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## G = C9×M4(2)  order 144 = 24·32

### Direct product of C9 and M4(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C9×M4(2)
 Chief series C1 — C2 — C6 — C12 — C36 — C72 — C9×M4(2)
 Lower central C1 — C2 — C9×M4(2)
 Upper central C1 — C36 — C9×M4(2)

Generators and relations for C9×M4(2)
G = < a,b,c | a9=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Smallest permutation representation of C9×M4(2)
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 66 50 13 42 28 59 21)(2 67 51 14 43 29 60 22)(3 68 52 15 44 30 61 23)(4 69 53 16 45 31 62 24)(5 70 54 17 37 32 63 25)(6 71 46 18 38 33 55 26)(7 72 47 10 39 34 56 27)(8 64 48 11 40 35 57 19)(9 65 49 12 41 36 58 20)
(10 27)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 64)(36 65)

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

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

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

C9×M4(2) is a maximal subgroup of   C36.53D4  C4.D36  C36.48D4  Dic18⋊C4  D36.C4  C8⋊D18  C8.D18

90 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 6C 6D 8A 8B 8C 8D 9A ··· 9F 12A 12B 12C 12D 12E 12F 18A ··· 18F 18G ··· 18L 24A ··· 24H 36A ··· 36L 36M ··· 36R 72A ··· 72X order 1 2 2 3 3 4 4 4 6 6 6 6 8 8 8 8 9 ··· 9 12 12 12 12 12 12 18 ··· 18 18 ··· 18 24 ··· 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 2 1 1 1 1 2 1 1 2 2 2 2 2 2 1 ··· 1 1 1 1 1 2 2 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + image C1 C2 C2 C3 C4 C4 C6 C6 C9 C12 C12 C18 C18 C36 C36 M4(2) C3×M4(2) C9×M4(2) kernel C9×M4(2) C72 C2×C36 C3×M4(2) C36 C2×C18 C24 C2×C12 M4(2) C12 C2×C6 C8 C2×C4 C4 C22 C9 C3 C1 # reps 1 2 1 2 2 2 4 2 6 4 4 12 6 12 12 2 4 12

Matrix representation of C9×M4(2) in GL2(𝔽37) generated by

 34 0 0 34
,
 0 14 26 0
,
 36 0 0 1
G:=sub<GL(2,GF(37))| [34,0,0,34],[0,26,14,0],[36,0,0,1] >;

C9×M4(2) in GAP, Magma, Sage, TeX

C_9\times M_4(2)
% in TeX

G:=Group("C9xM4(2)");
// GroupNames label

G:=SmallGroup(144,24);
// by ID

G=gap.SmallGroup(144,24);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-2,72,889,122,165]);
// Polycyclic

G:=Group<a,b,c|a^9=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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