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## G = Dic6⋊23D4order 192 = 26·3

### 1st semidirect product of Dic6 and D4 acting through Inn(Dic6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic6⋊23D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×C3⋊D4 — Dic3⋊4D4 — Dic6⋊23D4
 Lower central C3 — C2×C6 — Dic6⋊23D4
 Upper central C1 — C22 — C4×D4

Generators and relations for Dic623D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, ac=ca, ad=da, cbc-1=a6b, bd=db, dcd=c-1 >

Subgroups: 728 in 290 conjugacy classes, 107 normal (51 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, Q85D4, C4×Dic6, C427S3, Dic34D4, C23.11D6, D6⋊Q8, C12.48D4, C127D4, C23.14D6, D4×C12, C22×Dic6, C2×C4○D12, Dic623D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, 2- 1+4, C4○D12, S3×D4, S3×C23, Q85D4, C2×C4○D12, C2×S3×D4, Q8○D12, Dic623D4

Smallest permutation representation of Dic623D4
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 58 7 52)(2 57 8 51)(3 56 9 50)(4 55 10 49)(5 54 11 60)(6 53 12 59)(13 84 19 78)(14 83 20 77)(15 82 21 76)(16 81 22 75)(17 80 23 74)(18 79 24 73)(25 41 31 47)(26 40 32 46)(27 39 33 45)(28 38 34 44)(29 37 35 43)(30 48 36 42)(61 85 67 91)(62 96 68 90)(63 95 69 89)(64 94 70 88)(65 93 71 87)(66 92 72 86)
(1 43 19 72)(2 44 20 61)(3 45 21 62)(4 46 22 63)(5 47 23 64)(6 48 24 65)(7 37 13 66)(8 38 14 67)(9 39 15 68)(10 40 16 69)(11 41 17 70)(12 42 18 71)(25 80 94 60)(26 81 95 49)(27 82 96 50)(28 83 85 51)(29 84 86 52)(30 73 87 53)(31 74 88 54)(32 75 89 55)(33 76 90 56)(34 77 91 57)(35 78 92 58)(36 79 93 59)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 91)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 25)(22 26)(23 27)(24 28)(37 84)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 71)(58 72)(59 61)(60 62)

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 D6 C4○D4 C4○D12 2- 1+4 S3×D4 Q8○D12 kernel Dic6⋊23D4 C4×Dic6 C42⋊7S3 Dic3⋊4D4 C23.11D6 D6⋊Q8 C12.48D4 C12⋊7D4 C23.14D6 D4×C12 C22×Dic6 C2×C4○D12 C4×D4 Dic6 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C2×C6 C22 C6 C4 C2 # reps 1 1 1 2 2 2 1 1 2 1 1 1 1 4 1 2 1 2 1 4 8 1 2 2

Matrix representation of Dic623D4 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 3 3 0 0 10 6
,
 1 0 0 0 0 1 0 0 0 0 2 2 0 0 4 11
,
 6 3 0 0 5 7 0 0 0 0 2 9 0 0 4 11
,
 6 3 0 0 10 7 0 0 0 0 12 0 0 0 0 12
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,3,10,0,0,3,6],[1,0,0,0,0,1,0,0,0,0,2,4,0,0,2,11],[6,5,0,0,3,7,0,0,0,0,2,4,0,0,9,11],[6,10,0,0,3,7,0,0,0,0,12,0,0,0,0,12] >;

Dic623D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_{23}D_4
% in TeX

G:=Group("Dic6:23D4");
// GroupNames label

G:=SmallGroup(192,1111);
// by ID

G=gap.SmallGroup(192,1111);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,387,100,675,570,6278]);
// 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,a*d=d*a,c*b*c^-1=a^6*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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