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## G = C4×C9⋊A4order 432 = 24·33

### Direct product of C4 and C9⋊A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C4×C9⋊A4
 Chief series C1 — C22 — C2×C6 — C22×C6 — C22×C18 — C2×C9⋊A4 — C4×C9⋊A4
 Lower central C22 — C2×C6 — C4×C9⋊A4
 Upper central C1 — C12 — C36

Generators and relations for C4×C9⋊A4
G = < a,b,c,d,e | a4=b9=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b7, ece-1=cd=dc, ede-1=c >

Subgroups: 186 in 69 conjugacy classes, 30 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C2×C4, C23, C9, C9, C32, C12, C12, A4, C2×C6, C2×C6, C22×C4, C18, C18, C3×C6, C2×C12, C2×A4, C22×C6, 3- 1+2, C36, C36, C3.A4, C2×C18, C2×C18, C3×C12, C3×A4, C4×A4, C22×C12, C2×3- 1+2, C2×C36, C2×C3.A4, C22×C18, C6×A4, C4×3- 1+2, C9⋊A4, C4×C3.A4, C22×C36, C12×A4, C2×C9⋊A4, C4×C9⋊A4
Quotients: C1, C2, C3, C4, C6, C32, C12, A4, C3×C6, C2×A4, 3- 1+2, C3×C12, C3×A4, C4×A4, C2×3- 1+2, C6×A4, C4×3- 1+2, C9⋊A4, C12×A4, C2×C9⋊A4, C4×C9⋊A4

Smallest permutation representation of C4×C9⋊A4
On 108 points
Generators in S108
(1 73 19 67)(2 74 20 68)(3 75 21 69)(4 76 22 70)(5 77 23 71)(6 78 24 72)(7 79 25 64)(8 80 26 65)(9 81 27 66)(10 62 103 49)(11 63 104 50)(12 55 105 51)(13 56 106 52)(14 57 107 53)(15 58 108 54)(16 59 100 46)(17 60 101 47)(18 61 102 48)(28 91 37 82)(29 92 38 83)(30 93 39 84)(31 94 40 85)(32 95 41 86)(33 96 42 87)(34 97 43 88)(35 98 44 89)(36 99 45 90)
(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)(73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)
(10 103)(11 104)(12 105)(13 106)(14 107)(15 108)(16 100)(17 101)(18 102)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)(46 59)(47 60)(48 61)(49 62)(50 63)(51 55)(52 56)(53 57)(54 58)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 97)(89 98)(90 99)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)(64 79)(65 80)(66 81)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 97)(89 98)(90 99)
(1 52 28)(2 47 35)(3 51 33)(4 46 31)(5 50 29)(6 54 36)(7 49 34)(8 53 32)(9 48 30)(10 97 79)(11 92 77)(12 96 75)(13 91 73)(14 95 80)(15 99 78)(16 94 76)(17 98 74)(18 93 81)(19 56 37)(20 60 44)(21 55 42)(22 59 40)(23 63 38)(24 58 45)(25 62 43)(26 57 41)(27 61 39)(64 103 88)(65 107 86)(66 102 84)(67 106 82)(68 101 89)(69 105 87)(70 100 85)(71 104 83)(72 108 90)

G:=sub<Sym(108)| (1,73,19,67)(2,74,20,68)(3,75,21,69)(4,76,22,70)(5,77,23,71)(6,78,24,72)(7,79,25,64)(8,80,26,65)(9,81,27,66)(10,62,103,49)(11,63,104,50)(12,55,105,51)(13,56,106,52)(14,57,107,53)(15,58,108,54)(16,59,100,46)(17,60,101,47)(18,61,102,48)(28,91,37,82)(29,92,38,83)(30,93,39,84)(31,94,40,85)(32,95,41,86)(33,96,42,87)(34,97,43,88)(35,98,44,89)(36,99,45,90), (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,100)(17,101)(18,102)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(46,59)(47,60)(48,61)(49,62)(50,63)(51,55)(52,56)(53,57)(54,58)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(64,79)(65,80)(66,81)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99), (1,52,28)(2,47,35)(3,51,33)(4,46,31)(5,50,29)(6,54,36)(7,49,34)(8,53,32)(9,48,30)(10,97,79)(11,92,77)(12,96,75)(13,91,73)(14,95,80)(15,99,78)(16,94,76)(17,98,74)(18,93,81)(19,56,37)(20,60,44)(21,55,42)(22,59,40)(23,63,38)(24,58,45)(25,62,43)(26,57,41)(27,61,39)(64,103,88)(65,107,86)(66,102,84)(67,106,82)(68,101,89)(69,105,87)(70,100,85)(71,104,83)(72,108,90)>;

