Copied to
clipboard

## G = C3×Q16⋊S3order 288 = 25·32

### Direct product of C3 and Q16⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×Q16⋊S3
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×S3×Q8 — C3×Q16⋊S3
 Lower central C3 — C6 — C12 — C3×Q16⋊S3
 Upper central C1 — C6 — C12 — C3×Q16

Generators and relations for C3×Q16⋊S3
G = < a,b,c,d,e | a3=b8=d3=e2=1, c2=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=b5, cd=dc, ece=b4c, ede=d-1 >

Subgroups: 314 in 129 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×4], C22 [×2], S3 [×2], C6 [×2], C6 [×3], C8, C8, C2×C4 [×3], D4 [×2], Q8 [×2], Q8 [×2], C32, Dic3, Dic3, C12 [×2], C12 [×9], D6, D6, C2×C6 [×2], M4(2), SD16 [×2], Q16, Q16, C2×Q8, C4○D4, C3×S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6, Dic6, C4×S3, C4×S3 [×2], D12, D12, C2×C12 [×3], C3×D4 [×2], C3×Q8 [×4], C3×Q8 [×4], C8.C22, C3×Dic3, C3×Dic3, C3×C12, C3×C12 [×2], S3×C6, S3×C6, C8⋊S3, C24⋊C2, Q82S3, C3⋊Q16, C3×M4(2), C3×SD16 [×2], C3×Q16 [×2], C3×Q16 [×2], S3×Q8, Q83S3, C6×Q8, C3×C4○D4, C3×C3⋊C8, C3×C24, C3×Dic6, C3×Dic6, S3×C12, S3×C12 [×2], C3×D12, C3×D12, Q8×C32 [×2], Q16⋊S3, C3×C8.C22, C3×C8⋊S3, C3×C24⋊C2, C3×Q82S3, C3×C3⋊Q16, C32×Q16, C3×S3×Q8, C3×Q83S3, C3×Q16⋊S3
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3×D4 [×2], C22×S3, C22×C6, C8.C22, S3×C6 [×3], S3×D4, C6×D4, S3×C2×C6, Q16⋊S3, C3×C8.C22, C3×S3×D4, C3×Q16⋊S3

Smallest permutation representation of C3×Q16⋊S3
On 96 points
Generators in S96
(1 73 49)(2 74 50)(3 75 51)(4 76 52)(5 77 53)(6 78 54)(7 79 55)(8 80 56)(9 90 57)(10 91 58)(11 92 59)(12 93 60)(13 94 61)(14 95 62)(15 96 63)(16 89 64)(17 65 28)(18 66 29)(19 67 30)(20 68 31)(21 69 32)(22 70 25)(23 71 26)(24 72 27)(33 85 42)(34 86 43)(35 87 44)(36 88 45)(37 81 46)(38 82 47)(39 83 48)(40 84 41)
(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 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 85 13 81)(10 84 14 88)(11 83 15 87)(12 82 16 86)(25 54 29 50)(26 53 30 49)(27 52 31 56)(28 51 32 55)(33 61 37 57)(34 60 38 64)(35 59 39 63)(36 58 40 62)(41 95 45 91)(42 94 46 90)(43 93 47 89)(44 92 48 96)(65 75 69 79)(66 74 70 78)(67 73 71 77)(68 80 72 76)
(1 49 73)(2 50 74)(3 51 75)(4 52 76)(5 53 77)(6 54 78)(7 55 79)(8 56 80)(9 90 57)(10 91 58)(11 92 59)(12 93 60)(13 94 61)(14 95 62)(15 96 63)(16 89 64)(17 28 65)(18 29 66)(19 30 67)(20 31 68)(21 32 69)(22 25 70)(23 26 71)(24 27 72)(33 85 42)(34 86 43)(35 87 44)(36 88 45)(37 81 46)(38 82 47)(39 83 48)(40 84 41)
(1 33)(2 38)(3 35)(4 40)(5 37)(6 34)(7 39)(8 36)(9 71)(10 68)(11 65)(12 70)(13 67)(14 72)(15 69)(16 66)(17 59)(18 64)(19 61)(20 58)(21 63)(22 60)(23 57)(24 62)(25 93)(26 90)(27 95)(28 92)(29 89)(30 94)(31 91)(32 96)(41 52)(42 49)(43 54)(44 51)(45 56)(46 53)(47 50)(48 55)(73 85)(74 82)(75 87)(76 84)(77 81)(78 86)(79 83)(80 88)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 6I 8A 8B 12A 12B 12C ··· 12I 12J 12K 12L ··· 12Q 12R 12S 24A ··· 24H 24I 24J order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 6 6 6 6 6 6 8 8 12 12 12 ··· 12 12 12 12 ··· 12 12 12 24 ··· 24 24 24 size 1 1 6 12 1 1 2 2 2 2 4 4 6 12 1 1 2 2 2 6 6 12 12 4 12 2 2 4 ··· 4 6 6 8 ··· 8 12 12 4 ··· 4 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 C3×S3 C3×D4 C3×D4 S3×C6 S3×C6 C8.C22 S3×D4 Q16⋊S3 C3×C8.C22 C3×S3×D4 C3×Q16⋊S3 kernel C3×Q16⋊S3 C3×C8⋊S3 C3×C24⋊C2 C3×Q8⋊2S3 C3×C3⋊Q16 C32×Q16 C3×S3×Q8 C3×Q8⋊3S3 Q16⋊S3 C8⋊S3 C24⋊C2 Q8⋊2S3 C3⋊Q16 C3×Q16 S3×Q8 Q8⋊3S3 C3×Q16 C3×Dic3 S3×C6 C24 C3×Q8 Q16 Dic3 D6 C8 Q8 C32 C6 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 4 1 1 2 2 2 4

Matrix representation of C3×Q16⋊S3 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 1 3 3 3 1 6 5 3 5 4 1 1 4 0 4 6
,
 4 4 4 4 6 3 3 5 5 3 6 4 4 4 1 1
,
 2 0 0 0 6 5 1 1 0 5 2 5 1 6 6 3
,
 5 4 2 0 4 2 6 1 1 0 4 5 1 2 5 3
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,1,5,4,3,6,4,0,3,5,1,4,3,3,1,6],[4,6,5,4,4,3,3,4,4,3,6,1,4,5,4,1],[2,6,0,1,0,5,5,6,0,1,2,6,0,1,5,3],[5,4,1,1,4,2,0,2,2,6,4,5,0,1,5,3] >;

C3×Q16⋊S3 in GAP, Magma, Sage, TeX

C_3\times Q_{16}\rtimes S_3
% in TeX

G:=Group("C3xQ16:S3");
// GroupNames label

G:=SmallGroup(288,689);
// by ID

G=gap.SmallGroup(288,689);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,1094,303,268,1271,648,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^8=d^3=e^2=1,c^2=b^4,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^5,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