p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.270D4, C42.731C23, C4.1042+ (1+4), C4⋊D8⋊8C2, D4⋊D4⋊4C2, C4⋊2Q16⋊9C2, C4⋊SD16⋊37C2, D4.7D4⋊4C2, C8.12D4⋊4C2, D4.D4⋊38C2, C4.114(C4○D8), C4⋊C4.151C23, C4⋊C8.287C22, (C2×C8).328C23, (C4×C8).113C22, (C2×C4).410C24, (C2×D8).25C22, (C22×C4).499D4, C23.286(C2×D4), C4⋊Q8.303C22, (C4×D4).105C22, (C2×D4).159C23, (C4×Q8).102C22, (C2×Q8).147C23, (C2×Q16).27C22, C42.12C4⋊36C2, C4⋊1D4.164C22, C4⋊D4.190C22, C22⋊C8.194C22, (C2×C42).877C22, (C2×SD16).86C22, C22.670(C22×D4), C22⋊Q8.195C22, D4⋊C4.108C22, C2.55(D8⋊C22), (C22×C4).1081C23, C42.78C22⋊8C2, C22.26C24⋊18C2, Q8⋊C4.101C22, C4.4D4.151C22, C42.C2.126C22, C23.36C23⋊10C2, C2.81(C22.29C24), C2.44(C2×C4○D8), (C2×C4).707(C2×D4), (C2×C4○D4).173C22, SmallGroup(128,1944)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 420 in 200 conjugacy classes, 86 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×12], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×16], Q8 [×6], C23, C23 [×3], C42 [×4], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C2×Q8 [×2], C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2×C42, C42⋊C2, C4×D4, C4×D4 [×3], C4×Q8, C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4, C4.4D4, C4.4D4 [×2], C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C2×D8 [×2], C2×SD16 [×4], C2×Q16 [×2], C2×C4○D4 [×2], C42.12C4, D4⋊D4 [×2], D4.7D4 [×2], C4⋊D8, C4⋊SD16, D4.D4, C4⋊2Q16, C42.78C22 [×2], C8.12D4 [×2], C23.36C23, C22.26C24, C42.270D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C22×D4, 2+ (1+4) [×2], C22.29C24, C2×C4○D8, D8⋊C22, C42.270D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1b2, cbc-1=dbd=a2b-1, dcd=b2c3 >
►On 64 points
(1 30 63 11)(2 31 64 12)(3 32 57 13)(4 25 58 14)(5 26 59 15)(6 27 60 16)(7 28 61 9)(8 29 62 10)(17 38 55 44)(18 39 56 45)(19 40 49 46)(20 33 50 47)(21 34 51 48)(22 35 52 41)(23 36 53 42)(24 37 54 43)
(1 24 5 20)(2 51 6 55)(3 18 7 22)(4 53 8 49)(9 41 13 45)(10 40 14 36)(11 43 15 47)(12 34 16 38)(17 64 21 60)(19 58 23 62)(25 42 29 46)(26 33 30 37)(27 44 31 48)(28 35 32 39)(50 63 54 59)(52 57 56 61)
(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 36)(2 35)(3 34)(4 33)(5 40)(6 39)(7 38)(8 37)(9 51)(10 50)(11 49)(12 56)(13 55)(14 54)(15 53)(16 52)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)
G:=sub<Sym(64)| (1,30,63,11)(2,31,64,12)(3,32,57,13)(4,25,58,14)(5,26,59,15)(6,27,60,16)(7,28,61,9)(8,29,62,10)(17,38,55,44)(18,39,56,45)(19,40,49,46)(20,33,50,47)(21,34,51,48)(22,35,52,41)(23,36,53,42)(24,37,54,43), (1,24,5,20)(2,51,6,55)(3,18,7,22)(4,53,8,49)(9,41,13,45)(10,40,14,36)(11,43,15,47)(12,34,16,38)(17,64,21,60)(19,58,23,62)(25,42,29,46)(26,33,30,37)(27,44,31,48)(28,35,32,39)(50,63,54,59)(52,57,56,61), (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), (1,36)(2,35)(3,34)(4,33)(5,40)(6,39)(7,38)(8,37)(9,51)(10,50)(11,49)(12,56)(13,55)(14,54)(15,53)(16,52)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)>;
