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## G = C3×C12.23D4order 288 = 25·32

### Direct product of C3 and C12.23D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C12.23D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C6×D12 — C3×C12.23D4
 Lower central C3 — C2×C6 — C3×C12.23D4
 Upper central C1 — C2×C6 — C6×Q8

Generators and relations for C3×C12.23D4
G = < a,b,c,d | a3=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=b6c-1 >

Subgroups: 426 in 171 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3 [×2], C6 [×2], C6 [×4], C6 [×5], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], C32, Dic3 [×2], C12 [×4], C12 [×12], D6 [×6], C2×C6 [×2], C2×C6 [×7], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3×C6, C3×C6 [×2], D12 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×5], C3×D4 [×2], C3×Q8 [×8], C22×S3 [×2], C22×C6 [×2], C4.4D4, C3×Dic3 [×2], C3×C12 [×2], C3×C12 [×2], S3×C6 [×6], C62, C4×Dic3, D6⋊C4 [×4], C4×C12, C3×C22⋊C4 [×4], C2×D12, C6×D4, C6×Q8 [×2], C6×Q8, C3×D12 [×2], C6×Dic3 [×2], C6×C12, C6×C12 [×2], Q8×C32 [×2], S3×C2×C6 [×2], C12.23D4, C3×C4.4D4, Dic3×C12, C3×D6⋊C4 [×4], C6×D12, Q8×C3×C6, C3×C12.23D4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C4○D4 [×2], C3×S3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C22×C6, C4.4D4, S3×C6 [×3], Q83S3 [×2], C2×C3⋊D4, C6×D4, C3×C4○D4 [×2], C3×C3⋊D4 [×2], S3×C2×C6, C12.23D4, C3×C4.4D4, C3×Q83S3 [×2], C6×C3⋊D4, C3×C12.23D4

Smallest permutation representation of C3×C12.23D4
On 96 points
Generators in S96
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 89 93)(86 90 94)(87 91 95)(88 92 96)
(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 48 73 14)(2 41 74 19)(3 46 75 24)(4 39 76 17)(5 44 77 22)(6 37 78 15)(7 42 79 20)(8 47 80 13)(9 40 81 18)(10 45 82 23)(11 38 83 16)(12 43 84 21)(25 66 92 49)(26 71 93 54)(27 64 94 59)(28 69 95 52)(29 62 96 57)(30 67 85 50)(31 72 86 55)(32 65 87 60)(33 70 88 53)(34 63 89 58)(35 68 90 51)(36 61 91 56)
(1 66)(2 65)(3 64)(4 63)(5 62)(6 61)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 87)(14 86)(15 85)(16 96)(17 95)(18 94)(19 93)(20 92)(21 91)(22 90)(23 89)(24 88)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(49 73)(50 84)(51 83)(52 82)(53 81)(54 80)(55 79)(56 78)(57 77)(58 76)(59 75)(60 74)

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D4 D6 C4○D4 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×C4○D4 C3×C3⋊D4 Q8⋊3S3 C3×Q8⋊3S3 kernel C3×C12.23D4 Dic3×C12 C3×D6⋊C4 C6×D12 Q8×C3×C6 C12.23D4 C4×Dic3 D6⋊C4 C2×D12 C6×Q8 C6×Q8 C3×C12 C2×C12 C3×C6 C2×Q8 C12 C12 C2×C4 C6 C4 C6 C2 # reps 1 1 4 1 1 2 2 8 2 2 1 2 3 4 2 4 4 6 8 8 2 4

Matrix representation of C3×C12.23D4 in GL4(𝔽13) generated by

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

C3×C12.23D4 in GAP, Magma, Sage, TeX

C_3\times C_{12}._{23}D_4
% in TeX

G:=Group("C3xC12.23D4");
// GroupNames label

G:=SmallGroup(288,718);
// by ID

G=gap.SmallGroup(288,718);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,590,555,268,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=b^6*c^-1>;
// generators/relations

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