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## G = D4×C33⋊C2order 432 = 24·33

### Direct product of D4 and C33⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — D4×C33⋊C2
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C2×C33⋊C2 — C22×C33⋊C2 — D4×C33⋊C2
 Lower central C33 — C32×C6 — D4×C33⋊C2
 Upper central C1 — C2 — D4

Generators and relations for D4×C33⋊C2
G = < a,b,c,d,e,f | a4=b2=c3=d3=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf=c-1, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 4880 in 756 conjugacy classes, 181 normal (13 characteristic)
C1, C2, C2 [×6], C3 [×13], C4, C4, C22 [×2], C22 [×7], S3 [×52], C6 [×13], C6 [×26], C2×C4, D4, D4 [×3], C23 [×2], C32 [×13], Dic3 [×13], C12 [×13], D6 [×91], C2×C6 [×26], C2×D4, C3⋊S3 [×52], C3×C6 [×13], C3×C6 [×26], C4×S3 [×13], D12 [×13], C3⋊D4 [×26], C3×D4 [×13], C22×S3 [×26], C33, C3⋊Dic3 [×13], C3×C12 [×13], C2×C3⋊S3 [×91], C62 [×26], S3×D4 [×13], C33⋊C2 [×2], C33⋊C2 [×2], C32×C6, C32×C6 [×2], C4×C3⋊S3 [×13], C12⋊S3 [×13], C327D4 [×26], D4×C32 [×13], C22×C3⋊S3 [×26], C335C4, C32×C12, C2×C33⋊C2, C2×C33⋊C2 [×2], C2×C33⋊C2 [×4], C3×C62 [×2], D4×C3⋊S3 [×13], C4×C33⋊C2, C3312D4, C3315D4 [×2], D4×C33, C22×C33⋊C2 [×2], D4×C33⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3 [×13], D4 [×2], C23, D6 [×39], C2×D4, C3⋊S3 [×13], C22×S3 [×13], C2×C3⋊S3 [×39], S3×D4 [×13], C33⋊C2, C22×C3⋊S3 [×13], C2×C33⋊C2 [×3], D4×C3⋊S3 [×13], C22×C33⋊C2, D4×C33⋊C2

Smallest permutation representation of D4×C33⋊C2
On 108 points
Generators in S108
(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)
(1 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 16)(14 15)(17 18)(19 20)(21 22)(23 24)(25 28)(26 27)(29 30)(31 32)(33 34)(35 36)(37 38)(39 40)(41 44)(42 43)(45 48)(46 47)(49 52)(50 51)(53 54)(55 56)(57 58)(59 60)(61 64)(62 63)(65 68)(66 67)(69 72)(70 71)(73 74)(75 76)(77 78)(79 80)(81 84)(82 83)(85 88)(86 87)(89 90)(91 92)(93 96)(94 95)(97 100)(98 99)(101 104)(102 103)(105 108)(106 107)
(1 54 20)(2 55 17)(3 56 18)(4 53 19)(5 51 69)(6 52 70)(7 49 71)(8 50 72)(9 27 74)(10 28 75)(11 25 76)(12 26 73)(13 30 80)(14 31 77)(15 32 78)(16 29 79)(21 91 86)(22 92 87)(23 89 88)(24 90 85)(33 106 96)(34 107 93)(35 108 94)(36 105 95)(37 57 98)(38 58 99)(39 59 100)(40 60 97)(41 63 101)(42 64 102)(43 61 103)(44 62 104)(45 65 83)(46 66 84)(47 67 81)(48 68 82)
(1 49 60)(2 50 57)(3 51 58)(4 52 59)(5 38 18)(6 39 19)(7 40 20)(8 37 17)(9 43 85)(10 44 86)(11 41 87)(12 42 88)(13 81 34)(14 82 35)(15 83 36)(16 84 33)(21 28 62)(22 25 63)(23 26 64)(24 27 61)(29 46 106)(30 47 107)(31 48 108)(32 45 105)(53 70 100)(54 71 97)(55 72 98)(56 69 99)(65 95 78)(66 96 79)(67 93 80)(68 94 77)(73 102 89)(74 103 90)(75 104 91)(76 101 92)
(1 80 43)(2 77 44)(3 78 41)(4 79 42)(5 45 92)(6 46 89)(7 47 90)(8 48 91)(9 60 93)(10 57 94)(11 58 95)(12 59 96)(13 61 54)(14 62 55)(15 63 56)(16 64 53)(17 31 104)(18 32 101)(19 29 102)(20 30 103)(21 72 82)(22 69 83)(23 70 84)(24 71 81)(25 99 36)(26 100 33)(27 97 34)(28 98 35)(37 108 75)(38 105 76)(39 106 73)(40 107 74)(49 67 85)(50 68 86)(51 65 87)(52 66 88)
(1 3)(2 4)(5 97)(6 98)(7 99)(8 100)(9 65)(10 66)(11 67)(12 68)(13 101)(14 102)(15 103)(16 104)(17 53)(18 54)(19 55)(20 56)(21 106)(22 107)(23 108)(24 105)(25 47)(26 48)(27 45)(28 46)(29 62)(30 63)(31 64)(32 61)(33 91)(34 92)(35 89)(36 90)(37 70)(38 71)(39 72)(40 69)(41 80)(42 77)(43 78)(44 79)(49 58)(50 59)(51 60)(52 57)(73 82)(74 83)(75 84)(76 81)(85 95)(86 96)(87 93)(88 94)

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

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

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

75 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3M 4A 4B 6A ··· 6M 6N ··· 6AM 12A ··· 12M order 1 2 2 2 2 2 2 2 3 ··· 3 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 2 2 27 27 54 54 2 ··· 2 2 54 2 ··· 2 4 ··· 4 4 ··· 4

75 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D6 D6 S3×D4 kernel D4×C33⋊C2 C4×C33⋊C2 C33⋊12D4 C33⋊15D4 D4×C33 C22×C33⋊C2 D4×C32 C33⋊C2 C3×C12 C62 C32 # reps 1 1 1 2 1 2 13 2 13 26 13

Matrix representation of D4×C33⋊C2 in GL8(ℤ)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;

D4×C33⋊C2 in GAP, Magma, Sage, TeX

D_4\times C_3^3\rtimes C_2
% in TeX

G:=Group("D4xC3^3:C2");
// GroupNames label

G:=SmallGroup(432,724);
// by ID

G=gap.SmallGroup(432,724);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,1124,4037,14118]);
// Polycyclic

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

׿
×
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