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## G = (C2×C6)⋊8Q16order 192 = 26·3

### 2nd semidirect product of C2×C6 and Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C2×C6)⋊8Q16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C12.48D4 — (C2×C6)⋊8Q16
 Lower central C3 — C6 — C2×C12 — (C2×C6)⋊8Q16
 Upper central C1 — C22 — C22×C4 — C22×Q8

Generators and relations for (C2×C6)⋊8Q16
G = < a,b,c,d | a2=b6=c8=1, d2=c4, ab=ba, cac-1=ab3, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 328 in 148 conjugacy classes, 51 normal (25 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×7], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×10], Q8 [×4], Q8 [×8], C23, Dic3 [×2], C12 [×2], C12 [×5], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], Q16 [×4], C22×C4, C22×C4, C2×Q8 [×2], C2×Q8 [×6], C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×8], C3×Q8 [×4], C3×Q8 [×6], C22×C6, C22⋊C8, Q8⋊C4 [×2], C22⋊Q8, C2×Q16 [×2], C22×Q8, C2×C3⋊C8 [×2], Dic3⋊C4, C4⋊Dic3, C3⋊Q16 [×4], C6.D4, C2×Dic6, C22×C12, C22×C12, C6×Q8 [×2], C6×Q8 [×5], C22⋊Q16, C12.55D4, Q82Dic3 [×2], C12.48D4, C2×C3⋊Q16 [×2], Q8×C2×C6, (C2×C6)⋊8Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], Q16 [×2], C2×D4 [×3], C3⋊D4 [×6], C22×S3, C22≀C2, C2×Q16, C8.C22, C3⋊Q16 [×2], C2×C3⋊D4 [×3], C22⋊Q16, Q8.11D6, C2×C3⋊Q16, C244S3, (C2×C6)⋊8Q16

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

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C ··· 4G 4H 4I 6A ··· 6G 8A 8B 8C 8D 12A ··· 12L order 1 2 2 2 2 2 3 4 4 4 ··· 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 1 1 2 2 2 2 2 4 ··· 4 24 24 2 ··· 2 12 12 12 12 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 Q16 C3⋊D4 C3⋊D4 C3⋊D4 C8.C22 C3⋊Q16 Q8.11D6 kernel (C2×C6)⋊8Q16 C12.55D4 Q8⋊2Dic3 C12.48D4 C2×C3⋊Q16 Q8×C2×C6 C22×Q8 C2×C12 C3×Q8 C22×C6 C22×C4 C2×Q8 C2×C6 C2×C4 Q8 C23 C6 C22 C2 # reps 1 1 2 1 2 1 1 1 4 1 1 2 4 2 8 2 1 2 2

Matrix representation of (C2×C6)⋊8Q16 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 72
,
 1 0 0 0 0 1 0 0 0 0 65 0 0 0 0 9
,
 57 16 0 0 57 57 0 0 0 0 0 1 0 0 72 0
,
 61 1 0 0 1 12 0 0 0 0 1 0 0 0 0 72
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,65,0,0,0,0,9],[57,57,0,0,16,57,0,0,0,0,0,72,0,0,1,0],[61,1,0,0,1,12,0,0,0,0,1,0,0,0,0,72] >;

(C2×C6)⋊8Q16 in GAP, Magma, Sage, TeX

(C_2\times C_6)\rtimes_8Q_{16}
% in TeX

G:=Group("(C2xC6):8Q16");
// GroupNames label

G:=SmallGroup(192,787);
// by ID

G=gap.SmallGroup(192,787);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,184,1123,297,136,6278]);
// Polycyclic

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

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