direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4×C4○D8, C42○D8, C42○Q16, C42○SD16, C42.461D4, C42.686C23, C4○2(C4×D8), C4○2(C4×Q16), C42○(C4×D8), C42○(C2×D8), (C4×D8)⋊47C2, D8⋊14(C2×C4), C4○2(C4×SD16), C42○(C2×Q16), C42○(C4×Q16), (C4×Q16)⋊47C2, Q16⋊14(C2×C4), C4.185(C4×D4), C42○(C4×SD16), C42○(C2×SD16), C22.4(C4×D4), C42○(C2.D8), C42○(C4.Q8), (C4×SD16)⋊65C2, SD16⋊10(C2×C4), C8.45(C22×C4), C4.16(C23×C4), D4.2(C22×C4), Q8.2(C22×C4), C42○(D4⋊C4), C4⋊C4.356C23, (C2×C8).478C23, (C4×C8).407C22, (C2×C4).196C24, C42○(Q8⋊C4), (C22×C4).563D4, C23.380(C2×D4), (C2×D8).172C22, (C4×D4).289C22, (C2×D4).366C23, (C4×Q8).272C22, (C2×Q8).339C23, C2.D8.234C22, C4.Q8.185C22, C4○2(C23.24D4), C4○2(C23.25D4), C23.24D4⋊44C2, C23.25D4⋊34C2, (C22×C8).513C22, (C2×Q16).168C22, C22.140(C22×D4), D4⋊C4.215C22, (C22×C4).1512C23, (C2×C42).1114C22, Q8⋊C4.215C22, (C2×SD16).177C22, C42○(C23.24D4), C42○(C23.25D4), C42⋊C2.295C22, (C2×C4×C8)⋊28C2, C2.56(C2×C4×D4), (C2×C8)⋊30(C2×C4), (C4×C4○D4)⋊3C2, C4○D4⋊7(C2×C4), C4.4(C2×C4○D4), C2.6(C2×C4○D8), C42○(C2×C4○D8), (C2×C4○D8).21C2, (C2×C4).1576(C2×D4), (C2×C4).688(C4○D4), (C2×C4).467(C22×C4), (C2×C4○D4).289C22, SmallGroup(128,1671)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 396 in 250 conjugacy classes, 144 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×2], C22 [×10], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×26], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C42 [×2], C42 [×2], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×2], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×6], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C4×C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42 [×2], C42⋊C2 [×2], C42⋊C2 [×2], C4×D4 [×4], C4×D4 [×4], C4×Q8 [×4], C22×C8 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C2×C4×C8, C23.24D4 [×2], C23.25D4, C4×D8 [×2], C4×SD16 [×4], C4×Q16 [×2], C4×C4○D4 [×2], C2×C4○D8, C4×C4○D8
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], C4○D8 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C4○D8 [×2], C4×C4○D8
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >
►On 64 points
(1 31 59 10)(2 32 60 11)(3 25 61 12)(4 26 62 13)(5 27 63 14)(6 28 64 15)(7 29 57 16)(8 30 58 9)(17 36 55 45)(18 37 56 46)(19 38 49 47)(20 39 50 48)(21 40 51 41)(22 33 52 42)(23 34 53 43)(24 35 54 44)
(1 49 5 53)(2 50 6 54)(3 51 7 55)(4 52 8 56)(9 37 13 33)(10 38 14 34)(11 39 15 35)(12 40 16 36)(17 61 21 57)(18 62 22 58)(19 63 23 59)(20 64 24 60)(25 41 29 45)(26 42 30 46)(27 43 31 47)(28 44 32 48)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 62)(2 61)(3 60)(4 59)(5 58)(6 57)(7 64)(8 63)(9 27)(10 26)(11 25)(12 32)(13 31)(14 30)(15 29)(16 28)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 56)(24 55)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 48)
