p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.462C23, C4.422+ 1+4, (C8×D4)⋊17C2, (C4×D8)⋊12C2, D4.Q8⋊6C2, D4⋊5D4⋊8C2, D4⋊D4⋊9C2, C8⋊7D4⋊12C2, C8⋊8D4⋊36C2, C4⋊C8⋊71C22, (C4×C8)⋊13C22, C4⋊C4.261D4, C22⋊D8⋊10C2, (C2×D4).234D4, D4.2D4⋊7C2, C2.45(D4○D8), (C4×D4)⋊24C22, C4.Q8⋊38C22, C2.D8⋊12C22, D4.18(C4○D4), C4⋊D4⋊15C22, C4⋊C4.228C23, C22⋊C8⋊64C22, (C2×C8).182C23, (C2×C4).489C24, C22⋊C4.101D4, (C22×C8)⋊13C22, Q8⋊C4⋊9C22, C23.107(C2×D4), C42.C2⋊8C22, D4⋊C4⋊60C22, (C2×SD16)⋊51C22, (C2×D8).138C22, (C2×D4).221C23, C22.11(C4○D8), C23.19D4⋊5C2, C23.48D4⋊7C2, (C2×Q8).206C23, C2.125(D4⋊5D4), C42⋊C2⋊21C22, C22⋊Q8.69C22, C23.24D4⋊10C2, C4.4D4.59C22, C22.749(C22×D4), C22.47C24⋊1C2, (C22×C4).1133C23, C42.78C22⋊4C2, (C22×D4).408C22, C2.59(C2×C4○D8), C4.214(C2×C4○D4), (C2×C4).166(C2×D4), (C2×D4⋊C4)⋊41C2, (C2×C4⋊C4).659C22, (C2×C4○D4).197C22, SmallGroup(128,2029)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.462C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=a2, e2=b2, ab=ba, cac-1=eae-1=a-1b2, ad=da, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2b2c, de=ed >
Subgroups: 472 in 210 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22×C8, C2×D8, C2×SD16, C22×D4, C2×C4○D4, C2×D4⋊C4, C23.24D4, C8×D4, C4×D8, C22⋊D8, D4⋊D4, D4.2D4, C8⋊8D4, C8⋊7D4, D4.Q8, C23.19D4, C23.48D4, C42.78C22, D4⋊5D4, C22.47C24, C42.462C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, C2×C4○D8, D4○D8, C42.462C23
►On 32 points
(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 28 24 19)(2 25 21 20)(3 26 22 17)(4 27 23 18)(5 12 15 31)(6 9 16 32)(7 10 13 29)(8 11 14 30)
(1 29 3 31)(2 9 4 11)(5 19 7 17)(6 27 8 25)(10 22 12 24)(13 26 15 28)(14 20 16 18)(21 32 23 30)
(1 3)(2 4)(5 10)(6 11)(7 12)(8 9)(13 31)(14 32)(15 29)(16 30)(17 28)(18 25)(19 26)(20 27)(21 23)(22 24)
(1 23 24 4)(2 3 21 22)(5 14 15 8)(6 7 16 13)(9 10 32 29)(11 12 30 31)(17 25 26 20)(18 19 27 28)
G:=sub<Sym(32)| (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), (1,28,24,19)(2,25,21,20)(3,26,22,17)(4,27,23,18)(5,12,15,31)(6,9,16,32)(7,10,13,29)(8,11,14,30), (1,29,3,31)(2,9,4,11)(5,19,7,17)(6,27,8,25)(10,22,12,24)(13,26,15,28)(14,20,16,18)(21,32,23,30), (1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,31)(14,32)(15,29)(16,30)(17,28)(18,25)(19,26)(20,27)(21,23)(22,24), (1,23,24,4)(2,3,21,22)(5,14,15,8)(6,7,16,13)(9,10,32,29)(11,12,30,31)(17,25,26,20)(18,19,27,28)>;
G:=Group( (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), (1,28,24,19)(2,25,21,20)(3,26,22,17)(4,27,23,18)(5,12,15,31)(6,9,16,32)(7,10,13,29)(8,11,14,30), (1,29,3,31)(2,9,4,11)(5,19,7,17)(6,27,8,25)(10,22,12,24)(13,26,15,28)(14,20,16,18)(21,32,23,30), (1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,31)(14,32)(15,29)(16,30)(17,28)(18,25)(19,26)(20,27)(21,23)(22,24), (1,23,24,4)(2,3,21,22)(5,14,15,8)(6,7,16,13)(9,10,32,29)(11,12,30,31)(17,25,26,20)(18,19,27,28) );
G=PermutationGroup([[(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)], [(1,28,24,19),(2,25,21,20),(3,26,22,17),(4,27,23,18),(5,12,15,31),(6,9,16,32),(7,10,13,29),(8,11,14,30)], [(1,29,3,31),(2,9,4,11),(5,19,7,17),(6,27,8,25),(10,22,12,24),(13,26,15,28),(14,20,16,18),(21,32,23,30)], [(1,3),(2,4),(5,10),(6,11),(7,12),(8,9),(13,31),(14,32),(15,29),(16,30),(17,28),(18,25),(19,26),(20,27),(21,23),(22,24)], [(1,23,24,4),(2,3,21,22),(5,14,15,8),(6,7,16,13),(9,10,32,29),(11,12,30,31),(17,25,26,20),(18,19,27,28)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C4○D8 | 2+ 1+4 | D4○D8 |
| kernel | C42.462C23 | C2×D4⋊C4 | C23.24D4 | C8×D4 | C4×D8 | C22⋊D8 | D4⋊D4 | D4.2D4 | C8⋊8D4 | C8⋊7D4 | D4.Q8 | C23.19D4 | C23.48D4 | C42.78C22 | D4⋊5D4 | C22.47C24 | C22⋊C4 | C4⋊C4 | C2×D4 | D4 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of C42.462C23 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 12 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 13 | 4 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,1,1,0,0,15,16],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[5,12,0,0,12,12,0,0,0,0,13,13,0,0,0,4],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,15,16] >;
C42.462C23 in GAP, Magma, Sage, TeX
C_4^2._{462}C_2^3 % in TeX
G:=Group("C4^2.462C2^3"); // GroupNames label
G:=SmallGroup(128,2029);
// by ID
G=gap.SmallGroup(128,2029);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2,e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1*b^2,a*d=d*a,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*b^2*c,d*e=e*d>;
// generators/relations