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## G = D4×C33order 216 = 23·33

### Direct product of C33 and D4

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

Aliases: D4×C33, C6210C6, C6.12C62, C4⋊(C32×C6), C123(C3×C6), (C3×C12)⋊9C6, (C3×C62)⋊1C2, (C32×C12)⋊7C2, C2.1(C3×C62), C222(C32×C6), (C32×C6).31C22, (C2×C6)⋊3(C3×C6), (C3×C6).36(C2×C6), SmallGroup(216,151)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C33
 Chief series C1 — C2 — C6 — C3×C6 — C32×C6 — C3×C62 — D4×C33
 Lower central C1 — C2 — D4×C33
 Upper central C1 — C32×C6 — D4×C33

Generators and relations for D4×C33
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 280 in 224 conjugacy classes, 168 normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, D4, C32, C12, C2×C6, C3×C6, C3×C6, C3×D4, C33, C3×C12, C62, C32×C6, C32×C6, D4×C32, C32×C12, C3×C62, D4×C33
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, C3×C6, C3×D4, C33, C62, C32×C6, D4×C32, C3×C62, D4×C33

Smallest permutation representation of D4×C33
On 108 points
Generators in S108
(1 56 66)(2 53 67)(3 54 68)(4 55 65)(5 108 96)(6 105 93)(7 106 94)(8 107 95)(9 76 71)(10 73 72)(11 74 69)(12 75 70)(13 37 104)(14 38 101)(15 39 102)(16 40 103)(17 60 79)(18 57 80)(19 58 77)(20 59 78)(21 33 100)(22 34 97)(23 35 98)(24 36 99)(25 91 87)(26 92 88)(27 89 85)(28 90 86)(29 43 82)(30 44 83)(31 41 84)(32 42 81)(45 64 49)(46 61 50)(47 62 51)(48 63 52)
(1 85 61)(2 86 62)(3 87 63)(4 88 64)(5 10 33)(6 11 34)(7 12 35)(8 9 36)(13 18 41)(14 19 42)(15 20 43)(16 17 44)(21 96 72)(22 93 69)(23 94 70)(24 95 71)(25 52 54)(26 49 55)(27 50 56)(28 51 53)(29 102 78)(30 103 79)(31 104 80)(32 101 77)(37 57 84)(38 58 81)(39 59 82)(40 60 83)(45 65 92)(46 66 89)(47 67 90)(48 68 91)(73 100 108)(74 97 105)(75 98 106)(76 99 107)
(1 81 6)(2 82 7)(3 83 8)(4 84 5)(9 87 40)(10 88 37)(11 85 38)(12 86 39)(13 72 92)(14 69 89)(15 70 90)(16 71 91)(17 24 48)(18 21 45)(19 22 46)(20 23 47)(25 103 76)(26 104 73)(27 101 74)(28 102 75)(29 106 53)(30 107 54)(31 108 55)(32 105 56)(33 64 57)(34 61 58)(35 62 59)(36 63 60)(41 96 65)(42 93 66)(43 94 67)(44 95 68)(49 80 100)(50 77 97)(51 78 98)(52 79 99)
(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)
(2 4)(5 7)(10 12)(13 15)(18 20)(21 23)(26 28)(29 31)(33 35)(37 39)(41 43)(45 47)(49 51)(53 55)(57 59)(62 64)(65 67)(70 72)(73 75)(78 80)(82 84)(86 88)(90 92)(94 96)(98 100)(102 104)(106 108)

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

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

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

D4×C33 is a maximal subgroup of   C3315D8  C3324SD16  C62.100D6

135 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3Z 4 6A ··· 6Z 6AA ··· 6BZ 12A ··· 12Z order 1 2 2 2 3 ··· 3 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 2 2 1 ··· 1 2 1 ··· 1 2 ··· 2 2 ··· 2

135 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C3 C6 C6 D4 C3×D4 kernel D4×C33 C32×C12 C3×C62 D4×C32 C3×C12 C62 C33 C32 # reps 1 1 2 26 26 52 1 26

Matrix representation of D4×C33 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 1 0 0 11 12
,
 12 0 0 0 0 12 0 0 0 0 1 1 0 0 0 12
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,11,0,0,1,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,1,12] >;

D4×C33 in GAP, Magma, Sage, TeX

D_4\times C_3^3
% in TeX

G:=Group("D4xC3^3");
// GroupNames label

G:=SmallGroup(216,151);
// by ID

G=gap.SmallGroup(216,151);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-2,1321]);
// Polycyclic

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

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