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## G = (C2×C6).Q16order 192 = 26·3

### 5th non-split extension by C2×C6 of Q16 acting via Q16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C2×C6).Q16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C2×C4⋊Dic3 — (C2×C6).Q16
 Lower central C3 — C6 — C2×C12 — (C2×C6).Q16
 Upper central C1 — C22 — C22×C4 — C22⋊Q8

Generators and relations for (C2×C6).Q16
G = < a,b,c,d | a2=b6=c8=1, d2=b3c4, ab=ba, cac-1=dad-1=ab3, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 256 in 104 conjugacy classes, 43 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×8], Q8 [×2], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C22×C4, C22×C4, C2×Q8, C3⋊C8 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], C22×C6, C22⋊C8, Q8⋊C4 [×2], C2.D8 [×2], C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C6×Q8, C23.48D4, C6.Q16 [×2], C12.55D4, Q82Dic3 [×2], C2×C4⋊Dic3, C3×C22⋊Q8, (C2×C6).Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], Q16 [×2], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C22.D4, C2×Q16, C8⋊C22, C3⋊Q16 [×2], D42S3 [×2], C2×C3⋊D4, C23.48D4, C23.23D6, C2×C3⋊Q16, D4⋊D6, (C2×C6).Q16

Smallest permutation representation of (C2×C6).Q16
On 96 points
Generators in S96
(1 5)(2 9)(3 7)(4 11)(6 13)(8 15)(10 14)(12 16)(17 60)(18 22)(19 62)(20 24)(21 64)(23 58)(25 71)(26 30)(27 65)(28 32)(29 67)(31 69)(33 91)(34 38)(35 93)(36 40)(37 95)(39 89)(41 45)(42 85)(43 47)(44 87)(46 81)(48 83)(49 53)(50 80)(51 55)(52 74)(54 76)(56 78)(57 61)(59 63)(66 70)(68 72)(73 77)(75 79)(82 86)(84 88)(90 94)(92 96)
(1 77 41 12 51 88)(2 81 52 13 42 78)(3 79 43 14 53 82)(4 83 54 15 44 80)(5 73 45 16 55 84)(6 85 56 9 46 74)(7 75 47 10 49 86)(8 87 50 11 48 76)(17 67 35 64 25 89)(18 90 26 57 36 68)(19 69 37 58 27 91)(20 92 28 59 38 70)(21 71 39 60 29 93)(22 94 30 61 40 72)(23 65 33 62 31 95)(24 96 32 63 34 66)
(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 16 58)(2 22 9 57)(3 21 10 64)(4 20 11 63)(5 19 12 62)(6 18 13 61)(7 17 14 60)(8 24 15 59)(25 79 71 49)(26 78 72 56)(27 77 65 55)(28 76 66 54)(29 75 67 53)(30 74 68 52)(31 73 69 51)(32 80 70 50)(33 84 91 41)(34 83 92 48)(35 82 93 47)(36 81 94 46)(37 88 95 45)(38 87 96 44)(39 86 89 43)(40 85 90 42)

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 size 1 1 1 1 2 2 2 2 2 4 8 8 12 12 12 12 2 2 2 4 4 12 12 12 12 4 4 4 4 8 8 8 8

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - + - - + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C4○D4 Q16 C3⋊D4 C3⋊D4 C8⋊C22 D4⋊2S3 C3⋊Q16 D4⋊D6 kernel (C2×C6).Q16 C6.Q16 C12.55D4 Q8⋊2Dic3 C2×C4⋊Dic3 C3×C22⋊Q8 C22⋊Q8 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×Q8 C12 C2×C6 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 1 1 1 1 1 1 1 1 4 4 2 2 1 2 2 2

Matrix representation of (C2×C6).Q16 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 13 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 72 72 0 0 0 0 1 0 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
,
 25 62 0 0 0 0 37 48 0 0 0 0 0 0 52 65 0 0 0 0 55 21 0 0 0 0 0 0 0 32 0 0 0 0 57 32
,
 30 60 0 0 0 0 13 43 0 0 0 0 0 0 17 3 0 0 0 0 25 56 0 0 0 0 0 0 13 71 0 0 0 0 12 60

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,13,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,37,0,0,0,0,62,48,0,0,0,0,0,0,52,55,0,0,0,0,65,21,0,0,0,0,0,0,0,57,0,0,0,0,32,32],[30,13,0,0,0,0,60,43,0,0,0,0,0,0,17,25,0,0,0,0,3,56,0,0,0,0,0,0,13,12,0,0,0,0,71,60] >;

(C2×C6).Q16 in GAP, Magma, Sage, TeX

(C_2\times C_6).Q_{16}
% in TeX

G:=Group("(C2xC6).Q16");
// GroupNames label

G:=SmallGroup(192,603);
// by ID

G=gap.SmallGroup(192,603);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,336,254,219,268,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^8=1,d^2=b^3*c^4,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^3,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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