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## G = C42.20D6order 192 = 26·3

### 20th non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.20D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C42⋊7S3 — C42.20D6
 Lower central C3 — C6 — C2×C12 — C42.20D6
 Upper central C1 — C22 — C42 — C8⋊C4

Generators and relations for C42.20D6
G = < a,b,c,d | a4=b4=1, c6=a2b, d2=a2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b-1c5 >

Subgroups: 344 in 100 conjugacy classes, 39 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, C24, Dic6, D12, C2×Dic3, C2×C12, C22×S3, C8⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, C2×Dic6, C2×Dic6, C2×D12, C42.28C22, C2.Dic12, C2.D24, C3×C8⋊C4, C122Q8, C427S3, C42.20D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4.4D4, C8⋊C22, C8.C22, C2×D12, C4○D12, C42.28C22, C427S3, C8⋊D6, C8.D6, C42.20D6

Smallest permutation representation of C42.20D6
On 96 points
Generators in S96
(1 28 84 68)(2 41 85 57)(3 30 86 70)(4 43 87 59)(5 32 88 72)(6 45 89 61)(7 34 90 50)(8 47 91 63)(9 36 92 52)(10 25 93 65)(11 38 94 54)(12 27 95 67)(13 40 96 56)(14 29 73 69)(15 42 74 58)(16 31 75 71)(17 44 76 60)(18 33 77 49)(19 46 78 62)(20 35 79 51)(21 48 80 64)(22 37 81 53)(23 26 82 66)(24 39 83 55)
(1 90 13 78)(2 91 14 79)(3 92 15 80)(4 93 16 81)(5 94 17 82)(6 95 18 83)(7 96 19 84)(8 73 20 85)(9 74 21 86)(10 75 22 87)(11 76 23 88)(12 77 24 89)(25 71 37 59)(26 72 38 60)(27 49 39 61)(28 50 40 62)(29 51 41 63)(30 52 42 64)(31 53 43 65)(32 54 44 66)(33 55 45 67)(34 56 46 68)(35 57 47 69)(36 58 48 70)
(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 18 84 77)(2 76 85 17)(3 16 86 75)(4 74 87 15)(5 14 88 73)(6 96 89 13)(7 12 90 95)(8 94 91 11)(9 10 92 93)(19 24 78 83)(20 82 79 23)(21 22 80 81)(25 36 65 52)(26 51 66 35)(27 34 67 50)(28 49 68 33)(29 32 69 72)(30 71 70 31)(37 48 53 64)(38 63 54 47)(39 46 55 62)(40 61 56 45)(41 44 57 60)(42 59 58 43)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 24A ··· 24H order 1 2 2 2 2 3 4 4 4 4 4 4 4 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 24 2 2 2 4 4 24 24 24 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 D12 C4○D12 C8⋊C22 C8.C22 C8⋊D6 C8.D6 kernel C42.20D6 C2.Dic12 C2.D24 C3×C8⋊C4 C12⋊2Q8 C42⋊7S3 C8⋊C4 C2×C12 C42 C2×C8 C12 C2×C4 C4 C6 C6 C2 C2 # reps 1 2 2 1 1 1 1 2 1 2 4 4 8 1 1 2 2

Matrix representation of C42.20D6 in GL6(𝔽73)

 1 71 0 0 0 0 1 72 0 0 0 0 0 0 11 0 58 65 0 0 0 11 8 66 0 0 53 2 62 0 0 0 71 51 0 62
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 14 0 0 0 0 59 66 0 0 0 0 0 0 7 14 0 0 0 0 59 66
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 0 0 2 2 0 0 0 0 71 0 0 0 33 40 0 0 0 0 33 66 0 0
,
 46 0 0 0 0 0 46 27 0 0 0 0 0 0 0 0 59 14 0 0 0 0 28 14 0 0 40 33 0 0 0 0 66 33 0 0

G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,71,72,0,0,0,0,0,0,11,0,53,71,0,0,0,11,2,51,0,0,58,8,62,0,0,0,65,66,0,62],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,59,0,0,0,0,14,66,0,0,0,0,0,0,7,59,0,0,0,0,14,66],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,33,33,0,0,0,0,40,66,0,0,2,71,0,0,0,0,2,0,0,0],[46,46,0,0,0,0,0,27,0,0,0,0,0,0,0,0,40,66,0,0,0,0,33,33,0,0,59,28,0,0,0,0,14,14,0,0] >;

C42.20D6 in GAP, Magma, Sage, TeX

C_4^2._{20}D_6
% in TeX

G:=Group("C4^2.20D6");
// GroupNames label

G:=SmallGroup(192,273);
// by ID

G=gap.SmallGroup(192,273);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,387,142,1123,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2*b,d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^-1*c^5>;
// generators/relations

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