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G = C9xC4oD4order 144 = 24·32

Direct product of C9 and C4oD4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C9xC4oD4, D4:2C18, Q8:3C18, C18.13C23, C36.21C22, C36o(D4xC9), C36o(Q8xC9), (C2xC36):7C2, (C2xC4):3C18, (D4xC9):5C2, (Q8xC9):5C2, C4.6(C2xC18), (C3xD4).6C6, C22.(C2xC18), (C2xC12).12C6, C12.22(C2xC6), (C3xQ8).10C6, C6.13(C22xC6), C2.3(C22xC18), (C2xC18).2C22, C3.(C3xC4oD4), C36o(C3xC4oD4), (C2xC6).4(C2xC6), (C3xC4oD4).2C3, SmallGroup(144,50)

Series: Derived Chief Lower central Upper central

C1C2 — C9xC4oD4
C1C3C6C18C2xC18D4xC9 — C9xC4oD4
C1C2 — C9xC4oD4
C1C36 — C9xC4oD4

Generators and relations for C9xC4oD4
 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, C3, C4, C4, C22, C6, C6, C2xC4, D4, Q8, C9, C12, C12, C2xC6, C4oD4, C18, C18, C2xC12, C3xD4, C3xQ8, C36, C36, C2xC18, C3xC4oD4, C2xC36, D4xC9, Q8xC9, C9xC4oD4
Quotients: C1, C2, C3, C22, C6, C23, C9, C2xC6, C4oD4, C18, C22xC6, C2xC18, C3xC4oD4, C22xC18, C9xC4oD4

Smallest permutation representation of C9xC4oD4
On 72 points
Generators in S72
(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 39 21 34)(2 40 22 35)(3 41 23 36)(4 42 24 28)(5 43 25 29)(6 44 26 30)(7 45 27 31)(8 37 19 32)(9 38 20 33)(10 62 66 48)(11 63 67 49)(12 55 68 50)(13 56 69 51)(14 57 70 52)(15 58 71 53)(16 59 72 54)(17 60 64 46)(18 61 65 47)
(1 52 21 57)(2 53 22 58)(3 54 23 59)(4 46 24 60)(5 47 25 61)(6 48 26 62)(7 49 27 63)(8 50 19 55)(9 51 20 56)(10 30 66 44)(11 31 67 45)(12 32 68 37)(13 33 69 38)(14 34 70 39)(15 35 71 40)(16 36 72 41)(17 28 64 42)(18 29 65 43)
(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 64)(18 65)(46 60)(47 61)(48 62)(49 63)(50 55)(51 56)(52 57)(53 58)(54 59)

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

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

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

C9xC4oD4 is a maximal subgroup of
Q8:3Dic9  D4.Dic9  D4.D18  D4:D18  D4.9D18  D4:8D18  D4.10D18  Q8.C54  C36.A4
C9xC4oD4 is a maximal quotient of
D4xC36  Q8xC36

90 conjugacy classes

class 1 2A2B2C2D3A3B4A4B4C4D4E6A6B6C···6H9A···9F12A12B12C12D12E···12J18A···18F18G···18X36A···36L36M···36AD
order122223344444666···69···91212121212···1218···1818···1836···3636···36
size112221111222112···21···111112···21···12···21···12···2

90 irreducible representations

dim111111111111222
type++++
imageC1C2C2C2C3C6C6C6C9C18C18C18C4oD4C3xC4oD4C9xC4oD4
kernelC9xC4oD4C2xC36D4xC9Q8xC9C3xC4oD4C2xC12C3xD4C3xQ8C4oD4C2xC4D4Q8C9C3C1
# reps133126626181862412

Matrix representation of C9xC4oD4 in GL2(F37) generated by

90
09
,
310
031
,
1536
422
,
10
3036
G:=sub<GL(2,GF(37))| [9,0,0,9],[31,0,0,31],[15,4,36,22],[1,30,0,36] >;

C9xC4oD4 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

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