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, C2.D8⋊57C22, C4.Q8⋊66C22, C4○3(D4.2D4), C4○3(C22⋊SD16), C22⋊SD16⋊37C2, D4.2D4⋊54C2, (C2×C4).303C24, (C2×C8).148C23, (C2×D4).87C23, C23.670(C2×D4), (C22×C4).720D4, (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(C23.46D4), C4○3(C23.48D4), C23.46D4⋊37C2, C4⋊D4.161C22, C23.48D4⋊38C2, 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
Generators and relations for C42.225D4
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 >
Subgroups: 460 in 227 conjugacy classes, 92 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C24, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C2×D8, C2×SD16, C23×C4, C22×D4, C42.12C4, C4×D8, C4×SD16, C22⋊D8, C22⋊SD16, D4.2D4, D4.Q8, C23.46D4, C23.48D4, C2×C4×D4, C23.36C23, C42.225D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C8⋊C22, C22×D4, C2×C4○D4, C22.19C24, C2×C4○D8, C2×C8⋊C22, C42.225D4
►On 32 points
(1 16 5 12)(2 8 6 4)(3 10 7 14)(9 15 13 11)(17 27 21 31)(18 24 22 20)(19 29 23 25)(26 32 30 28)
(1 29 10 17)(2 30 11 18)(3 31 12 19)(4 32 13 20)(5 25 14 21)(6 26 15 22)(7 27 16 23)(8 28 9 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 9)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)
G:=sub<Sym(32)| (1,16,5,12)(2,8,6,4)(3,10,7,14)(9,15,13,11)(17,27,21,31)(18,24,22,20)(19,29,23,25)(26,32,30,28), (1,29,10,17)(2,30,11,18)(3,31,12,19)(4,32,13,20)(5,25,14,21)(6,26,15,22)(7,27,16,23)(8,28,9,24), (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), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)>;
G:=Group( (1,16,5,12)(2,8,6,4)(3,10,7,14)(9,15,13,11)(17,27,21,31)(18,24,22,20)(19,29,23,25)(26,32,30,28), (1,29,10,17)(2,30,11,18)(3,31,12,19)(4,32,13,20)(5,25,14,21)(6,26,15,22)(7,27,16,23)(8,28,9,24), (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), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25) );
G=PermutationGroup([[(1,16,5,12),(2,8,6,4),(3,10,7,14),(9,15,13,11),(17,27,21,31),(18,24,22,20),(19,29,23,25),(26,32,30,28)], [(1,29,10,17),(2,30,11,18),(3,31,12,19),(4,32,13,20),(5,25,14,21),(6,26,15,22),(7,27,16,23),(8,28,9,24)], [(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)], [(1,9),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25)]])
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 |
Matrix representation of C42.225D4 ►in 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] >;
C42.225D4 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