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## G = (C2×Dic3)⋊C8order 192 = 26·3

### The semidirect product of C2×Dic3 and C8 acting via C8/C2=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C2×Dic3)⋊C8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C22×C12 — C2×Dic3⋊C4 — (C2×Dic3)⋊C8
 Lower central C3 — C6 — C2×C6 — (C2×Dic3)⋊C8
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for (C2×Dic3)⋊C8
G = < a,b,c,d | a2=b6=d8=1, c2=b3, ab=ba, ac=ca, dad-1=ab3, cbc-1=b-1, bd=db, dcd-1=ab3c >

Subgroups: 208 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×5], C22 [×3], C22 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×7], C23, Dic3 [×3], C12 [×2], C2×C6 [×3], C2×C6 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4 [×2], C3⋊C8, C24, C2×Dic3 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12, C22×C6, C22⋊C8, C22⋊C8, C2×C4⋊C4, C2×C3⋊C8, Dic3⋊C4 [×2], C2×C24, C22×Dic3 [×2], C22×C12, C22.M4(2), C12.55D4, C3×C22⋊C8, C2×Dic3⋊C4, (C2×Dic3)⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4 [×2], D6, C22⋊C4, C2×C8, M4(2), C4×S3, D12, C3⋊D4, C22⋊C8, C23⋊C4, C4.10D4, S3×C8, C8⋊S3, D6⋊C4, C22.M4(2), C23.6D6, D6⋊C8, C12.47D4, (C2×Dic3)⋊C8

Smallest permutation representation of (C2×Dic3)⋊C8
On 96 points
Generators in S96
(2 85)(4 87)(6 81)(8 83)(9 43)(11 45)(13 47)(15 41)(17 92)(19 94)(21 96)(23 90)(25 33)(27 35)(29 37)(31 39)(49 73)(51 75)(53 77)(55 79)(57 68)(59 70)(61 72)(63 66)
(1 56 60 84 80 71)(2 49 61 85 73 72)(3 50 62 86 74 65)(4 51 63 87 75 66)(5 52 64 88 76 67)(6 53 57 81 77 68)(7 54 58 82 78 69)(8 55 59 83 79 70)(9 17 25 43 92 33)(10 18 26 44 93 34)(11 19 27 45 94 35)(12 20 28 46 95 36)(13 21 29 47 96 37)(14 22 30 48 89 38)(15 23 31 41 90 39)(16 24 32 42 91 40)
(1 32 84 40)(2 33 85 25)(3 34 86 26)(4 27 87 35)(5 28 88 36)(6 37 81 29)(7 38 82 30)(8 31 83 39)(9 61 43 72)(10 62 44 65)(11 66 45 63)(12 67 46 64)(13 57 47 68)(14 58 48 69)(15 70 41 59)(16 71 42 60)(17 49 92 73)(18 50 93 74)(19 75 94 51)(20 76 95 52)(21 53 96 77)(22 54 89 78)(23 79 90 55)(24 80 91 56)
(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)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - - image C1 C2 C2 C2 C4 C8 S3 D4 D6 M4(2) D12 C3⋊D4 C4×S3 S3×C8 C8⋊S3 C23⋊C4 C4.10D4 C23.6D6 C12.47D4 kernel (C2×Dic3)⋊C8 C12.55D4 C3×C22⋊C8 C2×Dic3⋊C4 C22×Dic3 C2×Dic3 C22⋊C8 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C23 C22 C22 C6 C6 C2 C2 # reps 1 1 1 1 4 8 1 2 1 2 2 2 2 4 4 1 1 2 2

Matrix representation of (C2×Dic3)⋊C8 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 1 0 0 0 0 0 0 0 1 0 0 0 0 72 1
,
 72 0 0 0 0 0 0 1 0 0 0 0 0 0 19 59 0 0 0 0 5 54 0 0 0 0 0 0 19 59 0 0 0 0 5 54
,
 0 22 0 0 0 0 51 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 7 59 0 0 0 0 14 66 0 0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,19,5,0,0,0,0,59,54,0,0,0,0,0,0,19,5,0,0,0,0,59,54],[0,51,0,0,0,0,22,0,0,0,0,0,0,0,0,0,7,14,0,0,0,0,59,66,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C2×Dic3)⋊C8 in GAP, Magma, Sage, TeX

(C_2\times {\rm Dic}_3)\rtimes C_8
% in TeX

G:=Group("(C2xDic3):C8");
// GroupNames label

G:=SmallGroup(192,28);
// by ID

G=gap.SmallGroup(192,28);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,758,100,570,6278]);
// Polycyclic

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

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