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## G = (C3×C12)⋊4C8order 288 = 25·32

### 2nd semidirect product of C3×C12 and C8 acting via C8/C2=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — (C3×C12)⋊4C8
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C2×C32⋊2C8 — (C3×C12)⋊4C8
 Lower central C32 — C3×C6 — (C3×C12)⋊4C8
 Upper central C1 — C22 — C2×C4

Generators and relations for (C3×C12)⋊4C8
G = < a,b,c | a3=b12=c8=1, ab=ba, cac-1=a-1b4, cbc-1=ab7 >

Subgroups: 272 in 68 conjugacy classes, 24 normal (18 characteristic)
C1, C2 [×3], C3 [×2], C4 [×2], C4 [×3], C22, C6 [×6], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×8], C12 [×4], C2×C6 [×2], C42, C2×C8 [×2], C3×C6 [×3], C2×Dic3 [×4], C2×C12 [×2], C4⋊C8, C3⋊Dic3 [×2], C3⋊Dic3, C3×C12 [×2], C62, C4×Dic3 [×2], C322C8 [×2], C2×C3⋊Dic3 [×2], C6×C12, C4×C3⋊Dic3, C2×C322C8 [×2], (C3×C12)⋊4C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4, Q8, C4⋊C4, C2×C8, M4(2), C4⋊C8, C32⋊C4, C322C8 [×2], C2×C32⋊C4, C32⋊M4(2), C4⋊(C32⋊C4), C2×C322C8, (C3×C12)⋊4C8

Smallest permutation representation of (C3×C12)⋊4C8
On 96 points
Generators in S96
(25 29 33)(26 30 34)(27 31 35)(28 32 36)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(85 89 93)(86 90 94)(87 91 95)(88 92 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 69 82 89 17 56 48 28)(2 64 75 88 18 51 41 27)(3 71 80 87 19 58 46 26)(4 66 73 86 20 53 39 25)(5 61 78 85 21 60 44 36)(6 68 83 96 22 55 37 35)(7 63 76 95 23 50 42 34)(8 70 81 94 24 57 47 33)(9 65 74 93 13 52 40 32)(10 72 79 92 14 59 45 31)(11 67 84 91 15 54 38 30)(12 62 77 90 16 49 43 29)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 8A ··· 8H 12A ··· 12H order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 4 4 2 2 9 9 9 9 18 18 4 ··· 4 18 ··· 18 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 4 4 4 4 type + + + + - + - + image C1 C2 C2 C4 C4 C8 D4 Q8 M4(2) C32⋊C4 C32⋊2C8 C2×C32⋊C4 C32⋊M4(2) C4⋊(C32⋊C4) kernel (C3×C12)⋊4C8 C4×C3⋊Dic3 C2×C32⋊2C8 C2×C3⋊Dic3 C6×C12 C3×C12 C3⋊Dic3 C3⋊Dic3 C3×C6 C2×C4 C4 C22 C2 C2 # reps 1 1 2 2 2 8 1 1 2 2 4 2 4 4

Matrix representation of (C3×C12)⋊4C8 in GL8(𝔽73)

 1 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 72 72 0 0 0 0 0 0 1 0
,
 1 71 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 72 2 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 72 72 0 0 0 0 1 0 1 0
,
 10 0 0 0 0 0 0 0 10 63 0 0 0 0 0 0 0 0 59 9 0 0 0 0 0 0 27 14 0 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 72 72 71 72 0 0 0 0 65 20 1 0 0 0 0 0 20 62 1 0

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[1,1,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,72,72,0,1,0,0,0,0,1,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[10,10,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,59,27,0,0,0,0,0,0,9,14,0,0,0,0,0,0,0,0,0,72,65,20,0,0,0,0,0,72,20,62,0,0,0,0,72,71,1,1,0,0,0,0,1,72,0,0] >;

(C3×C12)⋊4C8 in GAP, Magma, Sage, TeX

(C_3\times C_{12})\rtimes_4C_8
% in TeX

G:=Group("(C3xC12):4C8");
// GroupNames label

G:=SmallGroup(288,424);
// by ID

G=gap.SmallGroup(288,424);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,64,100,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c|a^3=b^12=c^8=1,a*b=b*a,c*a*c^-1=a^-1*b^4,c*b*c^-1=a*b^7>;
// generators/relations

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