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## G = D4⋊Dic6order 192 = 26·3

### 2nd semidirect product of D4 and Dic6 acting via Dic6/Dic3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D4⋊Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×Dic3 — D4×Dic3 — D4⋊Dic6
 Lower central C3 — C6 — C2×C12 — D4⋊Dic6
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D4⋊Dic6
G = < a,b,c,d | a4=b2=c12=1, d2=c6, bab=cac-1=a-1, ad=da, cbc-1=a-1b, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 312 in 108 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, D4⋊C4, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C4⋊C4, C2×C24, C2×Dic6, C22×Dic3, C6×D4, D42Q8, C12.Q8, Dic3⋊C8, C8⋊Dic3, D4⋊Dic3, C3×D4⋊C4, C12⋊Q8, D4×Dic3, D4⋊Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, SD16, C2×D4, C2×Q8, C4○D4, Dic6, C22×S3, C22⋊Q8, C2×SD16, C8⋊C22, C2×Dic6, S3×D4, D42S3, D42Q8, Dic3.D4, D8⋊S3, S3×SD16, D4⋊Dic6

Smallest permutation representation of D4⋊Dic6
On 96 points
Generators in S96
(1 80 71 14)(2 15 72 81)(3 82 61 16)(4 17 62 83)(5 84 63 18)(6 19 64 73)(7 74 65 20)(8 21 66 75)(9 76 67 22)(10 23 68 77)(11 78 69 24)(12 13 70 79)(25 53 95 44)(26 45 96 54)(27 55 85 46)(28 47 86 56)(29 57 87 48)(30 37 88 58)(31 59 89 38)(32 39 90 60)(33 49 91 40)(34 41 92 50)(35 51 93 42)(36 43 94 52)
(1 74)(2 66)(3 76)(4 68)(5 78)(6 70)(7 80)(8 72)(9 82)(10 62)(11 84)(12 64)(13 19)(14 65)(15 21)(16 67)(17 23)(18 69)(20 71)(22 61)(24 63)(25 31)(26 60)(27 33)(28 50)(29 35)(30 52)(32 54)(34 56)(36 58)(37 94)(38 53)(39 96)(40 55)(41 86)(42 57)(43 88)(44 59)(45 90)(46 49)(47 92)(48 51)(73 79)(75 81)(77 83)(85 91)(87 93)(89 95)
(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 34 7 28)(2 33 8 27)(3 32 9 26)(4 31 10 25)(5 30 11 36)(6 29 12 35)(13 51 19 57)(14 50 20 56)(15 49 21 55)(16 60 22 54)(17 59 23 53)(18 58 24 52)(37 78 43 84)(38 77 44 83)(39 76 45 82)(40 75 46 81)(41 74 47 80)(42 73 48 79)(61 90 67 96)(62 89 68 95)(63 88 69 94)(64 87 70 93)(65 86 71 92)(66 85 72 91)

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 24 24 24 24 size 1 1 1 1 4 4 2 2 2 6 6 8 12 12 12 24 2 2 2 8 8 4 4 12 12 4 4 8 8 4 4 4 4

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + - + + + - + - + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 D6 SD16 C4○D4 Dic6 C8⋊C22 D4⋊2S3 S3×D4 D8⋊S3 S3×SD16 kernel D4⋊Dic6 C12.Q8 Dic3⋊C8 C8⋊Dic3 D4⋊Dic3 C3×D4⋊C4 C12⋊Q8 D4×Dic3 D4⋊C4 C2×Dic3 C3×D4 C4⋊C4 C2×C8 C2×D4 Dic3 C12 D4 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 4 2 4 1 1 1 2 2

Matrix representation of D4⋊Dic6 in GL6(𝔽73)

 0 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 6 6 0 0 0 0 6 67 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 0 72 0 0 0 0 1 0 0 0 0 0 0 0 45 3 0 0 0 0 31 28 0 0 0 0 0 0 57 20 0 0 0 0 20 16

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[6,6,0,0,0,0,6,67,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,45,31,0,0,0,0,3,28,0,0,0,0,0,0,57,20,0,0,0,0,20,16] >;

D4⋊Dic6 in GAP, Magma, Sage, TeX

D_4\rtimes {\rm Dic}_6
% in TeX

G:=Group("D4:Dic6");
// GroupNames label

G:=SmallGroup(192,320);
// by ID

G=gap.SmallGroup(192,320);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,254,219,226,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^12=1,d^2=c^6,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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