p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.227D4, C42.343C23, D4⋊3(C4○D4), D8⋊C4⋊5C2, C4⋊D8⋊21C2, C4⋊C8⋊11C22, D4⋊2Q8⋊5C2, (C4×D4)⋊7C22, C4⋊Q8⋊59C22, C22⋊D8⋊15C2, C8⋊C4⋊2C22, C4⋊C4.62C23, (C2×C8).36C23, C4.Q8⋊12C22, C4⋊1D4⋊35C22, (C2×C4).307C24, C42.6C4⋊1C2, (C2×D8).57C22, (C2×D4).90C23, C23.672(C2×D4), (C22×C4).447D4, D4⋊C4⋊21C22, C4.102(C8⋊C22), C23.46D4⋊5C2, C22⋊C8.20C22, C4⋊D4.164C22, C22.47(C8⋊C22), C22.26C24⋊6C2, (C2×C42).834C22, C22.567(C22×D4), (C22×C4).1023C23, (C22×D4).575C22, C2.108(C22.19C24), (C2×C4×D4)⋊64C2, C4.192(C2×C4○D4), (C2×C4).496(C2×D4), C2.32(C2×C8⋊C22), (C2×C4⋊C4).936C22, SmallGroup(128,1841)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 532 in 244 conjugacy classes, 92 normal (28 characteristic)
C1, C2 [×3], C2 [×8], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×24], C8 [×4], C2×C4 [×6], C2×C4 [×24], D4 [×4], D4 [×18], Q8 [×2], C23, C23 [×12], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], D8 [×8], C22×C4 [×3], C22×C4 [×11], C2×D4 [×4], C2×D4 [×9], C2×Q8, C4○D4 [×4], C24, C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×4], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4 [×6], C4×D4 [×3], C4⋊D4 [×2], C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×D8 [×4], C23×C4, C22×D4, C2×C4○D4, C42.6C4, D8⋊C4 [×4], C22⋊D8 [×2], C4⋊D8 [×2], D4⋊2Q8 [×2], C23.46D4 [×2], C2×C4×D4, C22.26C24, C42.227D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8⋊C22 [×4], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C8⋊C22 [×2], C42.227D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=dbd=a2b, dcd=a2c3 >
►On 32 points
(1 29 5 25)(2 8 6 4)(3 31 7 27)(9 15 13 11)(10 21 14 17)(12 23 16 19)(18 24 22 20)(26 32 30 28)
(1 12 27 21)(2 9 28 18)(3 14 29 23)(4 11 30 20)(5 16 31 17)(6 13 32 22)(7 10 25 19)(8 15 26 24)
(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)(2 29)(3 28)(4 27)(5 26)(6 25)(7 32)(8 31)(9 19)(10 18)(11 17)(12 24)(13 23)(14 22)(15 21)(16 20)
G:=sub<Sym(32)| (1,29,5,25)(2,8,6,4)(3,31,7,27)(9,15,13,11)(10,21,14,17)(12,23,16,19)(18,24,22,20)(26,32,30,28), (1,12,27,21)(2,9,28,18)(3,14,29,23)(4,11,30,20)(5,16,31,17)(6,13,32,22)(7,10,25,19)(8,15,26,24), (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)(2,29)(3,28)(4,27)(5,26)(6,25)(7,32)(8,31)(9,19)(10,18)(11,17)(12,24)(13,23)(14,22)(15,21)(16,20)>;
G:=Group( (1,29,5,25)(2,8,6,4)(3,31,7,27)(9,15,13,11)(10,21,14,17)(12,23,16,19)(18,24,22,20)(26,32,30,28), (1,12,27,21)(2,9,28,18)(3,14,29,23)(4,11,30,20)(5,16,31,17)(6,13,32,22)(7,10,25,19)(8,15,26,24), (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)(2,29)(3,28)(4,27)(5,26)(6,25)(7,32)(8,31)(9,19)(10,18)(11,17)(12,24)(13,23)(14,22)(15,21)(16,20) );
G=PermutationGroup([(1,29,5,25),(2,8,6,4),(3,31,7,27),(9,15,13,11),(10,21,14,17),(12,23,16,19),(18,24,22,20),(26,32,30,28)], [(1,12,27,21),(2,9,28,18),(3,14,29,23),(4,11,30,20),(5,16,31,17),(6,13,32,22),(7,10,25,19),(8,15,26,24)], [(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),(2,29),(3,28),(4,27),(5,26),(6,25),(7,32),(8,31),(9,19),(10,18),(11,17),(12,24),(13,23),(14,22),(15,21),(16,20)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,0,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4N | 4O | 4P | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 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 | C42.227D4 | C42.6C4 | D8⋊C4 | C22⋊D8 | C4⋊D8 | D4⋊2Q8 | C23.46D4 | C2×C4×D4 | C22.26C24 | 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._{227}D_4 % in TeX
G:=Group("C4^2.227D4"); // GroupNames label
G:=SmallGroup(128,1841);
// by ID
G=gap.SmallGroup(128,1841);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,521,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a*b^2,c*b*c^-1=d*b*d=a^2*b,d*c*d=a^2*c^3>;
// generators/relations