direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×2+ (1+4), C22.15C25, C23.109C24, C24.474C23, C42.536C23, D4○(C4×D4), Q8○(C4×Q8), Q8⋊9(C22×C4), (C4×D4)⋊88C22, D4⋊10(C22×C4), C2.11(C24×C4), C4.41(C23×C4), C23⋊2(C22×C4), C4⋊C4.516C23, (C2×C4).161C24, (C2×C42)⋊41C22, (C23×C4)⋊16C22, (C4×Q8)⋊102C22, C22.5(C23×C4), (C2×D4).499C23, (C2×Q8).482C23, C4○2(C22.11C24), C42⋊C2⋊82C22, C22.11C24⋊26C2, C2.3(C2×2+ (1+4)), C22⋊C4.128C23, C2.2(C2.C25), (C22×C4).1295C23, (C22×D4).580C22, (C2×2+ (1+4)).10C2, C4○2(C23.33C23), C23.33C23⋊34C2, C4⋊C4○(C4×D4), (C2×D4)○(C4×D4), (C4×D4)○(C4×D4), (C4×Q8)○(C4×Q8), (C2×C4×D4)⋊68C2, C22⋊C4○(C4×D4), C4○D4⋊10(C2×C4), (C4×C4○D4)⋊14C2, (C2×D4)⋊35(C2×C4), (C2×C4)⋊2(C22×C4), (C2×C4⋊C4)⋊121C22, (C2×C4)○(C2×2+ (1+4)), (C2×C22⋊C4)⋊78C22, (C2×C4)○(C22.11C24), (C2×C4○D4).318C22, (C2×C4)○(C23.33C23), SmallGroup(128,2161)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 1124 in 830 conjugacy classes, 686 normal (8 characteristic)
C1, C2 [×3], C2 [×18], C4 [×14], C4 [×15], C22, C22 [×18], C22 [×42], C2×C4, C2×C4 [×57], C2×C4 [×45], D4 [×72], Q8 [×8], C23 [×33], C23 [×12], C42 [×24], C22⋊C4 [×36], C4⋊C4 [×24], C22×C4 [×45], C22×C4 [×12], C2×D4 [×90], C2×Q8 [×2], C4○D4 [×48], C24 [×6], C2×C42 [×9], C2×C22⋊C4 [×18], C2×C4⋊C4 [×9], C42⋊C2 [×18], C4×D4 [×72], C4×Q8 [×8], C23×C4 [×6], C22×D4 [×9], C2×C4○D4 [×6], 2+ (1+4) [×16], C2×C4×D4 [×9], C4×C4○D4 [×6], C22.11C24 [×9], C23.33C23 [×6], C2×2+ (1+4), C4×2+ (1+4)
Quotients:
C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C24 [×31], C23×C4 [×30], 2+ (1+4) [×2], C25, C24×C4, C2×2+ (1+4), C2.C25, C4×2+ (1+4)
Generators and relations
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
►On 32 points
(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 17 11 14)(2 18 12 15)(3 19 9 16)(4 20 10 13)(5 25 32 22)(6 26 29 23)(7 27 30 24)(8 28 31 21)
(1 14)(2 15)(3 16)(4 13)(5 25)(6 26)(7 27)(8 28)(9 19)(10 20)(11 17)(12 18)(21 31)(22 32)(23 29)(24 30)
(1 27 11 24)(2 28 12 21)(3 25 9 22)(4 26 10 23)(5 19 32 16)(6 20 29 13)(7 17 30 14)(8 18 31 15)
(1 22)(2 23)(3 24)(4 21)(5 17)(6 18)(7 19)(8 20)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)
G:=sub<Sym(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), (1,17,11,14)(2,18,12,15)(3,19,9,16)(4,20,10,13)(5,25,32,22)(6,26,29,23)(7,27,30,24)(8,28,31,21), (1,14)(2,15)(3,16)(4,13)(5,25)(6,26)(7,27)(8,28)(9,19)(10,20)(11,17)(12,18)(21,31)(22,32)(23,29)(24,30), (1,27,11,24)(2,28,12,21)(3,25,9,22)(4,26,10,23)(5,19,32,16)(6,20,29,13)(7,17,30,14)(8,18,31,15), (1,22)(2,23)(3,24)(4,21)(5,17)(6,18)(7,19)(8,20)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)>;
G:=Group( (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), (1,17,11,14)(2,18,12,15)(3,19,9,16)(4,20,10,13)(5,25,32,22)(6,26,29,23)(7,27,30,24)(8,28,31,21), (1,14)(2,15)(3,16)(4,13)(5,25)(6,26)(7,27)(8,28)(9,19)(10,20)(11,17)(12,18)(21,31)(22,32)(23,29)(24,30), (1,27,11,24)(2,28,12,21)(3,25,9,22)(4,26,10,23)(5,19,32,16)(6,20,29,13)(7,17,30,14)(8,18,31,15), (1,22)(2,23)(3,24)(4,21)(5,17)(6,18)(7,19)(8,20)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30) );
G=PermutationGroup([(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)], [(1,17,11,14),(2,18,12,15),(3,19,9,16),(4,20,10,13),(5,25,32,22),(6,26,29,23),(7,27,30,24),(8,28,31,21)], [(1,14),(2,15),(3,16),(4,13),(5,25),(6,26),(7,27),(8,28),(9,19),(10,20),(11,17),(12,18),(21,31),(22,32),(23,29),(24,30)], [(1,27,11,24),(2,28,12,21),(3,25,9,22),(4,26,10,23),(5,19,32,16),(6,20,29,13),(7,17,30,14),(8,18,31,15)], [(1,22),(2,23),(3,24),(4,21),(5,17),(6,18),(7,19),(8,20),(9,27),(10,28),(11,25),(12,26),(13,31),(14,32),(15,29),(16,30)])
Matrix representation ►G ⊆ GL5(𝔽5)
| 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 2 | 4 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 2 | 2 | 0 | 0 |
| 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 |
| 0 | 4 | 2 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 2 | 0 |
| 0 | 0 | 0 | 3 | 1 |
| 0 | 4 | 0 | 1 | 0 |
| 0 | 3 | 4 | 2 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 2 | 0 |
| 0 | 0 | 0 | 3 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 2 | 0 |
G:=sub<GL(5,GF(5))| [2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,2,0,0,0,0,2,3,1,2,0,0,0,0,4,0,0,0,1,0],[4,0,0,0,0,0,2,1,0,4,0,2,3,1,2,0,0,0,0,1,0,0,0,1,0],[4,0,0,0,0,0,4,0,4,3,0,0,0,0,4,0,2,3,1,2,0,0,1,0,0],[4,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,2,3,1,2,0,0,1,0,0] >;
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2U | 4A | 4B | 4C | 4D | 4E | ··· | 4AT |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | 2+ (1+4) | C2.C25 |
| kernel | C4×2+ (1+4) | C2×C4×D4 | C4×C4○D4 | C22.11C24 | C23.33C23 | C2×2+ (1+4) | 2+ (1+4) | C4 | C2 |
| # reps | 1 | 9 | 6 | 9 | 6 | 1 | 32 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4\times 2_+^{(1+4)} % in TeX
G:=Group("C4xES+(2,2)"); // GroupNames label
G:=SmallGroup(128,2161);
// by ID
G=gap.SmallGroup(128,2161);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,387,1123,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=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