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## G = Dic6⋊9D4order 192 = 26·3

### 2nd semidirect product of Dic6 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic6⋊9D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C4×Dic6 — Dic6⋊9D4
 Lower central C3 — C6 — C2×C12 — Dic6⋊9D4
 Upper central C1 — C22 — C42 — C4⋊1D4

Generators and relations for Dic69D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, ac=ca, dad=a7, bc=cb, dbd=a9b, dcd=c-1 >

Subgroups: 352 in 128 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×3], C23 [×2], Dic3 [×3], C12 [×2], C12 [×2], C12, C2×C6, C2×C6 [×6], C42, C42, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8, C3⋊C8 [×2], Dic6 [×2], Dic6, C2×Dic3 [×2], C2×C12 [×3], C3×D4 [×8], C22×C6 [×2], D4⋊C4 [×2], C4⋊C8, C4×Q8, C41D4, C2×SD16 [×2], C2×C3⋊C8 [×2], C4×Dic3, Dic3⋊C4, C4⋊Dic3, D4.S3 [×4], C4×C12, C2×Dic6, C6×D4 [×2], C6×D4 [×2], C4⋊SD16, C12⋊C8, D4⋊Dic3 [×2], C4×Dic6, C2×D4.S3 [×2], C3×C41D4, Dic69D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], SD16 [×2], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C2×SD16, C8⋊C22, D4.S3 [×2], S3×D4, D42S3, C2×C3⋊D4, C4⋊SD16, D126C22, C2×D4.S3, D63D4, Dic69D4

Smallest permutation representation of Dic69D4
On 96 points
Generators in S96
(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 21 7 15)(2 20 8 14)(3 19 9 13)(4 18 10 24)(5 17 11 23)(6 16 12 22)(25 38 31 44)(26 37 32 43)(27 48 33 42)(28 47 34 41)(29 46 35 40)(30 45 36 39)(49 70 55 64)(50 69 56 63)(51 68 57 62)(52 67 58 61)(53 66 59 72)(54 65 60 71)(73 85 79 91)(74 96 80 90)(75 95 81 89)(76 94 82 88)(77 93 83 87)(78 92 84 86)
(1 49 91 25)(2 50 92 26)(3 51 93 27)(4 52 94 28)(5 53 95 29)(6 54 96 30)(7 55 85 31)(8 56 86 32)(9 57 87 33)(10 58 88 34)(11 59 89 35)(12 60 90 36)(13 62 77 42)(14 63 78 43)(15 64 79 44)(16 65 80 45)(17 66 81 46)(18 67 82 47)(19 68 83 48)(20 69 84 37)(21 70 73 38)(22 71 74 39)(23 72 75 40)(24 61 76 41)
(2 8)(4 10)(6 12)(13 16)(14 23)(15 18)(17 20)(19 22)(21 24)(25 49)(26 56)(27 51)(28 58)(29 53)(30 60)(31 55)(32 50)(33 57)(34 52)(35 59)(36 54)(37 66)(38 61)(39 68)(40 63)(41 70)(42 65)(43 72)(44 67)(45 62)(46 69)(47 64)(48 71)(73 76)(74 83)(75 78)(77 80)(79 82)(81 84)(86 92)(88 94)(90 96)

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

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

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

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 6F 6G 8A 8B 8C 8D 12A ··· 12F order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 8 8 8 8 12 ··· 12 size 1 1 1 1 8 8 2 2 2 2 2 4 12 12 12 12 2 2 2 8 8 8 8 12 12 12 12 4 ··· 4

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 SD16 C4○D4 C3⋊D4 C8⋊C22 D4.S3 S3×D4 D4⋊2S3 D12⋊6C22 kernel Dic6⋊9D4 C12⋊C8 D4⋊Dic3 C4×Dic6 C2×D4.S3 C3×C4⋊1D4 C4⋊1D4 Dic6 C2×C12 C42 C2×D4 C12 C12 C2×C4 C6 C4 C4 C4 C2 # reps 1 1 2 1 2 1 1 2 2 1 2 4 2 4 1 2 1 1 2

Matrix representation of Dic69D4 in GL6(𝔽73)

 9 0 0 0 0 0 0 65 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 27 0 0 0 0 46 0 0 0 0 0 0 0 6 67 0 0 0 0 67 67
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [9,0,0,0,0,0,0,65,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,0,6,67,0,0,0,0,67,67],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;

Dic69D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_9D_4
% in TeX

G:=Group("Dic6:9D4");
// GroupNames label

G:=SmallGroup(192,634);
// by ID

G=gap.SmallGroup(192,634);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^7,b*c=c*b,d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations

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