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## G = C4×C32⋊C9order 324 = 22·34

### Direct product of C4 and C32⋊C9

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C4×C32⋊C9
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C2×C32⋊C9 — C4×C32⋊C9
 Lower central C1 — C3 — C4×C32⋊C9
 Upper central C1 — C3×C12 — C4×C32⋊C9

Generators and relations for C4×C32⋊C9
G = < a,b,c,d | a4=b3=c3=d9=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Subgroups: 123 in 69 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C9, C32, C32, C32, C12, C12, C12, C18, C3×C6, C3×C6, C3×C6, C3×C9, C33, C36, C3×C12, C3×C12, C3×C12, C3×C18, C32×C6, C32⋊C9, C3×C36, C32×C12, C2×C32⋊C9, C4×C32⋊C9
Quotients: C1, C2, C3, C4, C6, C9, C32, C12, C18, C3×C6, C3×C9, He3, 3- 1+2, C36, C3×C12, C3×C18, C2×He3, C2×3- 1+2, C32⋊C9, C3×C36, C4×He3, C4×3- 1+2, C2×C32⋊C9, C4×C32⋊C9

Smallest permutation representation of C4×C32⋊C9
On 108 points
Generators in S108
(1 99 47 104)(2 91 48 105)(3 92 49 106)(4 93 50 107)(5 94 51 108)(6 95 52 100)(7 96 53 101)(8 97 54 102)(9 98 46 103)(10 87 33 64)(11 88 34 65)(12 89 35 66)(13 90 36 67)(14 82 28 68)(15 83 29 69)(16 84 30 70)(17 85 31 71)(18 86 32 72)(19 61 37 79)(20 62 38 80)(21 63 39 81)(22 55 40 73)(23 56 41 74)(24 57 42 75)(25 58 43 76)(26 59 44 77)(27 60 45 78)
(1 4 7)(2 36 60)(3 58 31)(5 30 63)(6 61 34)(8 33 57)(9 55 28)(10 75 54)(11 52 79)(12 15 18)(13 78 48)(14 46 73)(16 81 51)(17 49 76)(19 88 100)(20 23 26)(21 108 84)(22 82 103)(24 102 87)(25 85 106)(27 105 90)(29 32 35)(37 65 95)(38 41 44)(39 94 70)(40 68 98)(42 97 64)(43 71 92)(45 91 67)(47 50 53)(56 59 62)(66 69 72)(74 77 80)(83 86 89)(93 96 99)(101 104 107)
(1 62 32)(2 63 33)(3 55 34)(4 56 35)(5 57 36)(6 58 28)(7 59 29)(8 60 30)(9 61 31)(10 48 81)(11 49 73)(12 50 74)(13 51 75)(14 52 76)(15 53 77)(16 54 78)(17 46 79)(18 47 80)(19 85 103)(20 86 104)(21 87 105)(22 88 106)(23 89 107)(24 90 108)(25 82 100)(26 83 101)(27 84 102)(37 71 98)(38 72 99)(39 64 91)(40 65 92)(41 66 93)(42 67 94)(43 68 95)(44 69 96)(45 70 97)
(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)

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

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

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

132 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3N 4A 4B 6A ··· 6H 6I ··· 6N 9A ··· 9R 12A ··· 12P 12Q ··· 12AB 18A ··· 18R 36A ··· 36AJ order 1 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 ··· 1 3 ··· 3 1 1 1 ··· 1 3 ··· 3 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

132 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + image C1 C2 C3 C3 C4 C6 C6 C9 C12 C12 C18 C36 He3 3- 1+2 C2×He3 C2×3- 1+2 C4×He3 C4×3- 1+2 kernel C4×C32⋊C9 C2×C32⋊C9 C3×C36 C32×C12 C32⋊C9 C3×C18 C32×C6 C3×C12 C3×C9 C33 C3×C6 C32 C12 C12 C6 C6 C3 C3 # reps 1 1 6 2 2 6 2 18 12 4 18 36 2 4 2 4 4 8

Matrix representation of C4×C32⋊C9 in GL4(𝔽37) generated by

 6 0 0 0 0 31 0 0 0 0 31 0 0 0 0 31
,
 10 0 0 0 0 26 0 21 0 0 10 7 0 0 0 1
,
 1 0 0 0 0 26 0 0 0 0 26 0 0 0 0 26
,
 9 0 0 0 0 21 1 0 0 3 0 0 0 12 0 16
G:=sub<GL(4,GF(37))| [6,0,0,0,0,31,0,0,0,0,31,0,0,0,0,31],[10,0,0,0,0,26,0,0,0,0,10,0,0,21,7,1],[1,0,0,0,0,26,0,0,0,0,26,0,0,0,0,26],[9,0,0,0,0,21,3,12,0,1,0,0,0,0,0,16] >;

C4×C32⋊C9 in GAP, Magma, Sage, TeX

C_4\times C_3^2\rtimes C_9
% in TeX

G:=Group("C4xC3^2:C9");
// GroupNames label

G:=SmallGroup(324,27);
// by ID

G=gap.SmallGroup(324,27);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,655,386]);
// Polycyclic

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

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