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## G = C2×C32⋊2Q16order 288 = 25·32

### Direct product of C2 and C32⋊2Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C2×C32⋊2Q16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — C32⋊2Q16 — C2×C32⋊2Q16
 Lower central C32 — C3×C6 — C3×C12 — C2×C32⋊2Q16
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C322Q16
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=ebe-1=b-1, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 370 in 131 conjugacy classes, 52 normal (14 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C6 [×6], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×6], C32, Dic3 [×4], C12 [×4], C12 [×6], C2×C6 [×2], C2×C6, C2×C8, Q16 [×4], C2×Q8 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×6], Dic6 [×4], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×Q8 [×6], C2×Q16, C3×Dic3 [×4], C3×C12 [×2], C62, C2×C3⋊C8 [×3], C3⋊Q16 [×8], C2×Dic6 [×2], C6×Q8 [×2], C324C8 [×2], C3×Dic6 [×4], C3×Dic6 [×2], C6×Dic3 [×2], C6×C12, C2×C3⋊Q16 [×2], C322Q16 [×4], C2×C324C8, C6×Dic6 [×2], C2×C322Q16
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], Q16 [×2], C2×D4, C3⋊D4 [×4], C22×S3 [×2], C2×Q16, S32, C3⋊Q16 [×4], C2×C3⋊D4 [×2], D6⋊S3 [×2], C2×S32, C2×C3⋊Q16 [×2], C322Q16 [×2], C2×D6⋊S3, C2×C322Q16

Smallest permutation representation of C2×C322Q16
On 96 points
Generators in S96
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 33)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(41 84)(42 85)(43 86)(44 87)(45 88)(46 81)(47 82)(48 83)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 65)(64 66)
(1 94 44)(2 45 95)(3 96 46)(4 47 89)(5 90 48)(6 41 91)(7 92 42)(8 43 93)(9 52 85)(10 86 53)(11 54 87)(12 88 55)(13 56 81)(14 82 49)(15 50 83)(16 84 51)(17 64 74)(18 75 57)(19 58 76)(20 77 59)(21 60 78)(22 79 61)(23 62 80)(24 73 63)(25 65 33)(26 34 66)(27 67 35)(28 36 68)(29 69 37)(30 38 70)(31 71 39)(32 40 72)
(1 94 44)(2 45 95)(3 96 46)(4 47 89)(5 90 48)(6 41 91)(7 92 42)(8 43 93)(9 52 85)(10 86 53)(11 54 87)(12 88 55)(13 56 81)(14 82 49)(15 50 83)(16 84 51)(17 74 64)(18 57 75)(19 76 58)(20 59 77)(21 78 60)(22 61 79)(23 80 62)(24 63 73)(25 33 65)(26 66 34)(27 35 67)(28 68 36)(29 37 69)(30 70 38)(31 39 71)(32 72 40)
(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 79 5 75)(2 78 6 74)(3 77 7 73)(4 76 8 80)(9 25 13 29)(10 32 14 28)(11 31 15 27)(12 30 16 26)(17 95 21 91)(18 94 22 90)(19 93 23 89)(20 92 24 96)(33 56 37 52)(34 55 38 51)(35 54 39 50)(36 53 40 49)(41 64 45 60)(42 63 46 59)(43 62 47 58)(44 61 48 57)(65 81 69 85)(66 88 70 84)(67 87 71 83)(68 86 72 82)

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

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

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

42 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 ··· 12H 12I ··· 12P 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 size 1 1 1 1 2 2 4 2 2 12 12 12 12 2 ··· 2 4 4 4 18 18 18 18 4 ··· 4 12 ··· 12

42 irreducible representations

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

Matrix representation of C2×C322Q16 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0
,
 16 57 0 0 0 0 16 16 0 0 0 0 0 0 72 0 0 0 0 0 1 1 0 0 0 0 0 0 48 11 0 0 0 0 36 25
,
 8 64 0 0 0 0 64 65 0 0 0 0 0 0 72 0 0 0 0 0 1 1 0 0 0 0 0 0 43 13 0 0 0 0 60 30

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[16,16,0,0,0,0,57,16,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,48,36,0,0,0,0,11,25],[8,64,0,0,0,0,64,65,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,43,60,0,0,0,0,13,30] >;

C2×C322Q16 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_2Q_{16}
% in TeX

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

G:=SmallGroup(288,482);
// by ID

G=gap.SmallGroup(288,482);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,120,675,346,80,1356,9414]);
// 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=e*b*e^-1=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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