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## G = (C2×C12)⋊C8order 192 = 26·3

### 1st semidirect product of C2×C12 and C8 acting via C8/C2=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C2×C12)⋊C8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C22×C12 — C12.55D4 — (C2×C12)⋊C8
 Lower central C3 — C6 — C2×C6 — (C2×C12)⋊C8
 Upper central C1 — C22 — C22×C4 — C2×C4⋊C4

Generators and relations for (C2×C12)⋊C8
G = < a,b,c | a2=b12=c8=1, ab=ba, cac-1=ab6, cbc-1=ab-1 >

Subgroups: 168 in 78 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C22×C4, C3⋊C8, C2×C12, C2×C12, C22×C6, C22⋊C8, C2×C4⋊C4, C2×C3⋊C8, C3×C4⋊C4, C22×C12, C22.M4(2), C12.55D4, C6×C4⋊C4, (C2×C12)⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Dic3, D6, C22⋊C4, C2×C8, M4(2), C3⋊C8, C2×Dic3, C3⋊D4, C22⋊C8, C23⋊C4, C4.10D4, C2×C3⋊C8, C4.Dic3, C6.D4, C22.M4(2), C12.55D4, C23.7D6, C12.10D4, (C2×C12)⋊C8

Smallest permutation representation of (C2×C12)⋊C8
On 96 points
Generators in S96
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(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 62 19 82 37 31 51 91)(2 61 14 75 38 30 58 96)(3 72 21 80 39 29 53 89)(4 71 16 73 40 28 60 94)(5 70 23 78 41 27 55 87)(6 69 18 83 42 26 50 92)(7 68 13 76 43 25 57 85)(8 67 20 81 44 36 52 90)(9 66 15 74 45 35 59 95)(10 65 22 79 46 34 54 88)(11 64 17 84 47 33 49 93)(12 63 24 77 48 32 56 86)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + + - image C1 C2 C2 C4 C8 S3 D4 Dic3 D6 M4(2) C3⋊C8 C3⋊D4 C4.Dic3 C23⋊C4 C4.10D4 C23.7D6 C12.10D4 kernel (C2×C12)⋊C8 C12.55D4 C6×C4⋊C4 C22×C12 C2×C12 C2×C4⋊C4 C2×C12 C22×C4 C22×C4 C2×C6 C2×C4 C2×C4 C22 C6 C6 C2 C2 # reps 1 2 1 4 8 1 2 2 1 2 4 4 4 1 1 2 2

Matrix representation of (C2×C12)⋊C8 in GL8(𝔽73)

 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 27 46 0 72
,
 72 0 0 0 0 0 0 0 71 1 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 54 65 0 0 0 0 0 0 0 0 61 1 0 0 0 0 0 0 1 12 0 0 0 0 0 0 27 46 62 71 0 0 0 0 41 0 61 11
,
 51 22 0 0 0 0 0 0 0 22 0 0 0 0 0 0 0 0 25 64 0 0 0 0 0 0 45 48 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 27 46 1 71 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 27

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,27,0,0,0,0,0,1,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[72,71,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,64,54,0,0,0,0,0,0,0,65,0,0,0,0,0,0,0,0,61,1,27,41,0,0,0,0,1,12,46,0,0,0,0,0,0,0,62,61,0,0,0,0,0,0,71,11],[51,0,0,0,0,0,0,0,22,22,0,0,0,0,0,0,0,0,25,45,0,0,0,0,0,0,64,48,0,0,0,0,0,0,0,0,0,27,0,1,0,0,0,0,0,46,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,71,0,27] >;

(C2×C12)⋊C8 in GAP, Magma, Sage, TeX

(C_2\times C_{12})\rtimes C_8
% in TeX

G:=Group("(C2xC12):C8");
// GroupNames label

G:=SmallGroup(192,87);
// by ID

G=gap.SmallGroup(192,87);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,387,100,1123,6278]);
// Polycyclic

G:=Group<a,b,c|a^2=b^12=c^8=1,a*b=b*a,c*a*c^-1=a*b^6,c*b*c^-1=a*b^-1>;
// generators/relations

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