p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊30D4, C24.373C23, C23.536C24, C22.3122+ 1+4, C22.2292- 1+4, C42⋊8C4⋊53C2, C23.68(C4○D4), C23⋊2D4.18C2, C23.4Q8⋊32C2, C23.Q8⋊40C2, C23.8Q8⋊87C2, C23.10D4⋊62C2, C23.23D4⋊72C2, (C2×C42).613C22, (C23×C4).139C22, (C22×C4).146C23, C22.361(C22×D4), (C22×D4).544C22, C23.83C23⋊62C2, C2.87(C22.19C24), C2.44(C22.29C24), C2.42(C22.32C24), C2.C42.261C22, C2.41(C22.33C24), C2.29(C22.31C24), (C2×C4×D4)⋊55C2, (C2×C4).395(C2×D4), (C2×C42⋊2C2)⋊15C2, (C2×C4⋊C4).893C22, C22.408(C2×C4○D4), (C2×C22⋊C4).224C22, SmallGroup(128,1368)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊30D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=ab2, dad=a-1, cbc-1=a2b, bd=db, dcd=c-1 >
Subgroups: 596 in 281 conjugacy classes, 96 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C42⋊2C2, C23×C4, C22×D4, C22×D4, C42⋊8C4, C23.8Q8, C23.23D4, C23⋊2D4, C23.10D4, C23.10D4, C23.Q8, C23.4Q8, C23.83C23, C2×C4×D4, C2×C42⋊2C2, C42⋊30D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.19C24, C22.29C24, C22.31C24, C22.32C24, C22.33C24, C42⋊30D4
►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 15 28 47)(2 16 25 48)(3 13 26 45)(4 14 27 46)(5 50 38 18)(6 51 39 19)(7 52 40 20)(8 49 37 17)(9 59 41 30)(10 60 42 31)(11 57 43 32)(12 58 44 29)(21 33 53 63)(22 34 54 64)(23 35 55 61)(24 36 56 62)
(1 51 43 61)(2 20 44 36)(3 49 41 63)(4 18 42 34)(5 29 54 16)(6 59 55 45)(7 31 56 14)(8 57 53 47)(9 33 26 17)(10 64 27 50)(11 35 28 19)(12 62 25 52)(13 39 30 23)(15 37 32 21)(22 48 38 58)(24 46 40 60)
(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 21)(14 24)(15 23)(16 22)(25 34)(26 33)(27 36)(28 35)(29 38)(30 37)(31 40)(32 39)(41 49)(42 52)(43 51)(44 50)(45 53)(46 56)(47 55)(48 54)
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), (1,15,28,47)(2,16,25,48)(3,13,26,45)(4,14,27,46)(5,50,38,18)(6,51,39,19)(7,52,40,20)(8,49,37,17)(9,59,41,30)(10,60,42,31)(11,57,43,32)(12,58,44,29)(21,33,53,63)(22,34,54,64)(23,35,55,61)(24,36,56,62), (1,51,43,61)(2,20,44,36)(3,49,41,63)(4,18,42,34)(5,29,54,16)(6,59,55,45)(7,31,56,14)(8,57,53,47)(9,33,26,17)(10,64,27,50)(11,35,28,19)(12,62,25,52)(13,39,30,23)(15,37,32,21)(22,48,38,58)(24,46,40,60), (1,61)(2,64)(3,63)(4,62)(5,58)(6,57)(7,60)(8,59)(9,17)(10,20)(11,19)(12,18)(13,21)(14,24)(15,23)(16,22)(25,34)(26,33)(27,36)(28,35)(29,38)(30,37)(31,40)(32,39)(41,49)(42,52)(43,51)(44,50)(45,53)(46,56)(47,55)(48,54)>;
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), (1,15,28,47)(2,16,25,48)(3,13,26,45)(4,14,27,46)(5,50,38,18)(6,51,39,19)(7,52,40,20)(8,49,37,17)(9,59,41,30)(10,60,42,31)(11,57,43,32)(12,58,44,29)(21,33,53,63)(22,34,54,64)(23,35,55,61)(24,36,56,62), (1,51,43,61)(2,20,44,36)(3,49,41,63)(4,18,42,34)(5,29,54,16)(6,59,55,45)(7,31,56,14)(8,57,53,47)(9,33,26,17)(10,64,27,50)(11,35,28,19)(12,62,25,52)(13,39,30,23)(15,37,32,21)(22,48,38,58)(24,46,40,60), (1,61)(2,64)(3,63)(4,62)(5,58)(6,57)(7,60)(8,59)(9,17)(10,20)(11,19)(12,18)(13,21)(14,24)(15,23)(16,22)(25,34)(26,33)(27,36)(28,35)(29,38)(30,37)(31,40)(32,39)(41,49)(42,52)(43,51)(44,50)(45,53)(46,56)(47,55)(48,54) );
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)], [(1,15,28,47),(2,16,25,48),(3,13,26,45),(4,14,27,46),(5,50,38,18),(6,51,39,19),(7,52,40,20),(8,49,37,17),(9,59,41,30),(10,60,42,31),(11,57,43,32),(12,58,44,29),(21,33,53,63),(22,34,54,64),(23,35,55,61),(24,36,56,62)], [(1,51,43,61),(2,20,44,36),(3,49,41,63),(4,18,42,34),(5,29,54,16),(6,59,55,45),(7,31,56,14),(8,57,53,47),(9,33,26,17),(10,64,27,50),(11,35,28,19),(12,62,25,52),(13,39,30,23),(15,37,32,21),(22,48,38,58),(24,46,40,60)], [(1,61),(2,64),(3,63),(4,62),(5,58),(6,57),(7,60),(8,59),(9,17),(10,20),(11,19),(12,18),(13,21),(14,24),(15,23),(16,22),(25,34),(26,33),(27,36),(28,35),(29,38),(30,37),(31,40),(32,39),(41,49),(42,52),(43,51),(44,50),(45,53),(46,56),(47,55),(48,54)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4S |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42⋊30D4 | C42⋊8C4 | C23.8Q8 | C23.23D4 | C23⋊2D4 | C23.10D4 | C23.Q8 | C23.4Q8 | C23.83C23 | C2×C4×D4 | C2×C42⋊2C2 | C42 | C23 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 1 | 3 | 2 | 1 | 1 | 1 | 1 | 4 | 8 | 3 | 1 |
Matrix representation of C42⋊30D4 ►in GL8(𝔽5)
| 4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 | 1 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 4 |
| 0 | 0 | 0 | 0 | 0 | 1 | 4 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 4 | 4 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 4 | 4 | 0 |
G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,4,4,0,0,0,0,0,0,0,1,0,0,0,0,2,1,1,1,0,0,0,0,0,4,0,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[4,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,0,0,3,4,4,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,3,4,4,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0] >;
C42⋊30D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{30}D_4 % in TeX
G:=Group("C4^2:30D4"); // GroupNames label
G:=SmallGroup(128,1368);
// by ID
G=gap.SmallGroup(128,1368);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,185,192]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations