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## G = C3×C4⋊C4⋊S3order 288 = 25·32

### Direct product of C3 and C4⋊C4⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C4⋊C4⋊S3
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C3×D6⋊C4 — C3×C4⋊C4⋊S3
 Lower central C3 — C2×C6 — C3×C4⋊C4⋊S3
 Upper central C1 — C2×C6 — C3×C4⋊C4

Generators and relations for C3×C4⋊C4⋊S3
G = < a,b,c,d,e | a3=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=bc2, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 306 in 133 conjugacy classes, 58 normal (all characteristic)
C1, C2 [×3], C2, C3 [×2], C3, C4 [×6], C22, C22 [×3], S3, C6 [×6], C6 [×4], C2×C4 [×3], C2×C4 [×3], C23, C32, Dic3 [×3], C12 [×13], D6 [×3], C2×C6 [×2], C2×C6 [×4], C42, C22⋊C4 [×3], C4⋊C4, C4⋊C4 [×2], C3×S3, C3×C6 [×3], C2×Dic3 [×3], C2×C12 [×6], C2×C12 [×6], C22×S3, C22×C6, C422C2, C3×Dic3 [×3], C3×C12 [×3], S3×C6 [×3], C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4 [×3], C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×2], C3×C4⋊C4 [×3], C6×Dic3 [×3], C6×C12 [×3], S3×C2×C6, C4⋊C4⋊S3, C3×C422C2, Dic3×C12, C3×Dic3⋊C4, C3×C4⋊Dic3, C3×D6⋊C4 [×3], C32×C4⋊C4, C3×C4⋊C4⋊S3
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], C23, D6 [×3], C2×C6 [×7], C4○D4 [×3], C3×S3, C22×S3, C22×C6, C422C2, S3×C6 [×3], C4○D12, D42S3, Q83S3, C3×C4○D4 [×3], S3×C2×C6, C4⋊C4⋊S3, C3×C422C2, C3×C4○D12, C3×D42S3, C3×Q83S3, C3×C4⋊C4⋊S3

Smallest permutation representation of C3×C4⋊C4⋊S3
On 96 points
Generators in S96
(1 35 39)(2 36 40)(3 33 37)(4 34 38)(5 10 14)(6 11 15)(7 12 16)(8 9 13)(17 22 89)(18 23 90)(19 24 91)(20 21 92)(25 45 30)(26 46 31)(27 47 32)(28 48 29)(41 50 64)(42 51 61)(43 52 62)(44 49 63)(53 68 70)(54 65 71)(55 66 72)(56 67 69)(57 74 87)(58 75 88)(59 76 85)(60 73 86)(77 95 84)(78 96 81)(79 93 82)(80 94 83)
(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)
(1 45 5 42)(2 48 6 41)(3 47 7 44)(4 46 8 43)(9 52 34 31)(10 51 35 30)(11 50 36 29)(12 49 33 32)(13 62 38 26)(14 61 39 25)(15 64 40 28)(16 63 37 27)(17 73 81 53)(18 76 82 56)(19 75 83 55)(20 74 84 54)(21 87 77 65)(22 86 78 68)(23 85 79 67)(24 88 80 66)(57 95 71 92)(58 94 72 91)(59 93 69 90)(60 96 70 89)
(1 39 35)(2 40 36)(3 37 33)(4 38 34)(5 14 10)(6 15 11)(7 16 12)(8 13 9)(17 22 89)(18 23 90)(19 24 91)(20 21 92)(25 30 45)(26 31 46)(27 32 47)(28 29 48)(41 64 50)(42 61 51)(43 62 52)(44 63 49)(53 68 70)(54 65 71)(55 66 72)(56 67 69)(57 74 87)(58 75 88)(59 76 85)(60 73 86)(77 95 84)(78 96 81)(79 93 82)(80 94 83)
(1 71)(2 58)(3 69)(4 60)(5 57)(6 72)(7 59)(8 70)(9 53)(10 74)(11 55)(12 76)(13 68)(14 87)(15 66)(16 85)(17 50)(18 30)(19 52)(20 32)(21 27)(22 64)(23 25)(24 62)(26 80)(28 78)(29 81)(31 83)(33 56)(34 73)(35 54)(36 75)(37 67)(38 86)(39 65)(40 88)(41 89)(42 93)(43 91)(44 95)(45 90)(46 94)(47 92)(48 96)(49 84)(51 82)(61 79)(63 77)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A ··· 6F 6G ··· 6O 6P 6Q 12A 12B 12C 12D 12E ··· 12Z 12AA ··· 12AH 12AI 12AJ order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 12 12 size 1 1 1 1 12 1 1 2 2 2 2 2 4 4 6 6 6 6 12 1 ··· 1 2 ··· 2 12 12 2 2 2 2 4 ··· 4 6 ··· 6 12 12

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D6 C4○D4 C3×S3 S3×C6 C4○D12 C3×C4○D4 C3×C4○D12 D4⋊2S3 Q8⋊3S3 C3×D4⋊2S3 C3×Q8⋊3S3 kernel C3×C4⋊C4⋊S3 Dic3×C12 C3×Dic3⋊C4 C3×C4⋊Dic3 C3×D6⋊C4 C32×C4⋊C4 C4⋊C4⋊S3 C4×Dic3 Dic3⋊C4 C4⋊Dic3 D6⋊C4 C3×C4⋊C4 C3×C4⋊C4 C2×C12 C3×C6 C4⋊C4 C2×C4 C6 C6 C2 C6 C6 C2 C2 # reps 1 1 1 1 3 1 2 2 2 2 6 2 1 3 6 2 6 4 12 8 1 1 2 2

Matrix representation of C3×C4⋊C4⋊S3 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 10 1 0 0 0 0 0 1 0 0 12 0
,
 8 0 0 0 0 8 0 0 0 0 0 1 0 0 1 0
,
 3 0 0 0 4 9 0 0 0 0 1 0 0 0 0 1
,
 10 2 0 0 9 3 0 0 0 0 0 5 0 0 8 0
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[12,10,0,0,0,1,0,0,0,0,0,12,0,0,1,0],[8,0,0,0,0,8,0,0,0,0,0,1,0,0,1,0],[3,4,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[10,9,0,0,2,3,0,0,0,0,0,8,0,0,5,0] >;

C3×C4⋊C4⋊S3 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes C_4\rtimes S_3
% in TeX

G:=Group("C3xC4:C4:S3");
// GroupNames label

G:=SmallGroup(288,669);
// by ID

G=gap.SmallGroup(288,669);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,176,590,555,268,9414]);
// Polycyclic

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

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