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G = C4○D4×D9order 288 = 25·32

Direct product of C4○D4 and D9

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4○D4×D9, D47D18, Q87D18, D3610C22, C36.25C23, C18.11C24, D18.6C23, Dic1810C22, Dic9.10C23, (D4×D9)⋊6C2, (C2×C4)⋊7D18, (Q8×D9)⋊6C2, D42D96C2, (C2×C36)⋊4C22, Q83D96C2, (C3×D4).38D6, (C4×D9)⋊6C22, (D4×C9)⋊8C22, C9⋊D44C22, (C3×Q8).62D6, D365C27C2, (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, C94(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

C1C18 — C4○D4×D9
C1C3C9C18D18C22×D9C2×C4×D9 — C4○D4×D9
C9C18 — C4○D4×D9
C1C4C4○D4

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 [×8], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×10], S3 [×5], C6, C6 [×3], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], C9, Dic3 [×4], C12, C12 [×3], D6 [×10], C2×C6 [×3], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], D9 [×2], D9 [×3], C18, C18 [×3], Dic6 [×3], C4×S3 [×10], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×3], C2×C4○D4, Dic9, Dic9 [×3], C36, C36 [×3], D18, D18 [×3], D18 [×6], C2×C18 [×3], S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, C3×C4○D4, Dic18 [×3], C4×D9, C4×D9 [×9], D36 [×3], C2×Dic9 [×3], C9⋊D4 [×6], C2×C36 [×3], D4×C9 [×3], Q8×C9, C22×D9 [×3], S3×C4○D4, C2×C4×D9 [×3], D365C2 [×3], D4×D9 [×3], D42D9 [×3], Q8×D9, Q83D9, C9×C4○D4, C4○D4×D9
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, D9, C22×S3 [×7], C2×C4○D4, D18 [×7], S3×C23, C22×D9 [×7], S3×C4○D4, C23×D9, C4○D4×D9

Smallest permutation representation of C4○D4×D9
On 72 points
Generators in S72
(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 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J6A6B6C6D9A9B9C12A12B12C12D12E18A18B18C18D···18L36A···36F36G···36O
order1222222222344444444446666999121212121218181818···1836···3636···36
size1122299181818211222991818182444222224442224···42···24···4

60 irreducible representations

dim1111111122222222244
type++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D6D6D6C4○D4D9D18D18D18S3×C4○D4C4○D4×D9
kernelC4○D4×D9C2×C4×D9D365C2D4×D9D42D9Q8×D9Q83D9C9×C4○D4C3×C4○D4C2×C12C3×D4C3×Q8D9C4○D4C2×C4D4Q8C3C1
# reps1333311113314399326

Matrix representation of C4○D4×D9 in GL4(𝔽37) generated by

36000
03600
00310
00031
,
36000
03600
003613
00341
,
1000
0100
0010
00336
,
26600
312000
0010
0001
,
311100
17600
0010
0001
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

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