direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×Q8○M4(2), C4.21C25, C8.22C24, M4(2)⋊14C23, M4(2)○(C2×Q8), Q8○(C2×M4(2)), M4(2)○(C2×D4), D4○(C2×M4(2)), (C2×C8)⋊11C23, C8○D4⋊20C22, C4○(Q8○M4(2)), C4.43(C23×C4), C24.90(C2×C4), C2.15(C24×C4), M4(2)○2(C4○D4), (C2×C4).605C24, (C22×C8)⋊58C22, C4○D4.36C23, (C22×D4).45C4, D4.27(C22×C4), C22.8(C23×C4), Q8.28(C22×C4), (C22×Q8).35C4, M4(2)○2(C2×M4(2)), (C2×M4(2))⋊80C22, (C22×M4(2))⋊28C2, (C23×C4).621C22, C23.112(C22×C4), (C22×C4).1220C23, (C2×C8○D4)⋊28C2, C4○D4○(C2×M4(2)), M4(2)○(C2×C4○D4), (C2×Q8)○(C2×M4(2)), C4○D4.35(C2×C4), (C2×C4○D4).33C4, (C2×D4).240(C2×C4), (C2×Q8).215(C2×C4), (C2×M4(2))○(C2×M4(2)), (C2×C4).284(C22×C4), (C22×C4).373(C2×C4), (C22×C4○D4).28C2, (C2×C4○D4).334C22, (C2×M4(2))○(C2×C4○D4), SmallGroup(128,2304)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 812 in 730 conjugacy classes, 684 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×14], C4 [×2], C4 [×14], C22, C22 [×14], C22 [×26], C8 [×16], C2×C4 [×2], C2×C4 [×70], D4 [×48], Q8 [×16], C23, C23 [×18], C23 [×6], C2×C8 [×56], M4(2) [×64], C22×C4, C22×C4 [×39], C2×D4 [×36], C2×Q8 [×12], C4○D4 [×64], C24 [×3], C22×C8 [×12], C2×M4(2) [×64], C8○D4 [×64], C23×C4 [×3], C22×D4 [×3], C22×Q8, C2×C4○D4 [×24], C22×M4(2) [×6], C2×C8○D4 [×8], Q8○M4(2) [×16], C22×C4○D4, C2×Q8○M4(2)
Quotients:
C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C24 [×31], C23×C4 [×30], C25, Q8○M4(2) [×2], C24×C4, C2×Q8○M4(2)
Generators and relations
G = < a,b,c,d,e | a2=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
►On 32 points
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(1 21 5 17)(2 22 6 18)(3 23 7 19)(4 24 8 20)(9 29 13 25)(10 30 14 26)(11 31 15 27)(12 32 16 28)
(1 32 5 28)(2 25 6 29)(3 26 7 30)(4 27 8 31)(9 22 13 18)(10 23 14 19)(11 24 15 20)(12 17 16 21)
(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)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)
G:=sub<Sym(32)| (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (1,21,5,17)(2,22,6,18)(3,23,7,19)(4,24,8,20)(9,29,13,25)(10,30,14,26)(11,31,15,27)(12,32,16,28), (1,32,5,28)(2,25,6,29)(3,26,7,30)(4,27,8,31)(9,22,13,18)(10,23,14,19)(11,24,15,20)(12,17,16,21), (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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)>;
G:=Group( (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (1,21,5,17)(2,22,6,18)(3,23,7,19)(4,24,8,20)(9,29,13,25)(10,30,14,26)(11,31,15,27)(12,32,16,28), (1,32,5,28)(2,25,6,29)(3,26,7,30)(4,27,8,31)(9,22,13,18)(10,23,14,19)(11,24,15,20)(12,17,16,21), (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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31) );
G=PermutationGroup([(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,25),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19)], [(1,21,5,17),(2,22,6,18),(3,23,7,19),(4,24,8,20),(9,29,13,25),(10,30,14,26),(11,31,15,27),(12,32,16,28)], [(1,32,5,28),(2,25,6,29),(3,26,7,30),(4,27,8,31),(9,22,13,18),(10,23,14,19),(11,24,15,20),(12,17,16,21)], [(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)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 4 | 12 | 4 |
| 0 | 8 | 1 | 0 | 5 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 8 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 13 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 13 |
| 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 10 | 13 | 9 |
| 0 | 0 | 0 | 15 | 1 |
| 0 | 4 | 11 | 0 | 11 |
| 0 | 8 | 1 | 0 | 5 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 14 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,8,0,0,0,4,1,0,0,0,12,0,4,8,0,4,5,0,13],[1,0,0,0,0,0,13,0,0,0,0,16,4,0,0,0,0,0,4,0,0,0,0,13,13],[1,0,0,0,0,0,12,0,4,8,0,10,0,11,1,0,13,15,0,0,0,9,1,11,5],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,14,0,0,16] >;
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2Q | 4A | 4B | 4C | 4D | 4E | ··· | 4R | 8A | ··· | 8AF |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 |
| type | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8○M4(2) |
| kernel | C2×Q8○M4(2) | C22×M4(2) | C2×C8○D4 | Q8○M4(2) | C22×C4○D4 | C22×D4 | C22×Q8 | C2×C4○D4 | C2 |
| # reps | 1 | 6 | 8 | 16 | 1 | 6 | 2 | 24 | 4 |
In GAP, Magma, Sage, TeX
C_2\times Q_8\circ M_{4(2)} % in TeX
G:=Group("C2xQ8oM4(2)"); // GroupNames label
G:=SmallGroup(128,2304);
// by ID
G=gap.SmallGroup(128,2304);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,-2,224,723,2019,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=e^2=1,c^2=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations