p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊47D4, C24.122C23, C23.756C24, C4⋊3(C4⋊D4), (C22×C4)⋊48D4, C42⋊9C4⋊40C2, C22⋊1(C4⋊1D4), (C22×C42)⋊17C2, C23.374(C2×D4), (C23×C4).683C22, C22.466(C22×D4), (C22×C4).1486C23, (C2×C42).1091C22, (C22×D4).313C22, C24.3C22⋊101C2, C2.56(C22.26C24), (C2×C4⋊D4)⋊42C2, (C2×C4⋊1D4)⋊12C2, (C2×C4).687(C2×D4), C2.49(C2×C4⋊D4), C2.16(C2×C4⋊1D4), (C2×C4).672(C4○D4), (C2×C4⋊C4).559C22, C22.597(C2×C4○D4), (C2×C22⋊C4).366C22, SmallGroup(128,1588)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 948 in 462 conjugacy classes, 144 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×12], C4 [×10], C22, C22 [×10], C22 [×40], C2×C4 [×24], C2×C4 [×42], D4 [×40], C23, C23 [×6], C23 [×32], C42 [×4], C42 [×6], C22⋊C4 [×24], C4⋊C4 [×12], C22×C4 [×22], C22×C4 [×12], C2×D4 [×60], C24, C24 [×4], C2×C42, C2×C42 [×3], C2×C42 [×4], C2×C22⋊C4 [×12], C2×C4⋊C4 [×6], C4⋊D4 [×24], C4⋊1D4 [×4], C23×C4 [×3], C22×D4 [×10], C42⋊9C4, C24.3C22 [×6], C22×C42, C2×C4⋊D4 [×6], C2×C4⋊1D4, C42⋊47D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×16], C23 [×15], C2×D4 [×24], C4○D4 [×6], C24, C4⋊D4 [×12], C4⋊1D4 [×4], C22×D4 [×4], C2×C4○D4 [×3], C2×C4⋊D4 [×3], C2×C4⋊1D4, C22.26C24 [×3], C42⋊47D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-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 49 41 61)(2 50 42 62)(3 51 43 63)(4 52 44 64)(5 60 20 12)(6 57 17 9)(7 58 18 10)(8 59 19 11)(13 34 25 21)(14 35 26 22)(15 36 27 23)(16 33 28 24)(29 53 45 38)(30 54 46 39)(31 55 47 40)(32 56 48 37)
(1 34 11 55)(2 33 12 54)(3 36 9 53)(4 35 10 56)(5 30 50 16)(6 29 51 15)(7 32 52 14)(8 31 49 13)(17 45 63 27)(18 48 64 26)(19 47 61 25)(20 46 62 28)(21 59 40 41)(22 58 37 44)(23 57 38 43)(24 60 39 42)
(1 61)(2 64)(3 63)(4 62)(5 58)(6 57)(7 60)(8 59)(9 17)(10 20)(11 19)(12 18)(13 40)(14 39)(15 38)(16 37)(21 31)(22 30)(23 29)(24 32)(25 55)(26 54)(27 53)(28 56)(33 48)(34 47)(35 46)(36 45)(41 49)(42 52)(43 51)(44 50)
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), (1,49,41,61)(2,50,42,62)(3,51,43,63)(4,52,44,64)(5,60,20,12)(6,57,17,9)(7,58,18,10)(8,59,19,11)(13,34,25,21)(14,35,26,22)(15,36,27,23)(16,33,28,24)(29,53,45,38)(30,54,46,39)(31,55,47,40)(32,56,48,37), (1,34,11,55)(2,33,12,54)(3,36,9,53)(4,35,10,56)(5,30,50,16)(6,29,51,15)(7,32,52,14)(8,31,49,13)(17,45,63,27)(18,48,64,26)(19,47,61,25)(20,46,62,28)(21,59,40,41)(22,58,37,44)(23,57,38,43)(24,60,39,42), (1,61)(2,64)(3,63)(4,62)(5,58)(6,57)(7,60)(8,59)(9,17)(10,20)(11,19)(12,18)(13,40)(14,39)(15,38)(16,37)(21,31)(22,30)(23,29)(24,32)(25,55)(26,54)(27,53)(28,56)(33,48)(34,47)(35,46)(36,45)(41,49)(42,52)(43,51)(44,50)>;
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), (1,49,41,61)(2,50,42,62)(3,51,43,63)(4,52,44,64)(5,60,20,12)(6,57,17,9)(7,58,18,10)(8,59,19,11)(13,34,25,21)(14,35,26,22)(15,36,27,23)(16,33,28,24)(29,53,45,38)(30,54,46,39)(31,55,47,40)(32,56,48,37), (1,34,11,55)(2,33,12,54)(3,36,9,53)(4,35,10,56)(5,30,50,16)(6,29,51,15)(7,32,52,14)(8,31,49,13)(17,45,63,27)(18,48,64,26)(19,47,61,25)(20,46,62,28)(21,59,40,41)(22,58,37,44)(23,57,38,43)(24,60,39,42), (1,61)(2,64)(3,63)(4,62)(5,58)(6,57)(7,60)(8,59)(9,17)(10,20)(11,19)(12,18)(13,40)(14,39)(15,38)(16,37)(21,31)(22,30)(23,29)(24,32)(25,55)(26,54)(27,53)(28,56)(33,48)(34,47)(35,46)(36,45)(41,49)(42,52)(43,51)(44,50) );
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)], [(1,49,41,61),(2,50,42,62),(3,51,43,63),(4,52,44,64),(5,60,20,12),(6,57,17,9),(7,58,18,10),(8,59,19,11),(13,34,25,21),(14,35,26,22),(15,36,27,23),(16,33,28,24),(29,53,45,38),(30,54,46,39),(31,55,47,40),(32,56,48,37)], [(1,34,11,55),(2,33,12,54),(3,36,9,53),(4,35,10,56),(5,30,50,16),(6,29,51,15),(7,32,52,14),(8,31,49,13),(17,45,63,27),(18,48,64,26),(19,47,61,25),(20,46,62,28),(21,59,40,41),(22,58,37,44),(23,57,38,43),(24,60,39,42)], [(1,61),(2,64),(3,63),(4,62),(5,58),(6,57),(7,60),(8,59),(9,17),(10,20),(11,19),(12,18),(13,40),(14,39),(15,38),(16,37),(21,31),(22,30),(23,29),(24,32),(25,55),(26,54),(27,53),(28,56),(33,48),(34,47),(35,46),(36,45),(41,49),(42,52),(43,51),(44,50)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 3 | 3 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,4,3],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,1,2],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,3,0,0,0,0,0,3],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,2,3,0,0,0,0,4,3] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | ··· | 4X | 4Y | 4Z | 4AA | 4AB |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 |
| kernel | C42⋊47D4 | C42⋊9C4 | C24.3C22 | C22×C42 | C2×C4⋊D4 | C2×C4⋊1D4 | C42 | C22×C4 | C2×C4 |
| # reps | 1 | 1 | 6 | 1 | 6 | 1 | 4 | 12 | 12 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{47}D_4 % in TeX
G:=Group("C4^2:47D4"); // GroupNames label
G:=SmallGroup(128,1588);
// by ID
G=gap.SmallGroup(128,1588);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,184,2019]);
// 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^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations