p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.225D4, C42.701C23, (C4×D8)⋊20C2, C4⋊C8⋊66C22, (C4×C8)⋊14C22, (C4×SD16)⋊1C2, (C4×D4)⋊6C22, C4○3(D4.Q8), D4.Q8⋊55C2, C4○3(C22⋊D8), (C4×Q8)⋊6C22, D4.3(C4○D4), C22⋊D8.5C2, C4⋊C4.58C23, C4.Q8⋊66C22, C2.D8⋊57C22, C4○3(C22⋊SD16), C4○3(D4.2D4), D4.2D4⋊54C2, C22⋊SD16⋊37C2, (C2×C8).148C23, (C2×C4).303C24, (C2×D4).87C23, (C22×C4).720D4, C23.670(C2×D4), (C2×Q8).74C23, D4⋊C4⋊79C22, C4.148(C8⋊C22), Q8⋊C4⋊70C22, (C2×D8).123C22, C22.29(C4○D8), C4.4D4⋊54C22, C42.C2⋊31C22, C42.12C4⋊29C2, C4○3(C23.48D4), C4○3(C23.46D4), C23.46D4⋊37C2, C23.48D4⋊38C2, C4⋊D4.161C22, C22⋊C8.175C22, (C2×C42).830C22, C22.563(C22×D4), C22⋊Q8.166C22, C23.36C23⋊2C2, (C22×C4).1019C23, (C2×SD16).139C22, (C22×D4).574C22, C2.104(C22.19C24), (C2×C4×D4)⋊63C2, C2.24(C2×C4○D8), (C2×C4)○(D4.Q8), C4.188(C2×C4○D4), (C2×C4).494(C2×D4), C2.30(C2×C8⋊C22), (C2×C4⋊C4).934C22, (C2×C4)○(C23.48D4), SmallGroup(128,1837)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 460 in 227 conjugacy classes, 92 normal (42 characteristic)
C1, C2 [×3], C2 [×7], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×21], C8 [×4], C2×C4 [×6], C2×C4 [×23], D4 [×4], D4 [×10], Q8 [×2], C23, C23 [×11], C42 [×4], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], D8 [×4], SD16 [×4], C22×C4 [×3], C22×C4 [×10], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C24, C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4 [×4], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C2×D8 [×2], C2×SD16 [×2], C23×C4, C22×D4, C42.12C4, C4×D8 [×2], C4×SD16 [×2], C22⋊D8, C22⋊SD16, D4.2D4 [×2], D4.Q8 [×2], C23.46D4, C23.48D4, C2×C4×D4, C23.36C23, C42.225D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4○D8 [×2], C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C4○D8, C2×C8⋊C22, C42.225D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=ab2, dad=a-1b2, bc=cb, dbd=a2b, dcd=a2c3 >
►On 32 points
(1 31 5 27)(2 8 6 4)(3 25 7 29)(9 17 13 21)(10 16 14 12)(11 19 15 23)(18 24 22 20)(26 32 30 28)
(1 15 25 17)(2 16 26 18)(3 9 27 19)(4 10 28 20)(5 11 29 21)(6 12 30 22)(7 13 31 23)(8 14 32 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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)
G:=sub<Sym(32)| (1,31,5,27)(2,8,6,4)(3,25,7,29)(9,17,13,21)(10,16,14,12)(11,19,15,23)(18,24,22,20)(26,32,30,28), (1,15,25,17)(2,16,26,18)(3,9,27,19)(4,10,28,20)(5,11,29,21)(6,12,30,22)(7,13,31,23)(8,14,32,24), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)>;
G:=Group( (1,31,5,27)(2,8,6,4)(3,25,7,29)(9,17,13,21)(10,16,14,12)(11,19,15,23)(18,24,22,20)(26,32,30,28), (1,15,25,17)(2,16,26,18)(3,9,27,19)(4,10,28,20)(5,11,29,21)(6,12,30,22)(7,13,31,23)(8,14,32,24), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19) );
G=PermutationGroup([(1,31,5,27),(2,8,6,4),(3,25,7,29),(9,17,13,21),(10,16,14,12),(11,19,15,23),(18,24,22,20),(26,32,30,28)], [(1,15,25,17),(2,16,26,18),(3,9,27,19),(4,10,28,20),(5,11,29,21),(6,12,30,22),(7,13,31,23),(8,14,32,24)], [(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)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,18),(10,17),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 0 | 4 | 13 |
| 16 | 2 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 3 | 11 |
| 1 | 15 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 14 | 0 |
G:=sub<GL(4,GF(17))| [16,16,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[4,0,0,0,0,4,0,0,0,0,4,4,0,0,9,13],[16,0,0,0,2,1,0,0,0,0,0,3,0,0,11,11],[1,0,0,0,15,16,0,0,0,0,0,14,0,0,11,0] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4P | 4Q | 4R | 4S | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 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 | 4 | 4 | 4 | 4 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | C42.225D4 | C42.12C4 | C4×D8 | C4×SD16 | C22⋊D8 | C22⋊SD16 | D4.2D4 | D4.Q8 | C23.46D4 | C23.48D4 | C2×C4×D4 | C23.36C23 | C42 | C22×C4 | D4 | C22 | C4 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{225}D_4 % in TeX
G:=Group("C4^2.225D4"); // GroupNames label
G:=SmallGroup(128,1837);
// by ID
G=gap.SmallGroup(128,1837);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,304,4037,1027,124]);
// 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*b^2,d*a*d=a^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=a^2*c^3>;
// generators/relations