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## G = C2×Dic3⋊5D4order 192 = 26·3

### Direct product of C2 and Dic3⋊5D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×Dic3⋊5D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×D12 — C2×Dic3⋊5D4
 Lower central C3 — C6 — C2×Dic3⋊5D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×Dic35D4
G = < a,b,c,d,e | a2=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1048 in 426 conjugacy classes, 175 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C4×Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C22×Dic3, C22×C12, C22×C12, S3×C23, C2×C4×D4, Dic35D4, C2×C4×Dic3, C2×D6⋊C4, C6×C4⋊C4, S3×C22×C4, C22×D12, C2×Dic35D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C24, C4×S3, C22×S3, C4×D4, C23×C4, C22×D4, C2×C4○D4, S3×C2×C4, S3×D4, Q83S3, S3×C23, C2×C4×D4, Dic35D4, S3×C22×C4, C2×S3×D4, C2×Q83S3, C2×Dic35D4

Smallest permutation representation of C2×Dic35D4
On 96 points
Generators in S96
(1 65)(2 66)(3 61)(4 62)(5 63)(6 64)(7 46)(8 47)(9 48)(10 43)(11 44)(12 45)(13 56)(14 57)(15 58)(16 59)(17 60)(18 55)(19 78)(20 73)(21 74)(22 75)(23 76)(24 77)(25 68)(26 69)(27 70)(28 71)(29 72)(30 67)(31 90)(32 85)(33 86)(34 87)(35 88)(36 89)(37 80)(38 81)(39 82)(40 83)(41 84)(42 79)(49 92)(50 93)(51 94)(52 95)(53 96)(54 91)
(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 12 4 9)(2 11 5 8)(3 10 6 7)(13 91 16 94)(14 96 17 93)(15 95 18 92)(19 83 22 80)(20 82 23 79)(21 81 24 84)(25 87 28 90)(26 86 29 89)(27 85 30 88)(31 68 34 71)(32 67 35 70)(33 72 36 69)(37 78 40 75)(38 77 41 74)(39 76 42 73)(43 64 46 61)(44 63 47 66)(45 62 48 65)(49 58 52 55)(50 57 53 60)(51 56 54 59)
(1 41 17 36)(2 42 18 31)(3 37 13 32)(4 38 14 33)(5 39 15 34)(6 40 16 35)(7 75 94 70)(8 76 95 71)(9 77 96 72)(10 78 91 67)(11 73 92 68)(12 74 93 69)(19 54 30 43)(20 49 25 44)(21 50 26 45)(22 51 27 46)(23 52 28 47)(24 53 29 48)(55 90 66 79)(56 85 61 80)(57 86 62 81)(58 87 63 82)(59 88 64 83)(60 89 65 84)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 18)(7 92)(8 91)(9 96)(10 95)(11 94)(12 93)(19 23)(20 22)(25 27)(28 30)(31 35)(32 34)(37 39)(40 42)(43 52)(44 51)(45 50)(46 49)(47 54)(48 53)(55 64)(56 63)(57 62)(58 61)(59 66)(60 65)(67 71)(68 70)(73 75)(76 78)(79 83)(80 82)(85 87)(88 90)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 4A ··· 4L 4M ··· 4T 4U 4V 4W 4X 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 6 ··· 6 2 2 ··· 2 3 ··· 3 6 6 6 6 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 C4○D4 C4×S3 S3×D4 Q8⋊3S3 kernel C2×Dic3⋊5D4 Dic3⋊5D4 C2×C4×Dic3 C2×D6⋊C4 C6×C4⋊C4 S3×C22×C4 C22×D12 C2×D12 C2×C4⋊C4 C2×Dic3 C4⋊C4 C22×C4 C2×C6 C2×C4 C22 C22 # reps 1 8 1 2 1 2 1 16 1 4 4 3 4 8 2 2

Matrix representation of C2×Dic35D4 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 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 12 0
,
 12 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 8 0
,
 1 0 0 0 0 0 0 5 0 0 0 5 0 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

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],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,1,0],[12,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,8,0],[1,0,0,0,0,0,0,5,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C2×Dic35D4 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3\rtimes_5D_4
% in TeX

G:=Group("C2xDic3:5D4");
// GroupNames label

G:=SmallGroup(192,1062);
// by ID

G=gap.SmallGroup(192,1062);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,297,80,6278]);
// Polycyclic

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

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