metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4:C4:36D6, (C2xD12):11C4, C4.62(C2xD12), (C2xC4).46D12, C42:C2:1S3, C4.23(D6:C4), D12.25(C2xC4), C6.D8:27C2, C12.142(C2xD4), (C2xC12).472D4, (C22xC6).74D4, C2.2(D4:D6), C12.64(C22xC4), (C22xC4).126D6, C6.105(C8:C22), C12.47(C22:C4), (C2xC12).328C23, C22.23(D6:C4), (C22xD12).12C2, C23.61(C3:D4), C3:2(C23.37D4), (C2xD12).234C22, (C22xC12).150C22, C4.51(S3xC2xC4), (C2xC3:C8):4C22, (C2xC4).44(C4xS3), C2.17(C2xD6:C4), (C3xC4:C4):41C22, (C2xC12).91(C2xC4), (C2xC6).457(C2xD4), C6.44(C2xC22:C4), (C2xC4.Dic3):9C2, (C3xC42:C2):1C2, C22.72(C2xC3:D4), (C2xC4).241(C3:D4), (C2xC6).14(C22:C4), (C2xC4).428(C22xS3), SmallGroup(192,560)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4:C4:36D6
G = < a,b,c,d | a4=b4=c6=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=a2b, dbd=ab-1, dcd=c-1 >
Subgroups: 680 in 190 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2xC4, C2xC4, C2xC4, D4, C23, C23, C12, C12, C12, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, M4(2), C22xC4, C2xD4, C24, C3:C8, D12, D12, C2xC12, C2xC12, C2xC12, C22xS3, C22xC6, D4:C4, C42:C2, C2xM4(2), C22xD4, C2xC3:C8, C4.Dic3, C4xC12, C3xC22:C4, C3xC4:C4, C2xD12, C2xD12, C22xC12, S3xC23, C23.37D4, C6.D8, C2xC4.Dic3, C3xC42:C2, C22xD12, C4:C4:36D6
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, C22:C4, C22xC4, C2xD4, C4xS3, D12, C3:D4, C22xS3, C2xC22:C4, C8:C22, D6:C4, S3xC2xC4, C2xD12, C2xC3:D4, C23.37D4, C2xD6:C4, D4:D6, C4:C4:36D6
►On 48 points
(1 9 19 29)(2 10 20 30)(3 11 21 25)(4 12 22 26)(5 7 23 27)(6 8 24 28)(13 41 33 44)(14 42 34 45)(15 37 35 46)(16 38 36 47)(17 39 31 48)(18 40 32 43)
(1 43 22 46)(2 41 23 38)(3 45 24 48)(4 37 19 40)(5 47 20 44)(6 39 21 42)(7 36 30 33)(8 17 25 14)(9 32 26 35)(10 13 27 16)(11 34 28 31)(12 15 29 18)
(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)
(1 3)(4 6)(7 27)(8 26)(9 25)(10 30)(11 29)(12 28)(13 38)(14 37)(15 42)(16 41)(17 40)(18 39)(19 21)(22 24)(31 43)(32 48)(33 47)(34 46)(35 45)(36 44)
G:=sub<Sym(48)| (1,9,19,29)(2,10,20,30)(3,11,21,25)(4,12,22,26)(5,7,23,27)(6,8,24,28)(13,41,33,44)(14,42,34,45)(15,37,35,46)(16,38,36,47)(17,39,31,48)(18,40,32,43), (1,43,22,46)(2,41,23,38)(3,45,24,48)(4,37,19,40)(5,47,20,44)(6,39,21,42)(7,36,30,33)(8,17,25,14)(9,32,26,35)(10,13,27,16)(11,34,28,31)(12,15,29,18), (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), (1,3)(4,6)(7,27)(8,26)(9,25)(10,30)(11,29)(12,28)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,21)(22,24)(31,43)(32,48)(33,47)(34,46)(35,45)(36,44)>;
G:=Group( (1,9,19,29)(2,10,20,30)(3,11,21,25)(4,12,22,26)(5,7,23,27)(6,8,24,28)(13,41,33,44)(14,42,34,45)(15,37,35,46)(16,38,36,47)(17,39,31,48)(18,40,32,43), (1,43,22,46)(2,41,23,38)(3,45,24,48)(4,37,19,40)(5,47,20,44)(6,39,21,42)(7,36,30,33)(8,17,25,14)(9,32,26,35)(10,13,27,16)(11,34,28,31)(12,15,29,18), (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), (1,3)(4,6)(7,27)(8,26)(9,25)(10,30)(11,29)(12,28)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,21)(22,24)(31,43)(32,48)(33,47)(34,46)(35,45)(36,44) );
G=PermutationGroup([[(1,9,19,29),(2,10,20,30),(3,11,21,25),(4,12,22,26),(5,7,23,27),(6,8,24,28),(13,41,33,44),(14,42,34,45),(15,37,35,46),(16,38,36,47),(17,39,31,48),(18,40,32,43)], [(1,43,22,46),(2,41,23,38),(3,45,24,48),(4,37,19,40),(5,47,20,44),(6,39,21,42),(7,36,30,33),(8,17,25,14),(9,32,26,35),(10,13,27,16),(11,34,28,31),(12,15,29,18)], [(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)], [(1,3),(4,6),(7,27),(8,26),(9,25),(10,30),(11,29),(12,28),(13,38),(14,37),(15,42),(16,41),(17,40),(18,39),(19,21),(22,24),(31,43),(32,48),(33,47),(34,46),(35,45),(36,44)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D6 | C4xS3 | D12 | C3:D4 | C3:D4 | C8:C22 | D4:D6 |
| kernel | C4:C4:36D6 | C6.D8 | C2xC4.Dic3 | C3xC42:C2 | C22xD12 | C2xD12 | C42:C2 | C2xC12 | C22xC6 | C4:C4 | C22xC4 | C2xC4 | C2xC4 | C2xC4 | C23 | C6 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 3 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of C4:C4:36D6 ►in GL6(F73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 59 | 0 | 0 |
| 0 | 0 | 14 | 66 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 14 |
| 0 | 0 | 0 | 0 | 59 | 7 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 14 | 66 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,7,14,0,0,0,0,59,66,0,0,0,0,0,0,66,59,0,0,0,0,14,7],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,72,0,0],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,0,0,1,1,0,0,0,0,0,0,7,14,0,0,0,0,7,66] >;
C4:C4:36D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_{36}D_6 % in TeX
G:=Group("C4:C4:36D6"); // GroupNames label
G:=SmallGroup(192,560);
// by ID
G=gap.SmallGroup(192,560);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,387,58,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d=a*b^-1,d*c*d=c^-1>;
// generators/relations