direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D4×D9, D4⋊7D18, Q8⋊7D18, D36⋊10C22, C36.25C23, C18.11C24, D18.6C23, Dic18⋊10C22, Dic9.10C23, (D4×D9)⋊6C2, (C2×C4)⋊7D18, (Q8×D9)⋊6C2, D4⋊2D9⋊6C2, (C2×C36)⋊4C22, Q8⋊3D9⋊6C2, (C3×D4).38D6, (C4×D9)⋊6C22, (D4×C9)⋊8C22, C9⋊D4⋊4C22, (C3×Q8).62D6, D36⋊5C2⋊7C2, (Q8×C9)⋊7C22, (C2×C12).104D6, (C2×C18).3C23, C6.48(S3×C23), C4.25(C22×D9), C2.12(C23×D9), C12.65(C22×S3), C22.2(C22×D9), (C2×Dic9)⋊10C22, (C22×D9).31C22, (C2×C4×D9)⋊6C2, C9⋊4(C2×C4○D4), C3.(S3×C4○D4), (C9×C4○D4)⋊3C2, (C3×C4○D4).15S3, (C2×C6).9(C22×S3), SmallGroup(288,362)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D4×D9
G = < a,b,c,d,e | a4=c2=d9=e2=1, b2=a2, ab=ba, ac=ca, ad=da, ae=ea, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 992 in 246 conjugacy classes, 104 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C9, Dic3, C12, C12, D6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, D9, D9, C18, C18, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C2×C4○D4, Dic9, Dic9, C36, C36, D18, D18, D18, C2×C18, S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C4○D4, Dic18, C4×D9, C4×D9, D36, C2×Dic9, C9⋊D4, C2×C36, D4×C9, Q8×C9, C22×D9, S3×C4○D4, C2×C4×D9, D36⋊5C2, D4×D9, D4⋊2D9, Q8×D9, Q8⋊3D9, C9×C4○D4, C4○D4×D9
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, D9, C22×S3, C2×C4○D4, D18, S3×C23, C22×D9, S3×C4○D4, C23×D9, C4○D4×D9
►On 72 points
(1 32 14 23)(2 33 15 24)(3 34 16 25)(4 35 17 26)(5 36 18 27)(6 28 10 19)(7 29 11 20)(8 30 12 21)(9 31 13 22)(37 64 46 55)(38 65 47 56)(39 66 48 57)(40 67 49 58)(41 68 50 59)(42 69 51 60)(43 70 52 61)(44 71 53 62)(45 72 54 63)
(1 41 14 50)(2 42 15 51)(3 43 16 52)(4 44 17 53)(5 45 18 54)(6 37 10 46)(7 38 11 47)(8 39 12 48)(9 40 13 49)(19 55 28 64)(20 56 29 65)(21 57 30 66)(22 58 31 67)(23 59 32 68)(24 60 33 69)(25 61 34 70)(26 62 35 71)(27 63 36 72)
(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 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)
(1 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 35)(29 34)(30 33)(31 32)(37 44)(38 43)(39 42)(40 41)(46 53)(47 52)(48 51)(49 50)(55 62)(56 61)(57 60)(58 59)(64 71)(65 70)(66 69)(67 68)
G:=sub<Sym(72)| (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22)(37,64,46,55)(38,65,47,56)(39,66,48,57)(40,67,49,58)(41,68,50,59)(42,69,51,60)(43,70,52,61)(44,71,53,62)(45,72,54,63), (1,41,14,50)(2,42,15,51)(3,43,16,52)(4,44,17,53)(5,45,18,54)(6,37,10,46)(7,38,11,47)(8,39,12,48)(9,40,13,49)(19,55,28,64)(20,56,29,65)(21,57,30,66)(22,58,31,67)(23,59,32,68)(24,60,33,69)(25,61,34,70)(26,62,35,71)(27,63,36,72), (37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(37,44)(38,43)(39,42)(40,41)(46,53)(47,52)(48,51)(49,50)(55,62)(56,61)(57,60)(58,59)(64,71)(65,70)(66,69)(67,68)>;
G:=Group( (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22)(37,64,46,55)(38,65,47,56)(39,66,48,57)(40,67,49,58)(41,68,50,59)(42,69,51,60)(43,70,52,61)(44,71,53,62)(45,72,54,63), (1,41,14,50)(2,42,15,51)(3,43,16,52)(4,44,17,53)(5,45,18,54)(6,37,10,46)(7,38,11,47)(8,39,12,48)(9,40,13,49)(19,55,28,64)(20,56,29,65)(21,57,30,66)(22,58,31,67)(23,59,32,68)(24,60,33,69)(25,61,34,70)(26,62,35,71)(27,63,36,72), (37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(37,44)(38,43)(39,42)(40,41)(46,53)(47,52)(48,51)(49,50)(55,62)(56,61)(57,60)(58,59)(64,71)(65,70)(66,69)(67,68) );
G=PermutationGroup([[(1,32,14,23),(2,33,15,24),(3,34,16,25),(4,35,17,26),(5,36,18,27),(6,28,10,19),(7,29,11,20),(8,30,12,21),(9,31,13,22),(37,64,46,55),(38,65,47,56),(39,66,48,57),(40,67,49,58),(41,68,50,59),(42,69,51,60),(43,70,52,61),(44,71,53,62),(45,72,54,63)], [(1,41,14,50),(2,42,15,51),(3,43,16,52),(4,44,17,53),(5,45,18,54),(6,37,10,46),(7,38,11,47),(8,39,12,48),(9,40,13,49),(19,55,28,64),(20,56,29,65),(21,57,30,66),(22,58,31,67),(23,59,32,68),(24,60,33,69),(25,61,34,70),(26,62,35,71),(27,63,36,72)], [(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,26),(20,25),(21,24),(22,23),(28,35),(29,34),(30,33),(31,32),(37,44),(38,43),(39,42),(40,41),(46,53),(47,52),(48,51),(49,50),(55,62),(56,61),(57,60),(58,59),(64,71),(65,70),(66,69),(67,68)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 18A | 18B | 18C | 18D | ··· | 18L | 36A | ··· | 36F | 36G | ··· | 36O |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 18 | 18 | 18 | 2 | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 18 | 18 | 18 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | D9 | D18 | D18 | D18 | S3×C4○D4 | C4○D4×D9 |
| kernel | C4○D4×D9 | C2×C4×D9 | D36⋊5C2 | D4×D9 | D4⋊2D9 | Q8×D9 | Q8⋊3D9 | C9×C4○D4 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | D9 | C4○D4 | C2×C4 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 4 | 3 | 9 | 9 | 3 | 2 | 6 |
Matrix representation of C4○D4×D9 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 31 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 13 |
| 0 | 0 | 34 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 3 | 36 |
| 26 | 6 | 0 | 0 |
| 31 | 20 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 31 | 11 | 0 | 0 |
| 17 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,31,0,0,0,0,31],[36,0,0,0,0,36,0,0,0,0,36,34,0,0,13,1],[1,0,0,0,0,1,0,0,0,0,1,3,0,0,0,36],[26,31,0,0,6,20,0,0,0,0,1,0,0,0,0,1],[31,17,0,0,11,6,0,0,0,0,1,0,0,0,0,1] >;
C4○D4×D9 in GAP, Magma, Sage, TeX
C_4\circ D_4\times D_9
% in TeX
G:=Group("C4oD4xD9"); // GroupNames label
G:=SmallGroup(288,362);
// by ID
G=gap.SmallGroup(288,362);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,346,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^2=d^9=e^2=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=d^-1>;
// generators/relations