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## G = C9×C22⋊C4order 144 = 24·32

### Direct product of C9 and C22⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C9×C22⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C18 — C2×C36 — C9×C22⋊C4
 Lower central C1 — C2 — C9×C22⋊C4
 Upper central C1 — C2×C18 — C9×C22⋊C4

Generators and relations for C9×C22⋊C4
G = < a,b,c,d | a9=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Smallest permutation representation of C9×C22⋊C4
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 34)(2 35)(3 36)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 19)(17 20)(18 21)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 46)(44 47)(45 48)(55 68)(56 69)(57 70)(58 71)(59 72)(60 64)(61 65)(62 66)(63 67)
(1 44)(2 45)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 62)(11 63)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 61)(19 72)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 50)(29 51)(30 52)(31 53)(32 54)(33 46)(34 47)(35 48)(36 49)
(1 70 34 14)(2 71 35 15)(3 72 36 16)(4 64 28 17)(5 65 29 18)(6 66 30 10)(7 67 31 11)(8 68 32 12)(9 69 33 13)(19 49 59 37)(20 50 60 38)(21 51 61 39)(22 52 62 40)(23 53 63 41)(24 54 55 42)(25 46 56 43)(26 47 57 44)(27 48 58 45)

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

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

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

C9×C22⋊C4 is a maximal subgroup of
C22.D36  C23.16D18  C222Dic18  C23.8D18  Dic94D4  C223D36  C23.9D18  D18⋊D4  Dic9.D4  C22.4D36  D4×C36

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 6A ··· 6F 6G 6H 6I 6J 9A ··· 9F 12A ··· 12H 18A ··· 18R 18S ··· 18AD 36A ··· 36X order 1 2 2 2 2 2 3 3 4 4 4 4 6 ··· 6 6 6 6 6 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 1 1 2 2 2 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 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 C3 C4 C6 C6 C9 C12 C18 C18 C36 D4 C3×D4 D4×C9 kernel C9×C22⋊C4 C2×C36 C22×C18 C3×C22⋊C4 C2×C18 C2×C12 C22×C6 C22⋊C4 C2×C6 C2×C4 C23 C22 C18 C6 C2 # reps 1 2 1 2 4 4 2 6 8 12 6 24 2 4 12

Matrix representation of C9×C22⋊C4 in GL4(𝔽37) generated by

 7 0 0 0 0 1 0 0 0 0 10 0 0 0 0 10
,
 36 0 0 0 0 1 0 0 0 0 1 0 0 0 22 36
,
 1 0 0 0 0 1 0 0 0 0 36 0 0 0 0 36
,
 1 0 0 0 0 31 0 0 0 0 15 2 0 0 36 22
G:=sub<GL(4,GF(37))| [7,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[36,0,0,0,0,1,0,0,0,0,1,22,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,31,0,0,0,0,15,36,0,0,2,22] >;

C9×C22⋊C4 in GAP, Magma, Sage, TeX

C_9\times C_2^2\rtimes C_4
% in TeX

G:=Group("C9xC2^2:C4");
// GroupNames label

G:=SmallGroup(144,21);
// by ID

G=gap.SmallGroup(144,21);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,144,169,230]);
// Polycyclic

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

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