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## G = C2×C4⋊Dic3order 96 = 25·3

### Direct product of C2 and C4⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C4⋊Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×C4⋊Dic3
 Lower central C3 — C6 — C2×C4⋊Dic3
 Upper central C1 — C23 — C22×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)])

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + - - + + - + image C1 C2 C2 C2 C4 S3 D4 Q8 Dic3 D6 D6 Dic6 D12 kernel C2×C4⋊Dic3 C4⋊Dic3 C22×Dic3 C22×C12 C2×C12 C22×C4 C2×C6 C2×C6 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 2 1 8 1 2 2 4 2 1 4 4

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

 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 7 0 0 6 10
,
 12 0 0 0 0 1 0 0 0 0 0 1 0 0 12 1
,
 5 0 0 0 0 1 0 0 0 0 11 4 0 0 2 2
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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