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G = C2×C4⋊Dic3order 96 = 25·3

Direct product of C2 and C4⋊Dic3

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

Aliases: C2×C4⋊Dic3, C23.35D6, C22.15D12, C22.5Dic6, C62(C4⋊C4), C127(C2×C4), (C2×C12)⋊5C4, C6.9(C2×Q8), (C2×C6).6Q8, (C2×C4)⋊3Dic3, C42(C2×Dic3), C6.15(C2×D4), C2.2(C2×D12), (C2×C6).20D4, (C2×C4).84D6, (C22×C4).8S3, C2.3(C2×Dic6), (C2×C6).43C23, C6.23(C22×C4), (C22×C12).7C2, (C2×C12).92C22, C2.4(C22×Dic3), (C22×C6).35C22, C22.21(C22×S3), (C22×Dic3).5C2, C22.14(C2×Dic3), (C2×Dic3).34C22, C33(C2×C4⋊C4), (C2×C6).34(C2×C4), SmallGroup(96,132)

Series: Derived Chief Lower central Upper central

C1C6 — C2×C4⋊Dic3
C1C3C6C2×C6C2×Dic3C22×Dic3 — C2×C4⋊Dic3
C3C6 — C2×C4⋊Dic3
C1C23C22×C4

Generators and relations for C2×C4⋊Dic3
 G = < a,b,c,d | a2=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 162 in 92 conjugacy classes, 65 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×4], C22, C22 [×6], C6 [×3], C6 [×4], C2×C4 [×6], C2×C4 [×8], C23, Dic3 [×4], C12 [×4], C2×C6, C2×C6 [×6], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×6], C22×C6, C2×C4⋊C4, C4⋊Dic3 [×4], C22×Dic3 [×2], C22×C12, C2×C4⋊Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, Dic3 [×4], D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic6 [×2], D12 [×2], C2×Dic3 [×6], C22×S3, C2×C4⋊C4, C4⋊Dic3 [×4], C2×Dic6, C2×D12, C22×Dic3, C2×C4⋊Dic3

Smallest permutation representation of C2×C4⋊Dic3
Regular action on 96 points
Generators in S96
(1 38)(2 39)(3 40)(4 41)(5 42)(6 37)(7 69)(8 70)(9 71)(10 72)(11 67)(12 68)(13 35)(14 36)(15 31)(16 32)(17 33)(18 34)(19 52)(20 53)(21 54)(22 49)(23 50)(24 51)(25 47)(26 48)(27 43)(28 44)(29 45)(30 46)(55 86)(56 87)(57 88)(58 89)(59 90)(60 85)(61 84)(62 79)(63 80)(64 81)(65 82)(66 83)(73 96)(74 91)(75 92)(76 93)(77 94)(78 95)
(1 29 17 24)(2 30 18 19)(3 25 13 20)(4 26 14 21)(5 27 15 22)(6 28 16 23)(7 84 96 88)(8 79 91 89)(9 80 92 90)(10 81 93 85)(11 82 94 86)(12 83 95 87)(31 49 42 43)(32 50 37 44)(33 51 38 45)(34 52 39 46)(35 53 40 47)(36 54 41 48)(55 67 65 77)(56 68 66 78)(57 69 61 73)(58 70 62 74)(59 71 63 75)(60 72 64 76)
(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 57 4 60)(2 56 5 59)(3 55 6 58)(7 54 10 51)(8 53 11 50)(9 52 12 49)(13 65 16 62)(14 64 17 61)(15 63 18 66)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 77 28 74)(26 76 29 73)(27 75 30 78)(31 80 34 83)(32 79 35 82)(33 84 36 81)(37 89 40 86)(38 88 41 85)(39 87 42 90)(43 92 46 95)(44 91 47 94)(45 96 48 93)

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

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

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

C2×C4⋊Dic3 is a maximal subgroup of
C12.C42  C12.9C42  M4(2)⋊Dic3  C6.(C4×Q8)  C2.(C4×D12)  C2.(C4×Dic6)  Dic3⋊C4⋊C4  (C2×C4)⋊Dic6  C6.(C4⋊Q8)  (C2×C4).17D12  (C2×C4).Dic6  (C22×C4).85D6  D6⋊C4⋊C4  D6⋊C43C4  (C2×C4).21D12  (C2×C12).33D4  C23.39D12  C23.40D12  C23.43D12  C22.D24  C124(C4⋊C4)  (C2×Dic6)⋊7C4  C4210Dic3  C4211Dic3  (C2×C4)⋊6D12  C24.17D6  C24.18D6  C24.58D6  C24.19D6  C24.21D6  C24.27D6  C12⋊(C4⋊C4)  Dic3×C4⋊C4  (C4×Dic3)⋊8C4  (C4×Dic3)⋊9C4  C4⋊C45Dic3  (C2×C4).44D12  (C2×C12).54D4  C4⋊C46Dic3  (C2×C12).55D4  C4⋊(D6⋊C4)  (C2×C12).56D4  C4⋊C4.232D6  (C2×C6).D8  C4⋊D4.S3  (C2×Q8).49D6  (C2×C6).Q16  C23.52D12  C23.54D12  C24.75D6  C24.30D6  (C6×Q8)⋊7C4  C4○D43Dic3  C2×C4×Dic6  C2×C4×D12  C2×S3×C4⋊C4  C42.90D6  C42.91D6  C42.105D6  D46Dic6  D46D12  C42.119D6  C6.732- 1+4  C6.1152+ 1+4  C6.1182+ 1+4  C6.772- 1+4  C6.852- 1+4  C2×D4×Dic3  C2×Q8×Dic3  C6.1442+ 1+4  C6.1082- 1+4
C2×C4⋊Dic3 is a maximal quotient of
C127M4(2)  C4210Dic3  C4211Dic3  C24.58D6  C4⋊C46Dic3  C42.43D6  C23.27D12  C23.52D12  C23.9Dic6  C24.75D6

36 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E···4L6A···6G12A···12H
order12···2344444···46···612···12
size11···1222226···62···22···2

36 irreducible representations

dim1111122222222
type++++++--++-+
imageC1C2C2C2C4S3D4Q8Dic3D6D6Dic6D12
kernelC2×C4⋊Dic3C4⋊Dic3C22×Dic3C22×C12C2×C12C22×C4C2×C6C2×C6C2×C4C2×C4C23C22C22
# reps1421812242144

Matrix representation of C2×C4⋊Dic3 in GL4(𝔽13) generated by

12000
01200
00120
00012
,
1000
01200
0037
00610
,
12000
0100
0001
00121
,
5000
0100
00114
0022
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,3,6,0,0,7,10],[12,0,0,0,0,1,0,0,0,0,0,12,0,0,1,1],[5,0,0,0,0,1,0,0,0,0,11,2,0,0,4,2] >;

C2×C4⋊Dic3 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes {\rm Dic}_3
% in TeX

G:=Group("C2xC4:Dic3");
// GroupNames label

G:=SmallGroup(96,132);
// by ID

G=gap.SmallGroup(96,132);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,362,86,2309]);
// Polycyclic

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

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