G:=Group( (1,73,19,67)(2,74,20,68)(3,75,21,69)(4,76,22,70)(5,77,23,71)(6,78,24,72)(7,79,25,64)(8,80,26,65)(9,81,27,66)(10,62,103,49)(11,63,104,50)(12,55,105,51)(13,56,106,52)(14,57,107,53)(15,58,108,54)(16,59,100,46)(17,60,101,47)(18,61,102,48)(28,91,37,82)(29,92,38,83)(30,93,39,84)(31,94,40,85)(32,95,41,86)(33,96,42,87)(34,97,43,88)(35,98,44,89)(36,99,45,90), (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,100)(17,101)(18,102)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(46,59)(47,60)(48,61)(49,62)(50,63)(51,55)(52,56)(53,57)(54,58)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(64,79)(65,80)(66,81)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99), (1,52,28)(2,47,35)(3,51,33)(4,46,31)(5,50,29)(6,54,36)(7,49,34)(8,53,32)(9,48,30)(10,97,79)(11,92,77)(12,96,75)(13,91,73)(14,95,80)(15,99,78)(16,94,76)(17,98,74)(18,93,81)(19,56,37)(20,60,44)(21,55,42)(22,59,40)(23,63,38)(24,58,45)(25,62,43)(26,57,41)(27,61,39)(64,103,88)(65,107,86)(66,102,84)(67,106,82)(68,101,89)(69,105,87)(70,100,85)(71,104,83)(72,108,90) );

G=PermutationGroup([[(1,73,19,67),(2,74,20,68),(3,75,21,69),(4,76,22,70),(5,77,23,71),(6,78,24,72),(7,79,25,64),(8,80,26,65),(9,81,27,66),(10,62,103,49),(11,63,104,50),(12,55,105,51),(13,56,106,52),(14,57,107,53),(15,58,108,54),(16,59,100,46),(17,60,101,47),(18,61,102,48),(28,91,37,82),(29,92,38,83),(30,93,39,84),(31,94,40,85),(32,95,41,86),(33,96,42,87),(34,97,43,88),(35,98,44,89),(36,99,45,90)], [(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),(73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108)], [(10,103),(11,104),(12,105),(13,106),(14,107),(15,108),(16,100),(17,101),(18,102),(28,37),(29,38),(30,39),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45),(46,59),(47,60),(48,61),(49,62),(50,63),(51,55),(52,56),(53,57),(54,58),(82,91),(83,92),(84,93),(85,94),(86,95),(87,96),(88,97),(89,98),(90,99)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(28,37),(29,38),(30,39),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45),(64,79),(65,80),(66,81),(67,73),(68,74),(69,75),(70,76),(71,77),(72,78),(82,91),(83,92),(84,93),(85,94),(86,95),(87,96),(88,97),(89,98),(90,99)], [(1,52,28),(2,47,35),(3,51,33),(4,46,31),(5,50,29),(6,54,36),(7,49,34),(8,53,32),(9,48,30),(10,97,79),(11,92,77),(12,96,75),(13,91,73),(14,95,80),(15,99,78),(16,94,76),(17,98,74),(18,93,81),(19,56,37),(20,60,44),(21,55,42),(22,59,40),(23,63,38),(24,58,45),(25,62,43),(26,57,41),(27,61,39),(64,103,88),(65,107,86),(66,102,84),(67,106,82),(68,101,89),(69,105,87),(70,100,85),(71,104,83),(72,108,90)]])

80 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 18A ··· 18N 18O 18P 18Q 18R 36A ··· 36P 36Q ··· 36X order 1 2 2 2 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 9 9 9 9 9 9 12 12 12 12 12 12 12 12 12 12 12 12 18 ··· 18 18 18 18 18 36 ··· 36 36 ··· 36 size 1 1 3 3 1 1 12 12 1 1 3 3 1 1 3 3 3 3 12 12 3 3 12 12 12 12 1 1 1 1 3 3 3 3 12 12 12 12 3 ··· 3 12 12 12 12 3 ··· 3 12 ··· 12

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C3 C4 C6 C6 C6 C12 C12 C12 A4 C2×A4 3- 1+2 C3×A4 C4×A4 C2×3- 1+2 C6×A4 C9⋊A4 C4×3- 1+2 C12×A4 C2×C9⋊A4 C4×C9⋊A4 kernel C4×C9⋊A4 C2×C9⋊A4 C4×C3.A4 C22×C36 C12×A4 C9⋊A4 C2×C3.A4 C22×C18 C6×A4 C3.A4 C2×C18 C3×A4 C36 C18 C22×C4 C12 C9 C23 C6 C4 C22 C3 C2 C1 # reps 1 1 4 2 2 2 4 2 2 8 4 4 1 1 2 2 2 2 2 6 4 4 6 12

Matrix representation of C4×C9⋊A4 in GL3(𝔽37) generated by

 6 0 0 0 6 0 0 0 6
,
 9 0 0 0 12 0 17 13 16
,
 1 0 0 0 36 0 11 0 36
,
 36 0 0 0 36 0 26 25 1
,
 0 1 0 11 12 35 10 22 25
G:=sub<GL(3,GF(37))| [6,0,0,0,6,0,0,0,6],[9,0,17,0,12,13,0,0,16],[1,0,11,0,36,0,0,0,36],[36,0,26,0,36,25,0,0,1],[0,11,10,1,12,22,0,35,25] >;

C4×C9⋊A4 in GAP, Magma, Sage, TeX

C_4\times C_9\rtimes A_4
% in TeX

G:=Group("C4xC9:A4");
// GroupNames label

G:=SmallGroup(432,326);
// by ID

G=gap.SmallGroup(432,326);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,-3,-2,2,126,450,142,4548,7951]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^9=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^7,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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