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G = C4×Dic6order 96 = 25·3

Direct product of C4 and Dic6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×Dic6, C123Q8, C42.3S3, C31(C4×Q8), C4.9(C4×S3), C6.1(C2×Q8), (C4×C12).5C2, C4(C4⋊Dic3), (C2×C4).72D6, C12.19(C2×C4), C4(Dic3⋊C4), C6.1(C4○D4), C6.1(C22×C4), (C2×C6).9C23, C2.1(C2×Dic6), C2.1(C4○D12), Dic3⋊C4.7C2, C4⋊Dic3.13C2, (C4×Dic3).7C2, Dic3.1(C2×C4), (C2×C12).84C22, (C2×Dic6).10C2, C22.8(C22×S3), (C2×Dic3).23C22, C2.4(S3×C2×C4), SmallGroup(96,75)

Series: Derived Chief Lower central Upper central

C1C6 — C4×Dic6
C1C3C6C2×C6C2×Dic3C2×Dic6 — C4×Dic6
C3C6 — C4×Dic6
C1C2×C4C42

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)])

C4×Dic6 is a maximal subgroup of
C4.8Dic12  Dic62C8  C2412Q8  C24⋊Q8  C42.16D6  Dic12⋊C4  C42.27D6  Dic6.3Q8  Dic6⋊C8  C42.198D6  C42.36D6  Dic68D4  C4⋊Dic12  Dic63Q8  Dic64Q8  C42.51D6  C42.59D6  C42.61D6  Dic6.4Q8  Dic69D4  C12⋊Q16  Dic65Q8  Dic66Q8  C42.274D6  C42.277D6  C42.87D6  C42.88D6  C42.89D6  C42.91D6  C42.93D6  C42.96D6  C42.98D6  C42.99D6  C42.102D6  D45Dic6  C42.105D6  C42.106D6  D46Dic6  C42.108D6  Dic623D4  Dic624D4  C42.229D6  C42.114D6  C42.115D6  Dic610Q8  C42.122D6  Q86Dic6  Q87Dic6  C4×S3×Q8  C42.125D6  C42.232D6  C42.134D6  C42.135D6  C42.136D6  C42.137D6  C42.139D6  Dic610D4  C42.143D6  Dic67Q8  D127Q8  C42.152D6  C42.154D6  C42.159D6  C42.160D6  C42.162D6  C42.164D6  C42.166D6  Dic611D4  Dic68Q8  Dic69Q8  D128Q8  D129Q8  C42.177D6  Dic35Dic6  Dic36Dic6  Dic55Dic6  Dic3017C4
C4×Dic6 is a maximal quotient of
(C2×C12)⋊Q8  C6.(C4×Q8)  C2.(C4×Dic6)  Dic3⋊C4⋊C4  C2412Q8  C24⋊Q8  C124(C4⋊C4)  (C2×Dic6)⋊7C4  (C2×C42).6S3  Dic35Dic6  Dic36Dic6  Dic55Dic6  Dic3017C4

36 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I···4P6A6B6C12A···12L
order12223444444444···466612···12
size11112111122226···62222···2

36 irreducible representations

dim11111112222222
type+++++++-+-
imageC1C2C2C2C2C2C4S3Q8D6C4○D4Dic6C4×S3C4○D12
kernelC4×Dic6C4×Dic3Dic3⋊C4C4⋊Dic3C4×C12C2×Dic6Dic6C42C12C2×C4C6C4C4C2
# reps12211181232444

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

12000
01200
0050
0005
,
121100
1100
00112
0010
,
8000
5500
0001
0010
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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