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## G = C2.Dic32order 288 = 25·32

### 2nd central stem extension by C2 of Dic32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2.Dic32
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C6×C3⋊C8 — C2.Dic32
 Lower central C32 — C3×C6 — C2.Dic32
 Upper central C1 — C2×C4

Generators and relations for C2.Dic32
G = < a,b,c,d | a12=c3=1, b4=a6, d2=a9, bab-1=a5, ac=ca, ad=da, bc=cb, dbd-1=a6b, dcd-1=c-1 >

Subgroups: 266 in 95 conjugacy classes, 48 normal (10 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×6], C6 [×3], C8 [×4], C2×C4, C2×C4 [×2], C32, Dic3 [×8], C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C42, C2×C8 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×4], C24 [×4], C2×Dic3 [×6], C2×C12 [×2], C2×C12, C8⋊C4, C3⋊Dic3 [×2], C3×C12 [×2], C62, C2×C3⋊C8 [×2], C4×Dic3 [×3], C2×C24 [×2], C3×C3⋊C8 [×4], C2×C3⋊Dic3 [×2], C6×C12, C24⋊C4 [×2], C6×C3⋊C8 [×2], C4×C3⋊Dic3, C2.Dic32
Quotients: C1, C2 [×3], C4 [×6], C22, S3 [×2], C2×C4 [×3], Dic3 [×4], D6 [×2], C42, M4(2) [×2], C4×S3 [×4], C2×Dic3 [×2], C8⋊C4, S32, C8⋊S3 [×4], C4×Dic3 [×2], S3×Dic3 [×2], C6.D6, C24⋊C4 [×2], C12.31D6 [×2], Dic32, C2.Dic32

Smallest permutation representation of C2.Dic32
On 96 points
Generators in S96
(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 51 72 35 7 57 66 29)(2 56 61 28 8 50 67 34)(3 49 62 33 9 55 68 27)(4 54 63 26 10 60 69 32)(5 59 64 31 11 53 70 25)(6 52 65 36 12 58 71 30)(13 91 42 78 19 85 48 84)(14 96 43 83 20 90 37 77)(15 89 44 76 21 95 38 82)(16 94 45 81 22 88 39 75)(17 87 46 74 23 93 40 80)(18 92 47 79 24 86 41 73)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 89 93)(86 90 94)(87 91 95)(88 92 96)
(1 88 10 85 7 94 4 91)(2 89 11 86 8 95 5 92)(3 90 12 87 9 96 6 93)(13 35 22 32 19 29 16 26)(14 36 23 33 20 30 17 27)(15 25 24 34 21 31 18 28)(37 52 46 49 43 58 40 55)(38 53 47 50 44 59 41 56)(39 54 48 51 45 60 42 57)(61 76 70 73 67 82 64 79)(62 77 71 74 68 83 65 80)(63 78 72 75 69 84 66 81)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 8A ··· 8H 12A ··· 12H 12I 12J 12K 12L 24A ··· 24P order 1 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 8 ··· 8 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 4 1 1 1 1 18 18 18 18 2 ··· 2 4 4 4 6 ··· 6 2 ··· 2 4 4 4 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + - + + - + image C1 C2 C2 C4 C4 S3 Dic3 D6 M4(2) C4×S3 C4×S3 C8⋊S3 S32 S3×Dic3 C6.D6 C12.31D6 kernel C2.Dic32 C6×C3⋊C8 C4×C3⋊Dic3 C3×C3⋊C8 C2×C3⋊Dic3 C2×C3⋊C8 C3⋊C8 C2×C12 C3×C6 C12 C2×C6 C6 C2×C4 C4 C22 C2 # reps 1 2 1 8 4 2 4 2 4 4 4 16 1 2 1 4

Matrix representation of C2.Dic32 in GL6(𝔽73)

 27 0 0 0 0 0 0 27 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 0 1 0 0 0 0 46 0 0 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 49 71 0 0 0 0 19 24 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [27,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,46,0,0,0,0,1,0,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[49,19,0,0,0,0,71,24,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2.Dic32 in GAP, Magma, Sage, TeX

C_2.{\rm Dic}_3^2
% in TeX

G:=Group("C2.Dic3^2");
// GroupNames label

G:=SmallGroup(288,203);
// by ID

G=gap.SmallGroup(288,203);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,92,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^3=1,b^4=a^6,d^2=a^9,b*a*b^-1=a^5,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations

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