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, (C23×C4).533C22, C22.198(C23×C4), (C22×C4).941C23, C42.7C22⋊27C2, C42⋊C2.86C22, 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
Subgroups: 348 in 211 conjugacy classes, 128 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×7], C4 [×2], C4 [×9], C22, C22 [×4], C22 [×17], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×13], D4 [×11], Q8, C23, C23 [×4], C23 [×5], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×6], M4(2) [×2], C22×C4 [×2], C22×C4 [×6], C22×C4 [×2], C2×D4, C2×D4 [×6], C2×Q8, C4○D4 [×2], C24, C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×12], C4⋊C8 [×4], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C22×C8 [×6], C2×M4(2) [×2], C23×C4, C2×C4○D4, C2×C22⋊C8 [×2], (C22×C8)⋊C2 [×2], C42.7C22 [×2], C8×D4 [×4], C8⋊9D4 [×4], C22.19C24, C42.298C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C8○D4 [×4], C23×C4, 2+ (1+4) [×2], C22.11C24, C2×C8○D4 [×2], C42.298C23
Generators and relations
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 >
►On 32 points
(1 23 27 16)(2 13 28 20)(3 17 29 10)(4 15 30 22)(5 19 31 12)(6 9 32 24)(7 21 25 14)(8 11 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 16)(2 20)(3 10)(4 22)(5 12)(6 24)(7 14)(8 18)(9 32)(11 26)(13 28)(15 30)(17 29)(19 31)(21 25)(23 27)
(1 5)(2 28)(3 7)(4 30)(6 32)(8 26)(9 20)(11 22)(13 24)(15 18)(25 29)(27 31)
G:=sub<Sym(32)| (1,23,27,16)(2,13,28,20)(3,17,29,10)(4,15,30,22)(5,19,31,12)(6,9,32,24)(7,21,25,14)(8,11,26,18), (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), (1,16)(2,20)(3,10)(4,22)(5,12)(6,24)(7,14)(8,18)(9,32)(11,26)(13,28)(15,30)(17,29)(19,31)(21,25)(23,27), (1,5)(2,28)(3,7)(4,30)(6,32)(8,26)(9,20)(11,22)(13,24)(15,18)(25,29)(27,31)>;
G:=Group( (1,23,27,16)(2,13,28,20)(3,17,29,10)(4,15,30,22)(5,19,31,12)(6,9,32,24)(7,21,25,14)(8,11,26,18), (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), (1,16)(2,20)(3,10)(4,22)(5,12)(6,24)(7,14)(8,18)(9,32)(11,26)(13,28)(15,30)(17,29)(19,31)(21,25)(23,27), (1,5)(2,28)(3,7)(4,30)(6,32)(8,26)(9,20)(11,22)(13,24)(15,18)(25,29)(27,31) );
G=PermutationGroup([(1,23,27,16),(2,13,28,20),(3,17,29,10),(4,15,30,22),(5,19,31,12),(6,9,32,24),(7,21,25,14),(8,11,26,18)], [(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)], [(1,16),(2,20),(3,10),(4,22),(5,12),(6,24),(7,14),(8,18),(9,32),(11,26),(13,28),(15,30),(17,29),(19,31),(21,25),(23,27)], [(1,5),(2,28),(3,7),(4,30),(6,32),(8,26),(9,20),(11,22),(13,24),(15,18),(25,29),(27,31)])
Matrix representation ►G ⊆ 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] >;
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 |
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