p-group, metabelian, nilpotent (class 2), monomial
Aliases: C43⋊12C2, C42⋊40D4, C24.123C23, C23.758C24, C4⋊1(C4.4D4), C4.9(C4⋊1D4), C22.468(C22×D4), (C22×C4).1265C23, (C2×C42).1093C22, (C22×D4).314C22, (C22×Q8).250C22, C24.3C22⋊102C2, C2.58(C22.26C24), (C2×C4⋊Q8)⋊27C2, (C2×C4).835(C2×D4), C2.17(C2×C4⋊1D4), (C2×C4⋊1D4).20C2, (C2×C4.4D4)⋊35C2, C2.34(C2×C4.4D4), (C2×C4).674(C4○D4), (C2×C4⋊C4).561C22, C22.599(C2×C4○D4), (C2×C22⋊C4).368C22, SmallGroup(128,1590)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C43⋊12C2
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=ac2, bc=cb, dbd=b-1, dcd=a2c >
Subgroups: 772 in 376 conjugacy classes, 132 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.4D4, C4⋊1D4, C4⋊Q8, C22×D4, C22×Q8, C43, C24.3C22, C2×C4.4D4, C2×C4⋊1D4, C2×C4⋊Q8, C43⋊12C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C4⋊1D4, C22×D4, C2×C4○D4, C2×C4.4D4, C2×C4⋊1D4, C22.26C24, C43⋊12C2
►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 59 51 47)(2 60 52 48)(3 57 49 45)(4 58 50 46)(5 14 36 26)(6 15 33 27)(7 16 34 28)(8 13 35 25)(9 29 21 17)(10 30 22 18)(11 31 23 19)(12 32 24 20)(37 41 61 53)(38 42 62 54)(39 43 63 55)(40 44 64 56)
(1 15 11 43)(2 16 12 44)(3 13 9 41)(4 14 10 42)(5 18 38 46)(6 19 39 47)(7 20 40 48)(8 17 37 45)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 35)(30 62 58 36)(31 63 59 33)(32 64 60 34)
(1 42)(2 15)(3 44)(4 13)(5 31)(6 60)(7 29)(8 58)(9 16)(10 41)(11 14)(12 43)(17 34)(18 61)(19 36)(20 63)(21 28)(22 53)(23 26)(24 55)(25 50)(27 52)(30 37)(32 39)(33 48)(35 46)(38 59)(40 57)(45 64)(47 62)(49 56)(51 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), (1,59,51,47)(2,60,52,48)(3,57,49,45)(4,58,50,46)(5,14,36,26)(6,15,33,27)(7,16,34,28)(8,13,35,25)(9,29,21,17)(10,30,22,18)(11,31,23,19)(12,32,24,20)(37,41,61,53)(38,42,62,54)(39,43,63,55)(40,44,64,56), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,18,38,46)(6,19,39,47)(7,20,40,48)(8,17,37,45)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,35)(30,62,58,36)(31,63,59,33)(32,64,60,34), (1,42)(2,15)(3,44)(4,13)(5,31)(6,60)(7,29)(8,58)(9,16)(10,41)(11,14)(12,43)(17,34)(18,61)(19,36)(20,63)(21,28)(22,53)(23,26)(24,55)(25,50)(27,52)(30,37)(32,39)(33,48)(35,46)(38,59)(40,57)(45,64)(47,62)(49,56)(51,54)>;
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), (1,59,51,47)(2,60,52,48)(3,57,49,45)(4,58,50,46)(5,14,36,26)(6,15,33,27)(7,16,34,28)(8,13,35,25)(9,29,21,17)(10,30,22,18)(11,31,23,19)(12,32,24,20)(37,41,61,53)(38,42,62,54)(39,43,63,55)(40,44,64,56), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,18,38,46)(6,19,39,47)(7,20,40,48)(8,17,37,45)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,35)(30,62,58,36)(31,63,59,33)(32,64,60,34), (1,42)(2,15)(3,44)(4,13)(5,31)(6,60)(7,29)(8,58)(9,16)(10,41)(11,14)(12,43)(17,34)(18,61)(19,36)(20,63)(21,28)(22,53)(23,26)(24,55)(25,50)(27,52)(30,37)(32,39)(33,48)(35,46)(38,59)(40,57)(45,64)(47,62)(49,56)(51,54) );
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)], [(1,59,51,47),(2,60,52,48),(3,57,49,45),(4,58,50,46),(5,14,36,26),(6,15,33,27),(7,16,34,28),(8,13,35,25),(9,29,21,17),(10,30,22,18),(11,31,23,19),(12,32,24,20),(37,41,61,53),(38,42,62,54),(39,43,63,55),(40,44,64,56)], [(1,15,11,43),(2,16,12,44),(3,13,9,41),(4,14,10,42),(5,18,38,46),(6,19,39,47),(7,20,40,48),(8,17,37,45),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,61,57,35),(30,62,58,36),(31,63,59,33),(32,64,60,34)], [(1,42),(2,15),(3,44),(4,13),(5,31),(6,60),(7,29),(8,58),(9,16),(10,41),(11,14),(12,43),(17,34),(18,61),(19,36),(20,63),(21,28),(22,53),(23,26),(24,55),(25,50),(27,52),(30,37),(32,39),(33,48),(35,46),(38,59),(40,57),(45,64),(47,62),(49,56),(51,54)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4AB | 4AC | 4AD | 4AE | 4AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 |
| kernel | C43⋊12C2 | C43 | C24.3C22 | C2×C4.4D4 | C2×C4⋊1D4 | C2×C4⋊Q8 | C42 | C2×C4 |
| # reps | 1 | 1 | 8 | 4 | 1 | 1 | 12 | 16 |
Matrix representation of C43⋊12C2 ►in GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 3 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,2,0,0,0,0,4,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,4,0,0,0,0,3,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,4,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1] >;
C43⋊12C2 in GAP, Magma, Sage, TeX
C_4^3\rtimes_{12}C_2 % in TeX
G:=Group("C4^3:12C2"); // GroupNames label
G:=SmallGroup(128,1590);
// by ID
G=gap.SmallGroup(128,1590);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,456,758,184,2019,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*c^2,b*c=c*b,d*b*d=b^-1,d*c*d=a^2*c>;
// generators/relations