,12,20)(6,28,13,21)(7,29,14,22)(8,30,15,23), (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), (1,15,5,11)(2,10,6,14)(3,13,7,9)(4,16,8,12)(17,29,21,25)(18,32,22,28)(19,27,23,31)(20,30,24,26)>;
G:=Group( (1,14,5,10)(2,4,6,8)(3,16,7,12)(9,11,13,15)(17,19,21,23)(18,31,22,27)(20,25,24,29)(26,28,30,32), (1,31,16,24)(2,32,9,17)(3,25,10,18)(4,26,11,19)(5,27,12,20)(6,28,13,21)(7,29,14,22)(8,30,15,23), (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), (1,15,5,11)(2,10,6,14)(3,13,7,9)(4,16,8,12)(17,29,21,25)(18,32,22,28)(19,27,23,31)(20,30,24,26) );
G=PermutationGroup([[(1,14,5,10),(2,4,6,8),(3,16,7,12),(9,11,13,15),(17,19,21,23),(18,31,22,27),(20,25,24,29),(26,28,30,32)], [(1,31,16,24),(2,32,9,17),(3,25,10,18),(4,26,11,19),(5,27,12,20),(6,28,13,21),(7,29,14,22),(8,30,15,23)], [(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)], [(1,15,5,11),(2,10,6,14),(3,13,7,9),(4,16,8,12),(17,29,21,25),(18,32,22,28),(19,27,23,31),(20,30,24,26)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4P | 4Q | 4R | 4S | 4T | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 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 | C2 | C2 | C2 | D4 | D4 | SD16 | C4○D4 | D8⋊C22 |
| kernel | C42.222D4 | C42.12C4 | C4×SD16 | C22⋊SD16 | D4.D4 | D4⋊2Q8 | C23.47D4 | C2×C4×D4 | C23.37C23 | C42 | C22×C4 | C2×C4 | D4 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 8 | 2 |
Matrix representation of C42.222D4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 5 | 5 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 12 | 12 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,0,1,0,0,16,0],[0,16,0,0,16,0,0,0,0,0,5,5,0,0,12,5],[0,1,0,0,1,0,0,0,0,0,5,12,0,0,12,12] >;
C42.222D4 in GAP, Magma, Sage, TeX
C_4^2._{222}D_4 % in TeX
G:=Group("C4^2.222D4"); // GroupNames label
G:=SmallGroup(128,1833);
// by ID
G=gap.SmallGroup(128,1833);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations