direct product, metabelian, nilpotent (class 2), monomial
Aliases: C4○D4×3- 1+2, C12.46C62, (C2×C36)⋊5C6, (D4×C9)⋊6C6, (Q8×C9)⋊10C6, (C6×C12).15C6, C36.30(C2×C6), C62.3(C2×C6), (C2×C6).12C62, C6.26(C2×C62), (D4×C32).6C6, C18.14(C22×C6), (Q8×C32).16C6, (D4×3- 1+2)⋊5C2, D4⋊2(C2×3- 1+2), (Q8×3- 1+2)⋊5C2, Q8⋊4(C2×3- 1+2), C2.4(C23×3- 1+2), C4.7(C22×3- 1+2), C22.(C22×3- 1+2), (C4×3- 1+2).23C22, (C2×3- 1+2).14C23, (C22×3- 1+2).12C22, C9⋊4(C3×C4○D4), (C9×C4○D4)⋊3C3, C32.(C3×C4○D4), (C2×C18).12(C2×C6), (C3×C12).26(C2×C6), (C2×C12).24(C3×C6), (C3×D4).14(C3×C6), (C3×Q8).26(C3×C6), C3.3(C32×C4○D4), (C3×C6).36(C22×C6), (C3×C4○D4).8C32, (C32×C4○D4).3C3, (C2×C4×3- 1+2)⋊5C2, (C2×C4)⋊3(C2×3- 1+2), SmallGroup(432,411)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D4×3- 1+2
G = < a,b,c,d,e | a4=c2=d9=e3=1, b2=a2, ab=ba, ac=ca, ad=da, ae=ea, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >
Subgroups: 230 in 160 conjugacy classes, 119 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, D4, Q8, C9, C32, C12, C12, C12, C2×C6, C2×C6, C4○D4, C18, C18, C3×C6, C3×C6, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, 3- 1+2, C36, C2×C18, C3×C12, C3×C12, C62, C3×C4○D4, C3×C4○D4, C2×3- 1+2, C2×3- 1+2, C2×C36, D4×C9, Q8×C9, C6×C12, D4×C32, Q8×C32, C4×3- 1+2, C4×3- 1+2, C22×3- 1+2, C9×C4○D4, C32×C4○D4, C2×C4×3- 1+2, D4×3- 1+2, Q8×3- 1+2, C4○D4×3- 1+2
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C4○D4, C3×C6, C22×C6, 3- 1+2, C62, C3×C4○D4, C2×3- 1+2, C2×C62, C22×3- 1+2, C32×C4○D4, C23×3- 1+2, C4○D4×3- 1+2
►On 72 points
(1 45 27 34)(2 37 19 35)(3 38 20 36)(4 39 21 28)(5 40 22 29)(6 41 23 30)(7 42 24 31)(8 43 25 32)(9 44 26 33)(10 55 71 53)(11 56 72 54)(12 57 64 46)(13 58 65 47)(14 59 66 48)(15 60 67 49)(16 61 68 50)(17 62 69 51)(18 63 70 52)
(1 63 27 52)(2 55 19 53)(3 56 20 54)(4 57 21 46)(5 58 22 47)(6 59 23 48)(7 60 24 49)(8 61 25 50)(9 62 26 51)(10 37 71 35)(11 38 72 36)(12 39 64 28)(13 40 65 29)(14 41 66 30)(15 42 67 31)(16 43 68 32)(17 44 69 33)(18 45 70 34)
(1 52)(2 53)(3 54)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)
(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 65 66 67 68 69 70 71 72)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(37 43 40)(38 41 44)(47 53 50)(48 51 54)(55 61 58)(56 59 62)(65 71 68)(66 69 72)
G:=sub<Sym(72)| (1,45,27,34)(2,37,19,35)(3,38,20,36)(4,39,21,28)(5,40,22,29)(6,41,23,30)(7,42,24,31)(8,43,25,32)(9,44,26,33)(10,55,71,53)(11,56,72,54)(12,57,64,46)(13,58,65,47)(14,59,66,48)(15,60,67,49)(16,61,68,50)(17,62,69,51)(18,63,70,52), (1,63,27,52)(2,55,19,53)(3,56,20,54)(4,57,21,46)(5,58,22,47)(6,59,23,48)(7,60,24,49)(8,61,25,50)(9,62,26,51)(10,37,71,35)(11,38,72,36)(12,39,64,28)(13,40,65,29)(14,41,66,30)(15,42,67,31)(16,43,68,32)(17,44,69,33)(18,45,70,34), (1,52)(2,53)(3,54)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72), (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)(64,65,66,67,68,69,70,71,72), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(37,43,40)(38,41,44)(47,53,50)(48,51,54)(55,61,58)(56,59,62)(65,71,68)(66,69,72)>;
G:=Group( (1,45,27,34)(2,37,19,35)(3,38,20,36)(4,39,21,28)(5,40,22,29)(6,41,23,30)(7,42,24,31)(8,43,25,32)(9,44,26,33)(10,55,71,53)(11,56,72,54)(12,57,64,46)(13,58,65,47)(14,59,66,48)(15,60,67,49)(16,61,68,50)(17,62,69,51)(18,63,70,52), (1,63,27,52)(2,55,19,53)(3,56,20,54)(4,57,21,46)(5,58,22,47)(6,59,23,48)(7,60,24,49)(8,61,25,50)(9,62,26,51)(10,37,71,35)(11,38,72,36)(12,39,64,28)(13,40,65,29)(14,41,66,30)(15,42,67,31)(16,43,68,32)(17,44,69,33)(18,45,70,34), (1,52)(2,53)(3,54)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72), (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)(64,65,66,67,68,69,70,71,72), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(37,43,40)(38,41,44)(47,53,50)(48,51,54)(55,61,58)(56,59,62)(65,71,68)(66,69,72) );
G=PermutationGroup([[(1,45,27,34),(2,37,19,35),(3,38,20,36),(4,39,21,28),(5,40,22,29),(6,41,23,30),(7,42,24,31),(8,43,25,32),(9,44,26,33),(10,55,71,53),(11,56,72,54),(12,57,64,46),(13,58,65,47),(14,59,66,48),(15,60,67,49),(16,61,68,50),(17,62,69,51),(18,63,70,52)], [(1,63,27,52),(2,55,19,53),(3,56,20,54),(4,57,21,46),(5,58,22,47),(6,59,23,48),(7,60,24,49),(8,61,25,50),(9,62,26,51),(10,37,71,35),(11,38,72,36),(12,39,64,28),(13,40,65,29),(14,41,66,30),(15,42,67,31),(16,43,68,32),(17,44,69,33),(18,45,70,34)], [(1,52),(2,53),(3,54),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72)], [(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),(64,65,66,67,68,69,70,71,72)], [(2,8,5),(3,6,9),(10,16,13),(11,14,17),(19,25,22),(20,23,26),(29,35,32),(30,33,36),(37,43,40),(38,41,44),(47,53,50),(48,51,54),(55,61,58),(56,59,62),(65,71,68),(66,69,72)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 6K | ··· | 6P | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 12O | ··· | 12T | 18A | ··· | 18F | 18G | ··· | 18X | 36A | ··· | 36L | 36M | ··· | 36AD |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 6 | ··· | 6 | 3 | ··· | 3 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 3 | ··· | 3 | 6 | ··· | 6 | 3 | ··· | 3 | 6 | ··· | 6 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 |
| type | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C4○D4 | C3×C4○D4 | C3×C4○D4 | 3- 1+2 | C2×3- 1+2 | C2×3- 1+2 | C2×3- 1+2 | C4○D4×3- 1+2 |
| kernel | C4○D4×3- 1+2 | C2×C4×3- 1+2 | D4×3- 1+2 | Q8×3- 1+2 | C9×C4○D4 | C32×C4○D4 | C2×C36 | D4×C9 | Q8×C9 | C6×C12 | D4×C32 | Q8×C32 | 3- 1+2 | C9 | C32 | C4○D4 | C2×C4 | D4 | Q8 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 18 | 18 | 6 | 6 | 6 | 2 | 2 | 12 | 4 | 2 | 6 | 6 | 2 | 4 |
Matrix representation of C4○D4×3- 1+2 ►in GL5(𝔽37)
| 31 | 0 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 36 | 36 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 | 0 |
| 35 | 36 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 11 | 1 | 0 |
| 0 | 0 | 27 | 0 | 27 |
| 0 | 0 | 25 | 0 | 26 |
| 26 | 0 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 11 |
| 0 | 0 | 0 | 10 | 1 |
| 0 | 0 | 0 | 0 | 26 |
G:=sub<GL(5,GF(37))| [31,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[36,2,0,0,0,36,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,35,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,11,27,25,0,0,1,0,0,0,0,0,27,26],[26,0,0,0,0,0,26,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,11,1,26] >;
C4○D4×3- 1+2 in GAP, Magma, Sage, TeX
C_4\circ D_4\times 3_-^{1+2} % in TeX
G:=Group("C4oD4xES-(3,1)"); // GroupNames label
G:=SmallGroup(432,411);
// by ID
G=gap.SmallGroup(432,411);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,1037,394,528,760]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^2=d^9=e^3=1,b^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^2*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^4>;
// generators/relations