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## G = C4×Dic6order 96 = 25·3

### Direct product of C4 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C4×Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C2×Dic6 — C4×Dic6
 Lower central C3 — C6 — C4×Dic6
 Upper central C1 — C2×C4 — C42

Generators and relations for C4×Dic6
G = < a,b,c | a4=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 122 in 70 conjugacy classes, 45 normal (21 characteristic)
C1, C2 [×3], C3, C4 [×4], C4 [×7], C22, C6 [×3], C2×C4 [×3], C2×C4 [×4], Q8 [×4], Dic3 [×4], Dic3 [×2], C12 [×4], C12, C2×C6, C42, C42 [×2], C4⋊C4 [×3], C2×Q8, Dic6 [×4], C2×Dic3 [×4], C2×C12 [×3], C4×Q8, C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C4×C12, C2×Dic6, C4×Dic6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, D6 [×3], C22×C4, C2×Q8, C4○D4, Dic6 [×2], C4×S3 [×2], C22×S3, C4×Q8, C2×Dic6, S3×C2×C4, C4○D12, C4×Dic6

Smallest permutation representation of C4×Dic6
Regular action on 96 points
Generators in S96
(1 91 29 46)(2 92 30 47)(3 93 31 48)(4 94 32 37)(5 95 33 38)(6 96 34 39)(7 85 35 40)(8 86 36 41)(9 87 25 42)(10 88 26 43)(11 89 27 44)(12 90 28 45)(13 61 79 57)(14 62 80 58)(15 63 81 59)(16 64 82 60)(17 65 83 49)(18 66 84 50)(19 67 73 51)(20 68 74 52)(21 69 75 53)(22 70 76 54)(23 71 77 55)(24 72 78 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)
(1 16 7 22)(2 15 8 21)(3 14 9 20)(4 13 10 19)(5 24 11 18)(6 23 12 17)(25 74 31 80)(26 73 32 79)(27 84 33 78)(28 83 34 77)(29 82 35 76)(30 81 36 75)(37 57 43 51)(38 56 44 50)(39 55 45 49)(40 54 46 60)(41 53 47 59)(42 52 48 58)(61 88 67 94)(62 87 68 93)(63 86 69 92)(64 85 70 91)(65 96 71 90)(66 95 72 89)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C4 S3 Q8 D6 C4○D4 Dic6 C4×S3 C4○D12 kernel C4×Dic6 C4×Dic3 Dic3⋊C4 C4⋊Dic3 C4×C12 C2×Dic6 Dic6 C42 C12 C2×C4 C6 C4 C4 C2 # reps 1 2 2 1 1 1 8 1 2 3 2 4 4 4

Matrix representation of C4×Dic6 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 5 0 0 0 0 5
,
 12 11 0 0 1 1 0 0 0 0 1 12 0 0 1 0
,
 8 0 0 0 5 5 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[12,1,0,0,11,1,0,0,0,0,1,1,0,0,12,0],[8,5,0,0,0,5,0,0,0,0,0,1,0,0,1,0] >;

C4×Dic6 in GAP, Magma, Sage, TeX

C_4\times {\rm Dic}_6
% in TeX

G:=Group("C4xDic6");
// GroupNames label

G:=SmallGroup(96,75);
// by ID

G=gap.SmallGroup(96,75);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,217,103,50,2309]);
// Polycyclic

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

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