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## G = C2×C9⋊D4order 144 = 24·32

### Direct product of C2 and C9⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C2×C9⋊D4
 Chief series C1 — C3 — C9 — C18 — D18 — C22×D9 — C2×C9⋊D4
 Lower central C9 — C18 — C2×C9⋊D4
 Upper central C1 — C22 — C23

Generators and relations for C2×C9⋊D4
G = < a,b,c,d | a2=b9=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 295 in 81 conjugacy classes, 35 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C9, Dic3, D6, C2×C6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C18, C2×Dic3, C3⋊D4, C22×S3, C22×C6, Dic9, D18, D18, C2×C18, C2×C18, C2×C18, C2×C3⋊D4, C2×Dic9, C9⋊D4, C22×D9, C22×C18, C2×C9⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, D18, C2×C3⋊D4, C9⋊D4, C22×D9, C2×C9⋊D4

Smallest permutation representation of C2×C9⋊D4
On 72 points
Generators in S72
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 37)(8 38)(9 39)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 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 22 13 31)(2 21 14 30)(3 20 15 29)(4 19 16 28)(5 27 17 36)(6 26 18 35)(7 25 10 34)(8 24 11 33)(9 23 12 32)(37 61 46 70)(38 60 47 69)(39 59 48 68)(40 58 49 67)(41 57 50 66)(42 56 51 65)(43 55 52 64)(44 63 53 72)(45 62 54 71)
(2 9)(3 8)(4 7)(5 6)(10 16)(11 15)(12 14)(17 18)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 36)(27 35)(37 43)(38 42)(39 41)(44 45)(46 52)(47 51)(48 50)(53 54)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 72)(63 71)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A ··· 6G 9A 9B 9C 18A ··· 18U order 1 2 2 2 2 2 2 2 3 4 4 6 ··· 6 9 9 9 18 ··· 18 size 1 1 1 1 2 2 18 18 2 18 18 2 ··· 2 2 2 2 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 D9 C3⋊D4 D18 C9⋊D4 kernel C2×C9⋊D4 C2×Dic9 C9⋊D4 C22×D9 C22×C18 C22×C6 C18 C2×C6 C23 C6 C22 C2 # reps 1 1 4 1 1 1 2 3 3 4 9 12

Matrix representation of C2×C9⋊D4 in GL4(𝔽37) generated by

 36 0 0 0 0 36 0 0 0 0 1 0 0 0 0 1
,
 36 36 0 0 1 0 0 0 0 0 17 26 0 0 11 6
,
 36 0 0 0 1 1 0 0 0 0 13 35 0 0 11 24
,
 1 0 0 0 36 36 0 0 0 0 11 20 0 0 31 26
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[36,1,0,0,36,0,0,0,0,0,17,11,0,0,26,6],[36,1,0,0,0,1,0,0,0,0,13,11,0,0,35,24],[1,36,0,0,0,36,0,0,0,0,11,31,0,0,20,26] >;

C2×C9⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes D_4
% in TeX

G:=Group("C2xC9:D4");
// GroupNames label

G:=SmallGroup(144,46);
// by ID

G=gap.SmallGroup(144,46);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,218,2404,208,3461]);
// Polycyclic

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

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