G:=Group( (1,30,63,11)(2,31,64,12)(3,32,57,13)(4,25,58,14)(5,26,59,15)(6,27,60,16)(7,28,61,9)(8,29,62,10)(17,38,55,44)(18,39,56,45)(19,40,49,46)(20,33,50,47)(21,34,51,48)(22,35,52,41)(23,36,53,42)(24,37,54,43), (1,24,5,20)(2,51,6,55)(3,18,7,22)(4,53,8,49)(9,41,13,45)(10,40,14,36)(11,43,15,47)(12,34,16,38)(17,64,21,60)(19,58,23,62)(25,42,29,46)(26,33,30,37)(27,44,31,48)(28,35,32,39)(50,63,54,59)(52,57,56,61), (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), (1,36)(2,35)(3,34)(4,33)(5,40)(6,39)(7,38)(8,37)(9,51)(10,50)(11,49)(12,56)(13,55)(14,54)(15,53)(16,52)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57) );
G=PermutationGroup([(1,30,63,11),(2,31,64,12),(3,32,57,13),(4,25,58,14),(5,26,59,15),(6,27,60,16),(7,28,61,9),(8,29,62,10),(17,38,55,44),(18,39,56,45),(19,40,49,46),(20,33,50,47),(21,34,51,48),(22,35,52,41),(23,36,53,42),(24,37,54,43)], [(1,24,5,20),(2,51,6,55),(3,18,7,22),(4,53,8,49),(9,41,13,45),(10,40,14,36),(11,43,15,47),(12,34,16,38),(17,64,21,60),(19,58,23,62),(25,42,29,46),(26,33,30,37),(27,44,31,48),(28,35,32,39),(50,63,54,59),(52,57,56,61)], [(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)], [(1,36),(2,35),(3,34),(4,33),(5,40),(6,39),(7,38),(8,37),(9,51),(10,50),(11,49),(12,56),(13,55),(14,54),(15,53),(16,52),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25),(41,64),(42,63),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 15 | 0 |
| 0 | 0 | 0 | 4 | 0 | 15 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 11 | 11 |
| 0 | 0 | 9 | 8 | 6 | 11 |
| 0 | 0 | 15 | 15 | 9 | 9 |
| 0 | 0 | 2 | 15 | 8 | 9 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 9 | 2 | 15 |
| 0 | 0 | 9 | 9 | 15 | 15 |
| 0 | 0 | 15 | 2 | 9 | 8 |
| 0 | 0 | 2 | 2 | 8 | 8 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,15,0,13,0,0,0,0,15,0,13],[3,14,0,0,0,0,3,3,0,0,0,0,0,0,8,9,15,2,0,0,8,8,15,15,0,0,11,6,9,8,0,0,11,11,9,9],[14,14,0,0,0,0,14,3,0,0,0,0,0,0,8,9,15,2,0,0,9,9,2,2,0,0,2,15,9,8,0,0,15,15,8,8] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4J | 4K | 4L | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 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 | D4 | D4 | C4○D8 | 2+ (1+4) | D8⋊C22 |
| kernel | C42.270D4 | C42.12C4 | D4⋊D4 | D4.7D4 | C4⋊D8 | C4⋊SD16 | D4.D4 | C4⋊2Q16 | C42.78C22 | C8.12D4 | C23.36C23 | C22.26C24 | C42 | C22×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{270}D_4 % in TeX
G:=Group("C4^2.270D4"); // GroupNames label
G:=SmallGroup(128,1944);
// by ID
G=gap.SmallGroup(128,1944);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,675,248,4037,1027,124]);
// 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,d*a*d=a^-1*b^2,c*b*c^-1=d*b*d=a^2*b^-1,d*c*d=b^2*c^3>;
// generators/relations