p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.264C23, C8○C22≀C2, (C2×C8)⋊41D4, (C8×D4)⋊36C2, C8○(C8⋊9D4), C8○(C4⋊D4), C8○(C22⋊Q8), C8⋊9D4⋊52C2, C4⋊C8⋊87C22, (C4×C8)⋊55C22, (C23×C8)⋊15C2, C4.131(C4×D4), C8.143(C2×D4), C22≀C2.8C4, C4⋊D4.34C4, C22⋊1(C8○D4), C22.44(C4×D4), C8⋊C4⋊58C22, C22⋊Q8.34C4, C22⋊C8⋊76C22, C8○(C24.4C4), (C2×C8).639C23, (C2×C4).648C24, C24.101(C2×C4), (C22×C8)⋊50C22, C8○(C22.D4), C4.194(C22×D4), C8○2M4(2)⋊30C2, C24.4C4⋊39C2, C8○(C22.19C24), (C4×D4).286C22, C23.32(C22×C4), (C2×M4(2))⋊76C22, C8○(C42.6C22), (C23×C4).699C22, C22.175(C23×C4), C22.D4.15C4, C42.6C22⋊37C2, (C22×C4).1276C23, C22.19C24.24C2, C42⋊C2.292C22, C2.46(C2×C4×D4), (C2×C8)○C22≀C2, (C2×C8○D4)⋊19C2, (C2×C8)○(C8⋊9D4), C2.16(C2×C8○D4), (C2×C8)○(C22⋊Q8), C4⋊C4.159(C2×C4), C4.299(C2×C4○D4), C8○((C22×C8)⋊C2), (C2×D4).171(C2×C4), (C2×C4).1411(C2×D4), C22⋊C4.34(C2×C4), (C2×C4).65(C22×C4), (C2×Q8).152(C2×C4), (C22×C8)⋊C2⋊38C2, (C2×C4).681(C4○D4), (C2×C8)○(C24.4C4), (C22×C4).385(C2×C4), (C2×C8)○(C22.19C24), (C2×C4○D4).283C22, (C2×C8)○(C42.6C22), (C2×C8)○((C22×C8)⋊C2), SmallGroup(128,1661)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.264C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2b-1, ab=ba, cac-1=a-1b2, ad=da, eae=ab2, bc=cb, bd=db, be=eb, dcd-1=a2b2c, ce=ec, ede=b2d >
Subgroups: 380 in 260 conjugacy classes, 144 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22×C8, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C23×C4, C2×C4○D4, C8○2M4(2), C24.4C4, (C22×C8)⋊C2, C42.6C22, C8×D4, C8⋊9D4, C22.19C24, C23×C8, C2×C8○D4, C42.264C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C8○D4, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C8○D4, C42.264C23
►On 32 points
(1 13 27 18)(2 14 28 19)(3 15 29 20)(4 16 30 21)(5 9 31 22)(6 10 32 23)(7 11 25 24)(8 12 26 17)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 20 13 24)(10 21 14 17)(11 22 15 18)(12 23 16 19)
(1 3 5 7)(2 26 6 30)(4 28 8 32)(9 20 13 24)(10 12 14 16)(11 22 15 18)(17 19 21 23)(25 27 29 31)
(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 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)
G:=sub<Sym(32)| (1,13,27,18)(2,14,28,19)(3,15,29,20)(4,16,30,21)(5,9,31,22)(6,10,32,23)(7,11,25,24)(8,12,26,17), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,20,13,24)(10,21,14,17)(11,22,15,18)(12,23,16,19), (1,3,5,7)(2,26,6,30)(4,28,8,32)(9,20,13,24)(10,12,14,16)(11,22,15,18)(17,19,21,23)(25,27,29,31), (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), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)>;
G:=Group( (1,13,27,18)(2,14,28,19)(3,15,29,20)(4,16,30,21)(5,9,31,22)(6,10,32,23)(7,11,25,24)(8,12,26,17), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,20,13,24)(10,21,14,17)(11,22,15,18)(12,23,16,19), (1,3,5,7)(2,26,6,30)(4,28,8,32)(9,20,13,24)(10,12,14,16)(11,22,15,18)(17,19,21,23)(25,27,29,31), (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), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21) );
G=PermutationGroup([[(1,13,27,18),(2,14,28,19),(3,15,29,20),(4,16,30,21),(5,9,31,22),(6,10,32,23),(7,11,25,24),(8,12,26,17)], [(1,25,5,29),(2,26,6,30),(3,27,7,31),(4,28,8,32),(9,20,13,24),(10,21,14,17),(11,22,15,18),(12,23,16,19)], [(1,3,5,7),(2,26,6,30),(4,28,8,32),(9,20,13,24),(10,12,14,16),(11,22,15,18),(17,19,21,23),(25,27,29,31)], [(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)], [(1,27),(2,32),(3,29),(4,26),(5,31),(6,28),(7,25),(8,30),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4P | 8A | ··· | 8H | 8I | ··· | 8T | 8U | ··· | 8AB |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | C4○D4 | C8○D4 |
| kernel | C42.264C23 | C8○2M4(2) | C24.4C4 | (C22×C8)⋊C2 | C42.6C22 | C8×D4 | C8⋊9D4 | C22.19C24 | C23×C8 | C2×C8○D4 | C22≀C2 | C4⋊D4 | C22⋊Q8 | C22.D4 | C2×C8 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 16 |
Matrix representation of C42.264C23 ►in GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 9 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [0,16,0,0,16,0,0,0,0,0,0,1,0,0,16,0],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,4,0,0,0,0,16,0,0,0,0,1],[0,9,0,0,9,0,0,0,0,0,0,1,0,0,16,0],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16] >;
C42.264C23 in GAP, Magma, Sage, TeX
C_4^2._{264}C_2^3 % in TeX
G:=Group("C4^2.264C2^3"); // GroupNames label
G:=SmallGroup(128,1661);
// by ID
G=gap.SmallGroup(128,1661);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=b^2,d^2=a^2*b^-1,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^2*b^2*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations