p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.298C23, C4.1712+ 1+4, (C8×D4)⋊47C2, (C4×C8)⋊5C22, C8⋊9D4⋊42C2, C4⋊C8⋊92C22, C22≀C2.6C4, C4⋊D4.25C4, C24.88(C2×C4), (C22×C8)⋊6C22, C8⋊C4⋊32C22, C22⋊Q8.25C4, C22⋊C8⋊81C22, (C2×C4).674C24, (C2×C8).435C23, C22.7(C8○D4), (C4×D4).301C22, C23.41(C22×C4), C22.D4.9C4, (C2×M4(2))⋊48C22, (C22×C4).941C23, C22.198(C23×C4), (C23×C4).533C22, C42⋊C2.86C22, C42.7C22⋊27C2, C22.19C24.13C2, C2.48(C22.11C24), C2.29(C2×C8○D4), C4⋊C4.168(C2×C4), (C2×C22⋊C8)⋊48C2, (C2×D4).183(C2×C4), C22⋊C4.43(C2×C4), (C2×C4).80(C22×C4), (C2×Q8).123(C2×C4), (C22×C8)⋊C2⋊33C2, (C22×C4).139(C2×C4), (C2×C4○D4).94C22, SmallGroup(128,1709)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.298C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, dad=a-1, eae=ab2, bc=cb, bd=db, be=eb, dcd=ece=a2b2c, ede=b2d >
Subgroups: 348 in 211 conjugacy classes, 128 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, 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, C2×M4(2), C23×C4, C2×C4○D4, C2×C22⋊C8, (C22×C8)⋊C2, C42.7C22, C8×D4, C8⋊9D4, C22.19C24, C42.298C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C8○D4, C23×C4, 2+ 1+4, C22.11C24, C2×C8○D4, C42.298C23
►On 32 points
(1 23 27 15)(2 12 28 20)(3 17 29 9)(4 14 30 22)(5 19 31 11)(6 16 32 24)(7 21 25 13)(8 10 26 18)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 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 15)(2 20)(3 9)(4 22)(5 11)(6 24)(7 13)(8 18)(10 26)(12 28)(14 30)(16 32)(17 29)(19 31)(21 25)(23 27)
(1 5)(2 28)(3 7)(4 30)(6 32)(8 26)(10 22)(12 24)(14 18)(16 20)(25 29)(27 31)
G:=sub<Sym(32)| (1,23,27,15)(2,12,28,20)(3,17,29,9)(4,14,30,22)(5,19,31,11)(6,16,32,24)(7,21,25,13)(8,10,26,18), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (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), (1,15)(2,20)(3,9)(4,22)(5,11)(6,24)(7,13)(8,18)(10,26)(12,28)(14,30)(16,32)(17,29)(19,31)(21,25)(23,27), (1,5)(2,28)(3,7)(4,30)(6,32)(8,26)(10,22)(12,24)(14,18)(16,20)(25,29)(27,31)>;
G:=Group( (1,23,27,15)(2,12,28,20)(3,17,29,9)(4,14,30,22)(5,19,31,11)(6,16,32,24)(7,21,25,13)(8,10,26,18), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (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), (1,15)(2,20)(3,9)(4,22)(5,11)(6,24)(7,13)(8,18)(10,26)(12,28)(14,30)(16,32)(17,29)(19,31)(21,25)(23,27), (1,5)(2,28)(3,7)(4,30)(6,32)(8,26)(10,22)(12,24)(14,18)(16,20)(25,29)(27,31) );
G=PermutationGroup([[(1,23,27,15),(2,12,28,20),(3,17,29,9),(4,14,30,22),(5,19,31,11),(6,16,32,24),(7,21,25,13),(8,10,26,18)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32)], [(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)], [(1,15),(2,20),(3,9),(4,22),(5,11),(6,24),(7,13),(8,18),(10,26),(12,28),(14,30),(16,32),(17,29),(19,31),(21,25),(23,27)], [(1,5),(2,28),(3,7),(4,30),(6,32),(8,26),(10,22),(12,24),(14,18),(16,20),(25,29),(27,31)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4O | 8A | ··· | 8P | 8Q | ··· | 8X |
| 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8○D4 | 2+ 1+4 |
| kernel | C42.298C23 | C2×C22⋊C8 | (C22×C8)⋊C2 | C42.7C22 | C8×D4 | C8⋊9D4 | C22.19C24 | C22≀C2 | C4⋊D4 | C22⋊Q8 | C22.D4 | C22 | C4 |
| # reps | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 4 | 4 | 4 | 4 | 16 | 2 |
Matrix representation of C42.298C23 ►in GL4(𝔽17) generated by
| 1 | 15 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 9 | 0 |
| 1 | 15 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 16 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [1,1,0,0,15,16,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,0,2,0,0,0,0,0,9,0,0,8,0],[1,0,0,0,15,16,0,0,0,0,0,1,0,0,1,0],[16,16,0,0,0,1,0,0,0,0,16,0,0,0,0,1] >;
C42.298C23 in GAP, Magma, Sage, TeX
C_4^2._{298}C_2^3 % in TeX
G:=Group("C4^2.298C2^3"); // GroupNames label
G:=SmallGroup(128,1709);
// by ID
G=gap.SmallGroup(128,1709);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,891,675,1018,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a^-1,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=e*c*e=a^2*b^2*c,e*d*e=b^2*d>;
// generators/relations