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## G = D4.Dic6order 192 = 26·3

### 1st non-split extension by D4 of 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, bd=db, dcd-1=a2c-1 >

Subgroups: 280 in 102 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, D4⋊C4, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C6.D4, C3×C4⋊C4, C2×C24, C22×Dic3, C6×D4, D4.Q8, C6.Q16, Dic3⋊C8, C8⋊Dic3, D4⋊Dic3, C3×D4⋊C4, C4.Dic6, D4×Dic3, D4.Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C22×S3, C22⋊Q8, C4○D8, C8⋊C22, C2×Dic6, S3×D4, D42S3, D4.Q8, Dic3.D4, D8⋊S3, Q8.7D6, D4.Dic6

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

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

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

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

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 C4○D4 Dic6 C4○D8 C8⋊C22 D4⋊2S3 S3×D4 D8⋊S3 Q8.7D6 kernel D4.Dic6 C6.Q16 Dic3⋊C8 C8⋊Dic3 D4⋊Dic3 C3×D4⋊C4 C4.Dic6 D4×Dic3 D4⋊C4 C2×Dic3 C3×D4 C4⋊C4 C2×C8 C2×D4 C12 D4 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 2 4 4 1 1 1 2 2

Matrix representation of D4.Dic6 in GL4(𝔽73) generated by

 1 71 0 0 1 72 0 0 0 0 1 0 0 0 0 1
,
 72 2 0 0 0 1 0 0 0 0 72 0 0 0 0 72
,
 61 12 0 0 67 12 0 0 0 0 66 66 0 0 7 59
,
 27 0 0 0 0 27 0 0 0 0 10 12 0 0 22 63
G:=sub<GL(4,GF(73))| [1,1,0,0,71,72,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,2,1,0,0,0,0,72,0,0,0,0,72],[61,67,0,0,12,12,0,0,0,0,66,7,0,0,66,59],[27,0,0,0,0,27,0,0,0,0,10,22,0,0,12,63] >;

D4.Dic6 in GAP, Magma, Sage, TeX

D_4.{\rm Dic}_6
% in TeX

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

G:=SmallGroup(192,322);
// by ID

G=gap.SmallGroup(192,322);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,926,219,58,851,438,102,6278]);
// 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,b*d=d*b,d*c*d^-1=a^2*c^-1>;
// generators/relations

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