Copied to
clipboard

G = C3×C4.Dic9order 432 = 24·33

Direct product of C3 and C4.Dic9

direct product, metacyclic, supersoluble, monomial

Aliases: C3×C4.Dic9, C36.7C12, C12.78D18, C12.9Dic9, C62.16Dic3, C9⋊C812C6, C4.(C3×Dic9), (C3×C36).3C4, (C6×C18).5C4, C4.15(C6×D9), (C3×C9)⋊8M4(2), C12.91(S3×C6), C36.38(C2×C6), (C6×C36).11C2, (C2×C36).18C6, (C6×C12).38S3, (C2×C12).18D9, C95(C3×M4(2)), (C2×C6).2Dic9, C6.7(C6×Dic3), C2.3(C6×Dic9), (C2×C18).10C12, C18.21(C2×C12), C22.(C3×Dic9), (C3×C12).215D6, C6.18(C2×Dic9), C12.1(C3×Dic3), (C3×C36).68C22, (C3×C12).18Dic3, C32.3(C4.Dic3), (C3×C9⋊C8)⋊12C2, (C2×C4).2(C3×D9), (C2×C12).22(C3×S3), (C3×C18).33(C2×C4), C3.1(C3×C4.Dic3), (C2×C6).13(C3×Dic3), (C3×C6).55(C2×Dic3), SmallGroup(432,125)

Series: Derived Chief Lower central Upper central

C1C18 — C3×C4.Dic9
C1C3C9C18C36C3×C36C3×C9⋊C8 — C3×C4.Dic9
C9C18 — C3×C4.Dic9
C1C12C2×C12

Generators and relations for C3×C4.Dic9
 G = < a,b,c,d | a3=b4=1, c18=b2, d2=c9, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c17 >

Subgroups: 158 in 76 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3 [×2], C3, C4 [×2], C22, C6 [×2], C6 [×5], C8 [×2], C2×C4, C9, C9, C32, C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, M4(2), C18, C18 [×4], C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], C2×C12 [×2], C2×C12, C3×C9, C36 [×2], C36 [×2], C2×C18, C2×C18, C3×C12 [×2], C62, C4.Dic3, C3×M4(2), C3×C18, C3×C18, C9⋊C8 [×2], C2×C36, C2×C36, C3×C3⋊C8 [×2], C6×C12, C3×C36 [×2], C6×C18, C4.Dic9, C3×C4.Dic3, C3×C9⋊C8 [×2], C6×C36, C3×C4.Dic9
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, Dic3 [×2], C12 [×2], D6, C2×C6, M4(2), D9, C3×S3, C2×Dic3, C2×C12, Dic9 [×2], D18, C3×Dic3 [×2], S3×C6, C4.Dic3, C3×M4(2), C3×D9, C2×Dic9, C6×Dic3, C3×Dic9 [×2], C6×D9, C4.Dic9, C3×C4.Dic3, C6×Dic9, C3×C4.Dic9

Smallest permutation representation of C3×C4.Dic9
On 72 points
Generators in S72
(1 25 13)(2 26 14)(3 27 15)(4 28 16)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)(37 49 61)(38 50 62)(39 51 63)(40 52 64)(41 53 65)(42 54 66)(43 55 67)(44 56 68)(45 57 69)(46 58 70)(47 59 71)(48 60 72)
(1 10 19 28)(2 11 20 29)(3 12 21 30)(4 13 22 31)(5 14 23 32)(6 15 24 33)(7 16 25 34)(8 17 26 35)(9 18 27 36)(37 64 55 46)(38 65 56 47)(39 66 57 48)(40 67 58 49)(41 68 59 50)(42 69 60 51)(43 70 61 52)(44 71 62 53)(45 72 63 54)
(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 66 10 39 19 48 28 57)(2 47 11 56 20 65 29 38)(3 64 12 37 21 46 30 55)(4 45 13 54 22 63 31 72)(5 62 14 71 23 44 32 53)(6 43 15 52 24 61 33 70)(7 60 16 69 25 42 34 51)(8 41 17 50 26 59 35 68)(9 58 18 67 27 40 36 49)

G:=sub<Sym(72)| (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,49,61)(38,50,62)(39,51,63)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69)(46,58,70)(47,59,71)(48,60,72), (1,10,19,28)(2,11,20,29)(3,12,21,30)(4,13,22,31)(5,14,23,32)(6,15,24,33)(7,16,25,34)(8,17,26,35)(9,18,27,36)(37,64,55,46)(38,65,56,47)(39,66,57,48)(40,67,58,49)(41,68,59,50)(42,69,60,51)(43,70,61,52)(44,71,62,53)(45,72,63,54), (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), (1,66,10,39,19,48,28,57)(2,47,11,56,20,65,29,38)(3,64,12,37,21,46,30,55)(4,45,13,54,22,63,31,72)(5,62,14,71,23,44,32,53)(6,43,15,52,24,61,33,70)(7,60,16,69,25,42,34,51)(8,41,17,50,26,59,35,68)(9,58,18,67,27,40,36,49)>;

G:=Group( (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,49,61)(38,50,62)(39,51,63)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69)(46,58,70)(47,59,71)(48,60,72), (1,10,19,28)(2,11,20,29)(3,12,21,30)(4,13,22,31)(5,14,23,32)(6,15,24,33)(7,16,25,34)(8,17,26,35)(9,18,27,36)(37,64,55,46)(38,65,56,47)(39,66,57,48)(40,67,58,49)(41,68,59,50)(42,69,60,51)(43,70,61,52)(44,71,62,53)(45,72,63,54), (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), (1,66,10,39,19,48,28,57)(2,47,11,56,20,65,29,38)(3,64,12,37,21,46,30,55)(4,45,13,54,22,63,31,72)(5,62,14,71,23,44,32,53)(6,43,15,52,24,61,33,70)(7,60,16,69,25,42,34,51)(8,41,17,50,26,59,35,68)(9,58,18,67,27,40,36,49) );

G=PermutationGroup([(1,25,13),(2,26,14),(3,27,15),(4,28,16),(5,29,17),(6,30,18),(7,31,19),(8,32,20),(9,33,21),(10,34,22),(11,35,23),(12,36,24),(37,49,61),(38,50,62),(39,51,63),(40,52,64),(41,53,65),(42,54,66),(43,55,67),(44,56,68),(45,57,69),(46,58,70),(47,59,71),(48,60,72)], [(1,10,19,28),(2,11,20,29),(3,12,21,30),(4,13,22,31),(5,14,23,32),(6,15,24,33),(7,16,25,34),(8,17,26,35),(9,18,27,36),(37,64,55,46),(38,65,56,47),(39,66,57,48),(40,67,58,49),(41,68,59,50),(42,69,60,51),(43,70,61,52),(44,71,62,53),(45,72,63,54)], [(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)], [(1,66,10,39,19,48,28,57),(2,47,11,56,20,65,29,38),(3,64,12,37,21,46,30,55),(4,45,13,54,22,63,31,72),(5,62,14,71,23,44,32,53),(6,43,15,52,24,61,33,70),(7,60,16,69,25,42,34,51),(8,41,17,50,26,59,35,68),(9,58,18,67,27,40,36,49)])

126 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B4C6A6B6C···6M8A8B8C8D9A···9I12A12B12C12D12E···12R18A···18AA24A···24H36A···36AJ
order12233333444666···688889···91212121212···1218···1824···2436···36
size11211222112112···2181818182···211112···22···218···182···2

126 irreducible representations

dim11111111112222222222222222222222
type++++-+-+-+-
imageC1C2C2C3C4C4C6C6C12C12S3Dic3D6Dic3M4(2)D9C3×S3Dic9D18C3×Dic3S3×C6Dic9C3×Dic3C3×M4(2)C4.Dic3C3×D9C3×Dic9C6×D9C3×Dic9C4.Dic9C3×C4.Dic3C3×C4.Dic9
kernelC3×C4.Dic9C3×C9⋊C8C6×C36C4.Dic9C3×C36C6×C18C9⋊C8C2×C36C36C2×C18C6×C12C3×C12C3×C12C62C3×C9C2×C12C2×C12C12C12C12C12C2×C6C2×C6C9C32C2×C4C4C4C22C3C3C1
# reps1212224244111123233223244666612824

Matrix representation of C3×C4.Dic9 in GL2(𝔽37) generated by

260
026
,
310
06
,
130
017
,
06
10
G:=sub<GL(2,GF(37))| [26,0,0,26],[31,0,0,6],[13,0,0,17],[0,1,6,0] >;

C3×C4.Dic9 in GAP, Magma, Sage, TeX

C_3\times C_4.{\rm Dic}_9
% in TeX

G:=Group("C3xC4.Dic9");
// GroupNames label

G:=SmallGroup(432,125);
// by ID

G=gap.SmallGroup(432,125);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,80,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=1,c^18=b^2,d^2=c^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^17>;
// generators/relations

׿
×
𝔽