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, (C4×SD16)⋊27C2, D4.1(C4○D4), 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, (C22×C4).802D4, C23.667(C2×D4), (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
Subgroups: 436 in 224 conjugacy classes, 98 normal (28 characteristic)
C1, C2 [×3], C2 [×6], C4 [×6], C4 [×9], C22, C22 [×2], C22 [×18], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×19], D4 [×4], D4 [×6], Q8 [×6], C23, C23 [×10], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], SD16 [×8], C22×C4 [×3], C22×C4 [×9], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C2×Q8, C24, C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×4], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4×D4 [×2], C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C2×SD16 [×4], C23×C4, C22×D4, C42.12C4, C4×SD16 [×4], C22⋊SD16 [×2], D4.D4 [×2], D4⋊2Q8 [×2], C23.47D4 [×2], C2×C4×D4, C23.37C23, C42.222D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C4○D4 [×4], C24, C2×SD16 [×6], C22×D4, C2×C4○D4 [×2], C22.19C24, C22×SD16, D8⋊C22, C42.222D4
Generators and relations
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 >
►On 32 points
(1 16 5 12)(2 4 6 8)(3 10 7 14)(9 11 13 15)(17 19 21 23)(18 25 22 29)(20 27 24 31)(26 28 30 32)
(1 25 10 24)(2 26 11 17)(3 27 12 18)(4 28 13 19)(5 29 14 20)(6 30 15 21)(7 31 16 22)(8 32 9 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 9 5 13)(2 12 6 16)(3 15 7 11)(4 10 8 14)(17 31 21 27)(18 26 22 30)(19 29 23 25)(20 32 24 28)
G:=sub<Sym(32)| (1,16,5,12)(2,4,6,8)(3,10,7,14)(9,11,13,15)(17,19,21,23)(18,25,22,29)(20,27,24,31)(26,28,30,32), (1,25,10,24)(2,26,11,17)(3,27,12,18)(4,28,13,19)(5,29,14,20)(6,30,15,21)(7,31,16,22)(8,32,9,23), (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), (1,9,5,13)(2,12,6,16)(3,15,7,11)(4,10,8,14)(17,31,21,27)(18,26,22,30)(19,29,23,25)(20,32,24,28)>;
G:=Group( (1,16,5,12)(2,4,6,8)(3,10,7,14)(9,11,13,15)(17,19,21,23)(18,25,22,29)(20,27,24,31)(26,28,30,32), (1,25,10,24)(2,26,11,17)(3,27,12,18)(4,28,13,19)(5,29,14,20)(6,30,15,21)(7,31,16,22)(8,32,9,23), (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), (1,9,5,13)(2,12,6,16)(3,15,7,11)(4,10,8,14)(17,31,21,27)(18,26,22,30)(19,29,23,25)(20,32,24,28) );
G=PermutationGroup([(1,16,5,12),(2,4,6,8),(3,10,7,14),(9,11,13,15),(17,19,21,23),(18,25,22,29),(20,27,24,31),(26,28,30,32)], [(1,25,10,24),(2,26,11,17),(3,27,12,18),(4,28,13,19),(5,29,14,20),(6,30,15,21),(7,31,16,22),(8,32,9,23)], [(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)], [(1,9,5,13),(2,12,6,16),(3,15,7,11),(4,10,8,14),(17,31,21,27),(18,26,22,30),(19,29,23,25),(20,32,24,28)])
Matrix representation ►G ⊆ 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] >;
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 |
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