G:=sub<Sym(64)| (1,31,59,10)(2,32,60,11)(3,25,61,12)(4,26,62,13)(5,27,63,14)(6,28,64,15)(7,29,57,16)(8,30,58,9)(17,36,55,45)(18,37,56,46)(19,38,49,47)(20,39,50,48)(21,40,51,41)(22,33,52,42)(23,34,53,43)(24,35,54,44), (1,49,5,53)(2,50,6,54)(3,51,7,55)(4,52,8,56)(9,37,13,33)(10,38,14,34)(11,39,15,35)(12,40,16,36)(17,61,21,57)(18,62,22,58)(19,63,23,59)(20,64,24,60)(25,41,29,45)(26,42,30,46)(27,43,31,47)(28,44,32,48), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,62)(2,61)(3,60)(4,59)(5,58)(6,57)(7,64)(8,63)(9,27)(10,26)(11,25)(12,32)(13,31)(14,30)(15,29)(16,28)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,56)(24,55)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,48)>;
G:=Group( (1,31,59,10)(2,32,60,11)(3,25,61,12)(4,26,62,13)(5,27,63,14)(6,28,64,15)(7,29,57,16)(8,30,58,9)(17,36,55,45)(18,37,56,46)(19,38,49,47)(20,39,50,48)(21,40,51,41)(22,33,52,42)(23,34,53,43)(24,35,54,44), (1,49,5,53)(2,50,6,54)(3,51,7,55)(4,52,8,56)(9,37,13,33)(10,38,14,34)(11,39,15,35)(12,40,16,36)(17,61,21,57)(18,62,22,58)(19,63,23,59)(20,64,24,60)(25,41,29,45)(26,42,30,46)(27,43,31,47)(28,44,32,48), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,62)(2,61)(3,60)(4,59)(5,58)(6,57)(7,64)(8,63)(9,27)(10,26)(11,25)(12,32)(13,31)(14,30)(15,29)(16,28)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,56)(24,55)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,48) );
G=PermutationGroup([(1,31,59,10),(2,32,60,11),(3,25,61,12),(4,26,62,13),(5,27,63,14),(6,28,64,15),(7,29,57,16),(8,30,58,9),(17,36,55,45),(18,37,56,46),(19,38,49,47),(20,39,50,48),(21,40,51,41),(22,33,52,42),(23,34,53,43),(24,35,54,44)], [(1,49,5,53),(2,50,6,54),(3,51,7,55),(4,52,8,56),(9,37,13,33),(10,38,14,34),(11,39,15,35),(12,40,16,36),(17,61,21,57),(18,62,22,58),(19,63,23,59),(20,64,24,60),(25,41,29,45),(26,42,30,46),(27,43,31,47),(28,44,32,48)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,62),(2,61),(3,60),(4,59),(5,58),(6,57),(7,64),(8,63),(9,27),(10,26),(11,25),(12,32),(13,31),(14,30),(15,29),(16,28),(17,54),(18,53),(19,52),(20,51),(21,50),(22,49),(23,56),(24,55),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(40,48)])
Matrix representation ►G ⊆ GL3(𝔽17) generated by
| 4 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 13 |
| 1 | 0 | 0 |
| 0 | 3 | 14 |
| 0 | 3 | 3 |
| 1 | 0 | 0 |
| 0 | 14 | 3 |
| 0 | 3 | 3 |
G:=sub<GL(3,GF(17))| [4,0,0,0,1,0,0,0,1],[1,0,0,0,13,0,0,0,13],[1,0,0,0,3,3,0,14,3],[1,0,0,0,14,3,0,3,3] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4L | 4M | ··· | 4R | 4S | ··· | 4AD | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | C4○D8 |
| kernel | C4×C4○D8 | C2×C4×C8 | C23.24D4 | C23.25D4 | C4×D8 | C4×SD16 | C4×Q16 | C4×C4○D4 | C2×C4○D8 | C4○D8 | C42 | C22×C4 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 1 | 16 | 2 | 2 | 4 | 16 |
In GAP, Magma, Sage, TeX
C_4\times C_4\circ D_8
% in TeX
G:=Group("C4xC4oD8"); // GroupNames label
G:=SmallGroup(128,1671);
// by ID
G=gap.SmallGroup(128,1671);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,248,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations