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## G = C2×Dic3⋊C4order 96 = 25·3

### Direct product of C2 and Dic3⋊C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×Dic3⋊C4
G = < a,b,c,d | a2=b6=d4=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 162 in 92 conjugacy classes, 57 normal (17 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×8], C22, C22 [×6], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×12], C23, Dic3 [×4], Dic3 [×2], C12 [×2], C2×C6, C2×C6 [×6], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C2×Dic3 [×8], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×2], C22×C6, C2×C4⋊C4, Dic3⋊C4 [×4], C22×Dic3 [×2], C22×C12, C2×Dic3⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4, Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, C2×C3⋊D4, C2×Dic3⋊C4

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

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

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

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

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 D6 D6 Dic6 C4×S3 C3⋊D4 kernel C2×Dic3⋊C4 Dic3⋊C4 C22×Dic3 C22×C12 C2×Dic3 C22×C4 C2×C6 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 4 2 1 8 1 2 2 2 1 4 4 4

Matrix representation of C2×Dic3⋊C4 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 12 12 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 12 0
,
 1 0 0 0 0 0 10 7 0 0 0 10 3 0 0 0 0 0 8 8 0 0 0 0 5
,
 8 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 11 9 0 0 0 4 2

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,1,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,1,0],[1,0,0,0,0,0,10,10,0,0,0,7,3,0,0,0,0,0,8,0,0,0,0,8,5],[8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,11,4,0,0,0,9,2] >;

C2×Dic3⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C2xDic3:C4");
// GroupNames label

G:=SmallGroup(96,130);
// by ID

G=gap.SmallGroup(96,130);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,362,50,2309]);
// Polycyclic

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

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