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## G = C2×C32⋊3Q16order 288 = 25·32

### Direct product of C2 and C32⋊3Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C2×C32⋊3Q16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — C32⋊3Q16 — C2×C32⋊3Q16
 Lower central C32 — C3×C6 — C3×C12 — C2×C32⋊3Q16
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C323Q16
G = < a,b,c,d,e | a2=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 482 in 139 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C6 [×2], C6 [×4], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×6], C32, Dic3 [×10], C12 [×4], C12 [×4], C2×C6 [×2], C2×C6, C2×C8, Q16 [×4], C2×Q8 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6 [×11], C2×Dic3 [×5], C2×C12 [×2], C2×C12 [×2], C3×Q8 [×3], C2×Q16, C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], C62, Dic12 [×4], C2×C3⋊C8, C3⋊Q16 [×4], C2×C24, C2×Dic6, C2×Dic6 [×3], C6×Q8, C3×C3⋊C8 [×2], C3×Dic6 [×2], C3×Dic6, C6×Dic3, C324Q8 [×2], C324Q8, C2×C3⋊Dic3, C6×C12, C2×Dic12, C2×C3⋊Q16, C323Q16 [×4], C6×C3⋊C8, C6×Dic6, C2×C324Q8, C2×C323Q16
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], Q16 [×2], C2×D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C2×Q16, S32, Dic12 [×2], C3⋊Q16 [×2], C2×D12, C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C2×Dic12, C2×C3⋊Q16, C323Q16 [×2], C2×C3⋊D12, C2×C323Q16

Smallest permutation representation of C2×C323Q16
On 96 points
Generators in S96
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 89)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 81)
(1 49 45)(2 46 50)(3 51 47)(4 48 52)(5 53 41)(6 42 54)(7 55 43)(8 44 56)(9 95 71)(10 72 96)(11 89 65)(12 66 90)(13 91 67)(14 68 92)(15 93 69)(16 70 94)(17 86 73)(18 74 87)(19 88 75)(20 76 81)(21 82 77)(22 78 83)(23 84 79)(24 80 85)(25 64 34)(26 35 57)(27 58 36)(28 37 59)(29 60 38)(30 39 61)(31 62 40)(32 33 63)
(1 49 45)(2 50 46)(3 51 47)(4 52 48)(5 53 41)(6 54 42)(7 55 43)(8 56 44)(9 71 95)(10 72 96)(11 65 89)(12 66 90)(13 67 91)(14 68 92)(15 69 93)(16 70 94)(17 86 73)(18 87 74)(19 88 75)(20 81 76)(21 82 77)(22 83 78)(23 84 79)(24 85 80)(25 64 34)(26 57 35)(27 58 36)(28 59 37)(29 60 38)(30 61 39)(31 62 40)(32 63 33)
(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 33 5 37)(2 40 6 36)(3 39 7 35)(4 38 8 34)(9 23 13 19)(10 22 14 18)(11 21 15 17)(12 20 16 24)(25 48 29 44)(26 47 30 43)(27 46 31 42)(28 45 32 41)(49 63 53 59)(50 62 54 58)(51 61 55 57)(52 60 56 64)(65 77 69 73)(66 76 70 80)(67 75 71 79)(68 74 72 78)(81 94 85 90)(82 93 86 89)(83 92 87 96)(84 91 88 95)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 24A ··· 24H order 1 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 4 2 2 12 12 36 36 2 ··· 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 12 12 12 12 6 ··· 6

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + - + + - + - + + + - image C1 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 Q16 D12 C3⋊D4 D12 C3⋊D4 Dic12 S32 C3⋊Q16 C3⋊D12 C2×S32 C3⋊D12 C32⋊3Q16 kernel C2×C32⋊3Q16 C32⋊3Q16 C6×C3⋊C8 C6×Dic6 C2×C32⋊4Q8 C2×C3⋊C8 C2×Dic6 C3×C12 C62 C3⋊C8 Dic6 C2×C12 C3×C6 C12 C12 C2×C6 C2×C6 C6 C2×C4 C6 C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 1 2 2 2 4 2 2 2 2 8 1 2 1 1 1 4

Matrix representation of C2×C323Q16 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 0 72 0 0 1 72 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 72
,
 0 72 0 0 72 0 0 0 0 0 23 5 0 0 68 18
,
 72 0 0 0 0 72 0 0 0 0 36 11 0 0 48 37
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,72],[0,72,0,0,72,0,0,0,0,0,23,68,0,0,5,18],[72,0,0,0,0,72,0,0,0,0,36,48,0,0,11,37] >;

C2×C323Q16 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_3Q_{16}
% in TeX

G:=Group("C2xC3^2:3Q16");
// GroupNames label

G:=SmallGroup(288,483);
// by ID

G=gap.SmallGroup(288,483);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,176,675,80,1356,9414]);
// Polycyclic

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

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