direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4×C8⋊C22, C42.441D4, C42.273C23, D8⋊4(C2×C4), (C4×D8)⋊27C2, C8⋊1(C22×C4), (C4×C8)⋊20C22, SD16⋊1(C2×C4), C4.133(C4×D4), D4⋊3(C22×C4), C4○2(D8⋊C4), D8⋊C4⋊33C2, Q8⋊3(C22×C4), (C4×SD16)⋊11C2, (C4×D4)⋊80C22, (C4×M4(2))⋊1C2, M4(2)⋊7(C2×C4), C4.21(C23×C4), (C4×Q8)⋊76C22, C22.46(C4×D4), C2.D8⋊65C22, C4.Q8⋊45C22, C8⋊C4⋊36C22, C4⋊C4.361C23, (C2×C8).412C23, (C2×C4).201C24, C23.643(C2×D4), (C22×C4).710D4, C4○2(SD16⋊C4), SD16⋊C4⋊54C2, D4⋊C4⋊88C22, Q8⋊C4⋊91C22, (C2×D4).370C23, (C2×D8).157C22, (C2×Q8).343C23, C4○2(M4(2)⋊C4), M4(2)⋊C4⋊46C2, C4○2(C23.37D4), C4○2(C23.36D4), C23.37D4⋊39C2, C2.4(D8⋊C22), C23.36D4⋊48C2, (C2×C42).766C22, (C22×C4).922C23, C22.145(C22×D4), (C2×SD16).107C22, (C22×D4).559C22, C42⋊C2.297C22, (C2×M4(2)).350C22, (C2×C4×D4)⋊58C2, C2.61(C2×C4×D4), C4○D4⋊9(C2×C4), (C4×C4○D4)⋊5C2, C4.9(C2×C4○D4), (C2×D4)⋊34(C2×C4), C2.6(C2×C8⋊C22), (C2×C4)○(D8⋊C4), (C2×C4).691(C2×D4), (C2×C8⋊C22).13C2, (C2×C4).68(C22×C4), (C2×C4)○(SD16⋊C4), (C2×C4).262(C4○D4), (C2×C4⋊C4).912C22, (C2×C4○D4).291C22, (C2×C4)○(C23.37D4), (C2×C4)○(C2×C8⋊C22), SmallGroup(128,1676)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 500 in 274 conjugacy classes, 142 normal (44 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×22], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×26], D4 [×6], D4 [×11], Q8 [×2], Q8, C23, C23 [×11], C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×4], M4(2) [×4], M4(2) [×2], D8 [×8], SD16 [×8], C22×C4 [×3], C22×C4 [×12], C2×D4, C2×D4 [×6], C2×D4 [×4], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×4], C4×D4 [×4], C4×Q8 [×2], C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C23×C4, C22×D4, C2×C4○D4, C4×M4(2), C23.36D4, C23.37D4, M4(2)⋊C4, C4×D8 [×2], C4×SD16 [×2], SD16⋊C4 [×2], D8⋊C4 [×2], C2×C4×D4, C4×C4○D4, C2×C8⋊C22, C4×C8⋊C22
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C8⋊C22 [×2], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C8⋊C22, D8⋊C22, C4×C8⋊C22
Generators and relations
G = < a,b,c,d | a4=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >
►On 32 points
(1 19 28 11)(2 20 29 12)(3 21 30 13)(4 22 31 14)(5 23 32 15)(6 24 25 16)(7 17 26 9)(8 18 27 10)
(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 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)(25 27)(26 30)(29 31)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)
G:=sub<Sym(32)| (1,19,28,11)(2,20,29,12)(3,21,30,13)(4,22,31,14)(5,23,32,15)(6,24,25,16)(7,17,26,9)(8,18,27,10), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,21)(18,24)(20,22)(25,27)(26,30)(29,31), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)>;
G:=Group( (1,19,28,11)(2,20,29,12)(3,21,30,13)(4,22,31,14)(5,23,32,15)(6,24,25,16)(7,17,26,9)(8,18,27,10), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,21)(18,24)(20,22)(25,27)(26,30)(29,31), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31) );
G=PermutationGroup([(1,19,28,11),(2,20,29,12),(3,21,30,13),(4,22,31,14),(5,23,32,15),(6,24,25,16),(7,17,26,9),(8,18,27,10)], [(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)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(17,21),(18,24),(20,22),(25,27),(26,30),(29,31)], [(2,6),(4,8),(10,14),(12,16),(18,22),(20,24),(25,29),(27,31)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 16 | 8 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 13 | 0 |
| 0 | 0 | 1 | 0 | 0 | 13 |
| 0 | 0 | 13 | 4 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 13 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 4 | 13 |
| 0 | 0 | 0 | 16 | 4 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 2 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 13 |
| 0 | 0 | 0 | 1 | 0 | 13 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[16,4,0,0,0,0,8,1,0,0,0,0,0,0,1,1,13,9,0,0,0,0,4,0,0,0,13,0,0,0,0,0,0,13,0,16],[1,13,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,4,4,1,2,0,0,13,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,13,13,0,16] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4X | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | C8⋊C22 | D8⋊C22 |
| kernel | C4×C8⋊C22 | C4×M4(2) | C23.36D4 | C23.37D4 | M4(2)⋊C4 | C4×D8 | C4×SD16 | SD16⋊C4 | D8⋊C4 | C2×C4×D4 | C4×C4○D4 | C2×C8⋊C22 | C8⋊C22 | C42 | C22×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 16 | 2 | 2 | 4 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4\times C_8\rtimes C_2^2
% in TeX
G:=Group("C4xC8:C2^2"); // GroupNames label
G:=SmallGroup(128,1676);
// by ID
G=gap.SmallGroup(128,1676);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations