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## G = C9×C4○D4order 144 = 24·32

### Direct product of C9 and C4○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C9×C4○D4
 Chief series C1 — C3 — C6 — C18 — C2×C18 — D4×C9 — C9×C4○D4
 Lower central C1 — C2 — C9×C4○D4
 Upper central C1 — C36 — C9×C4○D4

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, C3, C4, C4, C22, C6, C6, C2×C4, D4, Q8, C9, C12, C12, C2×C6, C4○D4, C18, C18, C2×C12, C3×D4, C3×Q8, C36, C36, C2×C18, C3×C4○D4, C2×C36, D4×C9, Q8×C9, C9×C4○D4
Quotients: C1, C2, C3, C22, C6, C23, C9, C2×C6, C4○D4, C18, C22×C6, C2×C18, C3×C4○D4, C22×C18, C9×C4○D4

Smallest permutation representation of C9×C4○D4
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)]])

C9×C4○D4 is a maximal subgroup of
Q83Dic9  D4.Dic9  D4.D18  D4⋊D18  D4.9D18  D48D18  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

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