p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊43D4, C23.752C24, C24.120C23, (C22×C42)⋊7C2, C42⋊5C4⋊39C2, C22⋊1(C42⋊2C2), C23.252(C4○D4), (C23×C4).652C22, C23.8Q8⋊147C2, C22.462(C22×D4), (C2×C42).1013C22, (C22×C4).1263C23, C23.23D4.79C2, (C22×D4).310C22, C24.C22⋊183C2, C2.95(C22.19C24), C23.63C23⋊203C2, C2.C42.449C22, C2.110(C23.36C23), (C2×C4).1206(C2×D4), (C2×C42⋊2C2)⋊29C2, (C2×C4).528(C4○D4), (C2×C4⋊C4).555C22, C2.25(C2×C42⋊2C2), C22.593(C2×C4○D4), (C2×C22⋊C4).362C22, SmallGroup(128,1584)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 516 in 291 conjugacy classes, 108 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×5], C4 [×19], C22, C22 [×10], C22 [×19], C2×C4 [×12], C2×C4 [×57], D4 [×4], C23, C23 [×6], C23 [×11], C42 [×4], C42 [×6], C22⋊C4 [×18], C4⋊C4 [×12], C22×C4, C22×C4 [×12], C22×C4 [×18], C2×D4 [×6], C24 [×2], C2.C42 [×12], C2×C42, C2×C42 [×3], C2×C42 [×4], C2×C22⋊C4 [×9], C2×C4⋊C4 [×6], C42⋊2C2 [×4], C23×C4 [×3], C22×D4, C42⋊5C4, C23.8Q8 [×3], C23.23D4 [×3], C23.63C23 [×3], C24.C22 [×3], C22×C42, C2×C42⋊2C2, C42⋊43D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×12], C24, C42⋊2C2 [×4], C22×D4, C2×C4○D4 [×6], C22.19C24 [×3], C2×C42⋊2C2, C23.36C23 [×3], C42⋊43D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=dbd=a2b-1, dcd=c-1 >
►On 64 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)(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 46 55 19)(2 47 56 20)(3 48 53 17)(4 45 54 18)(5 43 38 16)(6 44 39 13)(7 41 40 14)(8 42 37 15)(9 61 24 35)(10 62 21 36)(11 63 22 33)(12 64 23 34)(25 31 49 58)(26 32 50 59)(27 29 51 60)(28 30 52 57)
(1 30 11 13)(2 58 12 41)(3 32 9 15)(4 60 10 43)(5 47 51 64)(6 17 52 35)(7 45 49 62)(8 19 50 33)(14 56 31 23)(16 54 29 21)(18 25 36 40)(20 27 34 38)(22 44 55 57)(24 42 53 59)(26 63 37 46)(28 61 39 48)
(2 56)(4 54)(5 49)(6 26)(7 51)(8 28)(10 21)(12 23)(13 30)(14 58)(15 32)(16 60)(17 46)(18 20)(19 48)(25 38)(27 40)(29 43)(31 41)(33 61)(34 36)(35 63)(37 52)(39 50)(42 59)(44 57)(45 47)(62 64)
G:=sub<Sym(64)| (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), (1,46,55,19)(2,47,56,20)(3,48,53,17)(4,45,54,18)(5,43,38,16)(6,44,39,13)(7,41,40,14)(8,42,37,15)(9,61,24,35)(10,62,21,36)(11,63,22,33)(12,64,23,34)(25,31,49,58)(26,32,50,59)(27,29,51,60)(28,30,52,57), (1,30,11,13)(2,58,12,41)(3,32,9,15)(4,60,10,43)(5,47,51,64)(6,17,52,35)(7,45,49,62)(8,19,50,33)(14,56,31,23)(16,54,29,21)(18,25,36,40)(20,27,34,38)(22,44,55,57)(24,42,53,59)(26,63,37,46)(28,61,39,48), (2,56)(4,54)(5,49)(6,26)(7,51)(8,28)(10,21)(12,23)(13,30)(14,58)(15,32)(16,60)(17,46)(18,20)(19,48)(25,38)(27,40)(29,43)(31,41)(33,61)(34,36)(35,63)(37,52)(39,50)(42,59)(44,57)(45,47)(62,64)>;
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)(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), (1,46,55,19)(2,47,56,20)(3,48,53,17)(4,45,54,18)(5,43,38,16)(6,44,39,13)(7,41,40,14)(8,42,37,15)(9,61,24,35)(10,62,21,36)(11,63,22,33)(12,64,23,34)(25,31,49,58)(26,32,50,59)(27,29,51,60)(28,30,52,57), (1,30,11,13)(2,58,12,41)(3,32,9,15)(4,60,10,43)(5,47,51,64)(6,17,52,35)(7,45,49,62)(8,19,50,33)(14,56,31,23)(16,54,29,21)(18,25,36,40)(20,27,34,38)(22,44,55,57)(24,42,53,59)(26,63,37,46)(28,61,39,48), (2,56)(4,54)(5,49)(6,26)(7,51)(8,28)(10,21)(12,23)(13,30)(14,58)(15,32)(16,60)(17,46)(18,20)(19,48)(25,38)(27,40)(29,43)(31,41)(33,61)(34,36)(35,63)(37,52)(39,50)(42,59)(44,57)(45,47)(62,64) );
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),(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)], [(1,46,55,19),(2,47,56,20),(3,48,53,17),(4,45,54,18),(5,43,38,16),(6,44,39,13),(7,41,40,14),(8,42,37,15),(9,61,24,35),(10,62,21,36),(11,63,22,33),(12,64,23,34),(25,31,49,58),(26,32,50,59),(27,29,51,60),(28,30,52,57)], [(1,30,11,13),(2,58,12,41),(3,32,9,15),(4,60,10,43),(5,47,51,64),(6,17,52,35),(7,45,49,62),(8,19,50,33),(14,56,31,23),(16,54,29,21),(18,25,36,40),(20,27,34,38),(22,44,55,57),(24,42,53,59),(26,63,37,46),(28,61,39,48)], [(2,56),(4,54),(5,49),(6,26),(7,51),(8,28),(10,21),(12,23),(13,30),(14,58),(15,32),(16,60),(17,46),(18,20),(19,48),(25,38),(27,40),(29,43),(31,41),(33,61),(34,36),(35,63),(37,52),(39,50),(42,59),(44,57),(45,47),(62,64)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 4A | ··· | 4X | 4Y | ··· | 4AE |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | C4○D4 |
| kernel | C42⋊43D4 | C42⋊5C4 | C23.8Q8 | C23.23D4 | C23.63C23 | C24.C22 | C22×C42 | C2×C42⋊2C2 | C42 | C2×C4 | C23 |
| # reps | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 4 | 12 | 12 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{43}D_4 % in TeX
G:=Group("C4^2:43D4"); // GroupNames label
G:=SmallGroup(128,1584);
// by ID
G=gap.SmallGroup(128,1584);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,100,2019]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^2,c*b*c^-1=d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations