direct product, metabelian, nilpotent (class 2), monomial
Aliases: C2×D4×3- 1+2, C12.28C62, (D4×C18)⋊C3, C9⋊4(C6×D4), (C2×C36)⋊4C6, C36⋊4(C2×C6), (D4×C9)⋊5C6, C18⋊3(C3×D4), C32.(C6×D4), (C6×C12).12C6, (C22×C18)⋊5C6, (C2×C62).3C6, (C2×C6).9C62, C62.39(C2×C6), C6.22(C2×C62), (D4×C32).5C6, (C6×D4).3C32, C6.23(D4×C32), C18.12(C22×C6), C4⋊(C22×3- 1+2), C23⋊3(C2×3- 1+2), (C4×3- 1+2)⋊4C22, (C23×3- 1+2)⋊3C2, C2.2(C23×3- 1+2), C22⋊3(C22×3- 1+2), (C2×3- 1+2).12C23, (C22×3- 1+2)⋊4C22, (D4×C3×C6).C3, C3.3(D4×C3×C6), (C2×C18)⋊6(C2×C6), (C3×C6).38(C3×D4), (C3×C12).23(C2×C6), (C2×C12).18(C3×C6), (C3×D4).11(C3×C6), (C22×C6).18(C3×C6), (C3×C6).32(C22×C6), (C2×C4×3- 1+2)⋊4C2, (C2×C4)⋊2(C2×3- 1+2), SmallGroup(432,405)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4×3- 1+2
G = < a,b,c,d,e | a2=b4=c2=d9=e3=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >
Subgroups: 350 in 216 conjugacy classes, 133 normal (20 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, D4, C23, C9, C32, C12, C12, C2×C6, C2×C6, C2×C6, C2×D4, C18, C18, C3×C6, C3×C6, C3×C6, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C22×C6, 3- 1+2, C36, C2×C18, C2×C18, C3×C12, C62, C62, C62, C6×D4, C6×D4, C2×3- 1+2, C2×3- 1+2, C2×3- 1+2, C2×C36, D4×C9, C22×C18, C6×C12, D4×C32, C2×C62, C4×3- 1+2, C22×3- 1+2, C22×3- 1+2, C22×3- 1+2, D4×C18, D4×C3×C6, C2×C4×3- 1+2, D4×3- 1+2, C23×3- 1+2, C2×D4×3- 1+2
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2×C6, C2×D4, C3×C6, C3×D4, C22×C6, 3- 1+2, C62, C6×D4, C2×3- 1+2, D4×C32, C2×C62, C22×3- 1+2, D4×C3×C6, D4×3- 1+2, C23×3- 1+2, C2×D4×3- 1+2
►On 72 points
(1 45)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 53)(11 54)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 35)(20 36)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(55 71)(56 72)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)
(1 18 27 70)(2 10 19 71)(3 11 20 72)(4 12 21 64)(5 13 22 65)(6 14 23 66)(7 15 24 67)(8 16 25 68)(9 17 26 69)(28 57 39 46)(29 58 40 47)(30 59 41 48)(31 60 42 49)(32 61 43 50)(33 62 44 51)(34 63 45 52)(35 55 37 53)(36 56 38 54)
(1 70)(2 71)(3 72)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)
(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)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,53)(11,54)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,35)(20,36)(21,28)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(55,71)(56,72)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70), (1,18,27,70)(2,10,19,71)(3,11,20,72)(4,12,21,64)(5,13,22,65)(6,14,23,66)(7,15,24,67)(8,16,25,68)(9,17,26,69)(28,57,39,46)(29,58,40,47)(30,59,41,48)(31,60,42,49)(32,61,43,50)(33,62,44,51)(34,63,45,52)(35,55,37,53)(36,56,38,54), (1,70)(2,71)(3,72)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63), (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)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,53)(11,54)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,35)(20,36)(21,28)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(55,71)(56,72)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70), (1,18,27,70)(2,10,19,71)(3,11,20,72)(4,12,21,64)(5,13,22,65)(6,14,23,66)(7,15,24,67)(8,16,25,68)(9,17,26,69)(28,57,39,46)(29,58,40,47)(30,59,41,48)(31,60,42,49)(32,61,43,50)(33,62,44,51)(34,63,45,52)(35,55,37,53)(36,56,38,54), (1,70)(2,71)(3,72)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63), (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),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,53),(11,54),(12,46),(13,47),(14,48),(15,49),(16,50),(17,51),(18,52),(19,35),(20,36),(21,28),(22,29),(23,30),(24,31),(25,32),(26,33),(27,34),(55,71),(56,72),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70)], [(1,18,27,70),(2,10,19,71),(3,11,20,72),(4,12,21,64),(5,13,22,65),(6,14,23,66),(7,15,24,67),(8,16,25,68),(9,17,26,69),(28,57,39,46),(29,58,40,47),(30,59,41,48),(31,60,42,49),(32,61,43,50),(33,62,44,51),(34,63,45,52),(35,55,37,53),(36,56,38,54)], [(1,70),(2,71),(3,72),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,25),(17,26),(18,27),(28,46),(29,47),(30,48),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63)], [(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 | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6N | 6O | ··· | 6T | 6U | ··· | 6AB | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | ··· | 18R | 18S | ··· | 18AP | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 3 | ··· | 3 | 6 | ··· | 6 | 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 | D4 | C3×D4 | C3×D4 | 3- 1+2 | C2×3- 1+2 | C2×3- 1+2 | C2×3- 1+2 | D4×3- 1+2 |
| kernel | C2×D4×3- 1+2 | C2×C4×3- 1+2 | D4×3- 1+2 | C23×3- 1+2 | D4×C18 | D4×C3×C6 | C2×C36 | D4×C9 | C22×C18 | C6×C12 | D4×C32 | C2×C62 | C2×3- 1+2 | C18 | C3×C6 | C2×D4 | C2×C4 | D4 | C23 | C2 |
| # reps | 1 | 1 | 4 | 2 | 6 | 2 | 6 | 24 | 12 | 2 | 8 | 4 | 2 | 12 | 4 | 2 | 2 | 8 | 4 | 4 |
Matrix representation of C2×D4×3- 1+2 ►in GL5(𝔽37)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 1 | 1 | 0 | 0 | 0 |
| 35 | 36 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 10 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 26 |
| 0 | 0 | 1 | 0 | 0 |
| 10 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 26 | 0 |
| 0 | 0 | 0 | 0 | 10 |
G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,35,0,0,0,1,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[36,2,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[10,0,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,26,0],[10,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,26,0,0,0,0,0,10] >;
C2×D4×3- 1+2 in GAP, Magma, Sage, TeX
C_2\times D_4\times 3_-^{1+2} % in TeX
G:=Group("C2xD4xES-(3,1)"); // GroupNames label
G:=SmallGroup(432,405);
// by ID
G=gap.SmallGroup(432,405);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,1037,528,760]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^9=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^4>;
// generators/relations