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## G = Dic6⋊11D4order 192 = 26·3

### 4th semidirect product of Dic6 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic6⋊11D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×D12 — C4×D12 — Dic6⋊11D4
 Lower central C3 — C2×C6 — Dic6⋊11D4
 Upper central C1 — C22 — C4⋊1D4

Generators and relations for Dic611D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, ac=ca, dad=a7, bc=cb, dbd=a6b, dcd=c-1 >

Subgroups: 832 in 312 conjugacy classes, 107 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4×D4, C4×Q8, C4⋊D4, C41D4, C41D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C2×Dic6, S3×C2×C4, C2×D12, D42S3, C22×Dic3, C2×C3⋊D4, C6×D4, Q86D4, C4×Dic6, C4×D12, D4×Dic3, D63D4, C23.14D6, C123D4, C3×C41D4, C2×D42S3, Dic611D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, 2+ 1+4, S3×D4, D42S3, S3×C23, Q86D4, C2×S3×D4, C2×D42S3, D46D6, Dic611D4

Smallest permutation representation of Dic611D4
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 65 7 71)(2 64 8 70)(3 63 9 69)(4 62 10 68)(5 61 11 67)(6 72 12 66)(13 73 19 79)(14 84 20 78)(15 83 21 77)(16 82 22 76)(17 81 23 75)(18 80 24 74)(25 42 31 48)(26 41 32 47)(27 40 33 46)(28 39 34 45)(29 38 35 44)(30 37 36 43)(49 90 55 96)(50 89 56 95)(51 88 57 94)(52 87 58 93)(53 86 59 92)(54 85 60 91)
(1 14 46 60)(2 15 47 49)(3 16 48 50)(4 17 37 51)(5 18 38 52)(6 19 39 53)(7 20 40 54)(8 21 41 55)(9 22 42 56)(10 23 43 57)(11 24 44 58)(12 13 45 59)(25 89 63 82)(26 90 64 83)(27 91 65 84)(28 92 66 73)(29 93 67 74)(30 94 68 75)(31 95 69 76)(32 96 70 77)(33 85 71 78)(34 86 72 79)(35 87 61 80)(36 88 62 81)
(1 91)(2 86)(3 93)(4 88)(5 95)(6 90)(7 85)(8 92)(9 87)(10 94)(11 89)(12 96)(13 32)(14 27)(15 34)(16 29)(17 36)(18 31)(19 26)(20 33)(21 28)(22 35)(23 30)(24 25)(37 81)(38 76)(39 83)(40 78)(41 73)(42 80)(43 75)(44 82)(45 77)(46 84)(47 79)(48 74)(49 72)(50 67)(51 62)(52 69)(53 64)(54 71)(55 66)(56 61)(57 68)(58 63)(59 70)(60 65)

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F ··· 4M 4N 4O 6A 6B 6C 6D 6E 6F 6G 12A ··· 12F order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 ··· 4 4 4 6 6 6 6 6 6 6 12 ··· 12 size 1 1 1 1 4 4 4 4 12 12 2 2 2 2 2 4 6 ··· 6 12 12 2 2 2 8 8 8 8 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 2+ 1+4 S3×D4 D4⋊2S3 D4⋊6D6 kernel Dic6⋊11D4 C4×Dic6 C4×D12 D4×Dic3 D6⋊3D4 C23.14D6 C12⋊3D4 C3×C4⋊1D4 C2×D4⋊2S3 C4⋊1D4 Dic6 C42 C2×D4 C12 C6 C4 C4 C2 # reps 1 1 1 2 2 4 2 1 2 1 4 1 6 4 1 2 2 2

Matrix representation of Dic611D4 in GL6(𝔽13)

 5 0 0 0 0 0 0 8 0 0 0 0 0 0 0 12 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

Dic611D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_{11}D_4
% in TeX

G:=Group("Dic6:11D4");
// GroupNames label

G:=SmallGroup(192,1277);
// by ID

G=gap.SmallGroup(192,1277);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,100,675,570,185,6278]);
// 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^6*b,d*c*d=c^-1>;
// generators/relations

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