p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.228D4, C42.344C23, C4⋊C8⋊12C22, C4⋊Q8⋊60C22, (C4×Q8)⋊7C22, C8⋊C4⋊3C22, D4⋊Q8⋊22C2, D4.5(C4○D4), D4.D4⋊5C2, C4⋊C4.63C23, (C2×C8).37C23, C2.D8⋊23C22, SD16⋊C4⋊5C2, (C2×C4).308C24, C42.6C4⋊2C2, C22⋊SD16.1C2, (C22×C4).448D4, C23.673(C2×D4), (C2×Q8).76C23, C4.103(C8⋊C22), Q8⋊C4⋊23C22, (C4×D4).321C22, (C2×D4).403C23, C22⋊C8.21C22, D4⋊C4.29C22, C23.48D4⋊15C2, (C2×C42).835C22, (C2×SD16).10C22, C22.568(C22×D4), C22⋊Q8.169C22, (C22×C4).1024C23, C23.37C23⋊6C2, (C22×D4).576C22, C22.36(C8.C22), C2.109(C22.19C24), (C2×C4×D4).85C2, C4.193(C2×C4○D4), (C2×C4).497(C2×D4), C2.33(C2×C8⋊C22), C2.32(C2×C8.C22), (C2×C4⋊C4).937C22, SmallGroup(128,1842)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 436 in 222 conjugacy classes, 92 normal (28 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×18], C8 [×4], C2×C4 [×6], C2×C4 [×22], 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, C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2.D8 [×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, C42.6C4, SD16⋊C4 [×4], C22⋊SD16 [×2], D4.D4 [×2], D4⋊Q8 [×2], C23.48D4 [×2], C2×C4×D4, C23.37C23, C42.228D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8⋊C22 [×2], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C8⋊C22, C2×C8.C22, C42.228D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=a2b, dcd-1=c3 >
►On 32 points
(1 29 5 25)(2 4 6 8)(3 31 7 27)(9 11 13 15)(10 24 14 20)(12 18 16 22)(17 19 21 23)(26 28 30 32)
(1 12 31 20)(2 9 32 17)(3 14 25 22)(4 11 26 19)(5 16 27 24)(6 13 28 21)(7 10 29 18)(8 15 30 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 30 5 26)(2 25 6 29)(3 28 7 32)(4 31 8 27)(9 18 13 22)(10 21 14 17)(11 24 15 20)(12 19 16 23)
G:=sub<Sym(32)| (1,29,5,25)(2,4,6,8)(3,31,7,27)(9,11,13,15)(10,24,14,20)(12,18,16,22)(17,19,21,23)(26,28,30,32), (1,12,31,20)(2,9,32,17)(3,14,25,22)(4,11,26,19)(5,16,27,24)(6,13,28,21)(7,10,29,18)(8,15,30,23), (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), (1,30,5,26)(2,25,6,29)(3,28,7,32)(4,31,8,27)(9,18,13,22)(10,21,14,17)(11,24,15,20)(12,19,16,23)>;
G:=Group( (1,29,5,25)(2,4,6,8)(3,31,7,27)(9,11,13,15)(10,24,14,20)(12,18,16,22)(17,19,21,23)(26,28,30,32), (1,12,31,20)(2,9,32,17)(3,14,25,22)(4,11,26,19)(5,16,27,24)(6,13,28,21)(7,10,29,18)(8,15,30,23), (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), (1,30,5,26)(2,25,6,29)(3,28,7,32)(4,31,8,27)(9,18,13,22)(10,21,14,17)(11,24,15,20)(12,19,16,23) );
G=PermutationGroup([(1,29,5,25),(2,4,6,8),(3,31,7,27),(9,11,13,15),(10,24,14,20),(12,18,16,22),(17,19,21,23),(26,28,30,32)], [(1,12,31,20),(2,9,32,17),(3,14,25,22),(4,11,26,19),(5,16,27,24),(6,13,28,21),(7,10,29,18),(8,15,30,23)], [(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)], [(1,30,5,26),(2,25,6,29),(3,28,7,32),(4,31,8,27),(9,18,13,22),(10,21,14,17),(11,24,15,20),(12,19,16,23)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 14 | 8 | 1 | 2 |
| 0 | 0 | 7 | 3 | 16 | 16 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 15 | 16 | 0 |
| 0 | 0 | 4 | 2 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 | 2 | 0 |
| 0 | 0 | 12 | 0 | 15 | 15 |
| 0 | 0 | 15 | 7 | 10 | 2 |
| 0 | 0 | 13 | 13 | 2 | 15 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 15 | 15 | 0 |
| 0 | 0 | 4 | 2 | 0 | 15 |
| 0 | 0 | 3 | 7 | 9 | 2 |
| 0 | 0 | 3 | 7 | 13 | 15 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,16,1,0,0,0,0,0,0,1,16,14,7,0,0,2,16,8,3,0,0,0,0,1,16,0,0,0,0,2,16],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,8,4,0,0,0,1,15,2,0,0,0,0,16,0,0,0,0,0,0,16],[16,2,0,0,0,0,0,1,0,0,0,0,0,0,9,12,15,13,0,0,2,0,7,13,0,0,2,15,10,2,0,0,0,15,2,15],[16,2,0,0,0,0,0,1,0,0,0,0,0,0,8,4,3,3,0,0,15,2,7,7,0,0,15,0,9,13,0,0,0,15,2,15] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 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 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C42.228D4 | C42.6C4 | SD16⋊C4 | C22⋊SD16 | D4.D4 | D4⋊Q8 | C23.48D4 | C2×C4×D4 | C23.37C23 | C42 | C22×C4 | D4 | C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{228}D_4 % in TeX
G:=Group("C4^2.228D4"); // GroupNames label
G:=SmallGroup(128,1842);
// by ID
G=gap.SmallGroup(128,1842);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,521,80,4037,1027,124]);
// 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,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations