direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C4○D4, D4⋊2C18, Q8⋊3C18, C18.13C23, C36.21C22, C36○(D4×C9), C36○(Q8×C9), (C2×C36)⋊7C2, (C2×C4)⋊3C18, (D4×C9)⋊5C2, (Q8×C9)⋊5C2, C4.6(C2×C18), (C3×D4).6C6, C22.(C2×C18), (C2×C12).12C6, C12.22(C2×C6), (C3×Q8).10C6, C6.13(C22×C6), C2.3(C22×C18), (C2×C18).2C22, C3.(C3×C4○D4), C36○(C3×C4○D4), (C2×C6).4(C2×C6), (C3×C4○D4).2C3, SmallGroup(144,50)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C4○D4
G = < a,b,c,d | a9=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Subgroups: 69 in 60 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C6, C6 [×3], C2×C4 [×3], D4 [×3], Q8, C9, C12, C12 [×3], C2×C6 [×3], C4○D4, C18, C18 [×3], C2×C12 [×3], C3×D4 [×3], C3×Q8, C36, C36 [×3], C2×C18 [×3], C3×C4○D4, C2×C36 [×3], D4×C9 [×3], Q8×C9, C9×C4○D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, C9, C2×C6 [×7], C4○D4, C18 [×7], C22×C6, C2×C18 [×7], C3×C4○D4, C22×C18, C9×C4○D4
►On 72 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 65 66 67 68 69 70 71 72)
(1 40 22 34)(2 41 23 35)(3 42 24 36)(4 43 25 28)(5 44 26 29)(6 45 27 30)(7 37 19 31)(8 38 20 32)(9 39 21 33)(10 63 66 48)(11 55 67 49)(12 56 68 50)(13 57 69 51)(14 58 70 52)(15 59 71 53)(16 60 72 54)(17 61 64 46)(18 62 65 47)
(1 52 22 58)(2 53 23 59)(3 54 24 60)(4 46 25 61)(5 47 26 62)(6 48 27 63)(7 49 19 55)(8 50 20 56)(9 51 21 57)(10 30 66 45)(11 31 67 37)(12 32 68 38)(13 33 69 39)(14 34 70 40)(15 35 71 41)(16 36 72 42)(17 28 64 43)(18 29 65 44)
(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 64)(18 65)(46 61)(47 62)(48 63)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)
G:=sub<Sym(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), (1,40,22,34)(2,41,23,35)(3,42,24,36)(4,43,25,28)(5,44,26,29)(6,45,27,30)(7,37,19,31)(8,38,20,32)(9,39,21,33)(10,63,66,48)(11,55,67,49)(12,56,68,50)(13,57,69,51)(14,58,70,52)(15,59,71,53)(16,60,72,54)(17,61,64,46)(18,62,65,47), (1,52,22,58)(2,53,23,59)(3,54,24,60)(4,46,25,61)(5,47,26,62)(6,48,27,63)(7,49,19,55)(8,50,20,56)(9,51,21,57)(10,30,66,45)(11,31,67,37)(12,32,68,38)(13,33,69,39)(14,34,70,40)(15,35,71,41)(16,36,72,42)(17,28,64,43)(18,29,65,44), (10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,64)(18,65)(46,61)(47,62)(48,63)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)>;
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)(64,65,66,67,68,69,70,71,72), (1,40,22,34)(2,41,23,35)(3,42,24,36)(4,43,25,28)(5,44,26,29)(6,45,27,30)(7,37,19,31)(8,38,20,32)(9,39,21,33)(10,63,66,48)(11,55,67,49)(12,56,68,50)(13,57,69,51)(14,58,70,52)(15,59,71,53)(16,60,72,54)(17,61,64,46)(18,62,65,47), (1,52,22,58)(2,53,23,59)(3,54,24,60)(4,46,25,61)(5,47,26,62)(6,48,27,63)(7,49,19,55)(8,50,20,56)(9,51,21,57)(10,30,66,45)(11,31,67,37)(12,32,68,38)(13,33,69,39)(14,34,70,40)(15,35,71,41)(16,36,72,42)(17,28,64,43)(18,29,65,44), (10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,64)(18,65)(46,61)(47,62)(48,63)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60) );
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),(64,65,66,67,68,69,70,71,72)], [(1,40,22,34),(2,41,23,35),(3,42,24,36),(4,43,25,28),(5,44,26,29),(6,45,27,30),(7,37,19,31),(8,38,20,32),(9,39,21,33),(10,63,66,48),(11,55,67,49),(12,56,68,50),(13,57,69,51),(14,58,70,52),(15,59,71,53),(16,60,72,54),(17,61,64,46),(18,62,65,47)], [(1,52,22,58),(2,53,23,59),(3,54,24,60),(4,46,25,61),(5,47,26,62),(6,48,27,63),(7,49,19,55),(8,50,20,56),(9,51,21,57),(10,30,66,45),(11,31,67,37),(12,32,68,38),(13,33,69,39),(14,34,70,40),(15,35,71,41),(16,36,72,42),(17,28,64,43),(18,29,65,44)], [(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,64),(18,65),(46,61),(47,62),(48,63),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60)])
C9×C4○D4 is a maximal subgroup of
Q8⋊3Dic9 D4.Dic9 D4.D18 D4⋊D18 D4.9D18 D4⋊8D18 D4.10D18 Q8.C54 C36.A4
C9×C4○D4 is a maximal quotient of
D4×C36 Q8×C36
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6H | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 18A | ··· | 18F | 18G | ··· | 18X | 36A | ··· | 36L | 36M | ··· | 36AD |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | C4○D4 | C3×C4○D4 | C9×C4○D4 |
| kernel | C9×C4○D4 | C2×C36 | D4×C9 | Q8×C9 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C9 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 6 | 18 | 18 | 6 | 2 | 4 | 12 |
Matrix representation of C9×C4○D4 ►in GL2(𝔽37) generated by
| 9 | 0 |
| 0 | 9 |
| 31 | 0 |
| 0 | 31 |
| 15 | 36 |
| 4 | 22 |
| 1 | 0 |
| 30 | 36 |
G:=sub<GL(2,GF(37))| [9,0,0,9],[31,0,0,31],[15,4,36,22],[1,30,0,36] >;
C9×C4○D4 in GAP, Magma, Sage, TeX
C_9\times C_4\circ D_4
% in TeX
G:=Group("C9xC4oD4"); // GroupNames label
G:=SmallGroup(144,50);
// by ID
G=gap.SmallGroup(144,50);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-3,313,122,165]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations