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## G = Dic6⋊24D4order 192 = 26·3

### 2nd semidirect product of Dic6 and D4 acting through Inn(Dic6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic6⋊24D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×D12 — Dic3⋊5D4 — Dic6⋊24D4
 Lower central C3 — C2×C6 — Dic6⋊24D4
 Upper central C1 — C22 — C4×D4

Generators and relations for Dic624D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a6b, dcd=c-1 >

Subgroups: 888 in 312 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, 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, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4×D4, C4×D4, C4×Q8, C4⋊D4, C41D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C6×D4, Q86D4, C4×Dic6, C4⋊D12, Dic3⋊D4, Dic35D4, C127D4, C123D4, D4×C12, C2×C4○D12, Dic624D4
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, Q86D4, C2×C4○D12, C2×S3×D4, D4○D12, Dic624D4

Smallest permutation representation of Dic624D4
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 70 7 64)(2 69 8 63)(3 68 9 62)(4 67 10 61)(5 66 11 72)(6 65 12 71)(13 74 19 80)(14 73 20 79)(15 84 21 78)(16 83 22 77)(17 82 23 76)(18 81 24 75)(25 89 31 95)(26 88 32 94)(27 87 33 93)(28 86 34 92)(29 85 35 91)(30 96 36 90)(37 58 43 52)(38 57 44 51)(39 56 45 50)(40 55 46 49)(41 54 47 60)(42 53 48 59)
(1 76 57 29)(2 77 58 30)(3 78 59 31)(4 79 60 32)(5 80 49 33)(6 81 50 34)(7 82 51 35)(8 83 52 36)(9 84 53 25)(10 73 54 26)(11 74 55 27)(12 75 56 28)(13 40 93 66)(14 41 94 67)(15 42 95 68)(16 43 96 69)(17 44 85 70)(18 45 86 71)(19 46 87 72)(20 47 88 61)(21 48 89 62)(22 37 90 63)(23 38 91 64)(24 39 92 65)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 84)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 93)(26 94)(27 95)(28 96)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 61)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A ··· 4H 4I 4J 4K 4L 4M 4N 4O 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E ··· 12L 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 12 12 12 ··· 12 size 1 1 1 1 4 4 12 12 12 12 2 2 ··· 2 4 6 6 6 6 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 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 D6 C4○D4 C4○D12 2+ 1+4 S3×D4 D4○D12 kernel Dic6⋊24D4 C4×Dic6 C4⋊D12 Dic3⋊D4 Dic3⋊5D4 C12⋊7D4 C12⋊3D4 D4×C12 C2×C4○D12 C4×D4 Dic6 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C12 C4 C6 C4 C2 # reps 1 1 1 4 2 2 2 1 2 1 4 1 2 1 2 1 4 8 1 2 2

Matrix representation of Dic624D4 in GL4(𝔽13) generated by

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

Dic624D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_{24}D_4
% in TeX

G:=Group("Dic6:24D4");
// GroupNames label

G:=SmallGroup(192,1112);
// by ID

G=gap.SmallGroup(192,1112);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,387,100,675,570,6278]);
// 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,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations

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