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## G = (C2×C6).40D8order 192 = 26·3

### 17th non-split extension by C2×C6 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C2×C6).40D8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C12⋊7D4 — (C2×C6).40D8
 Lower central C3 — C6 — C2×C12 — (C2×C6).40D8
 Upper central C1 — C22 — C22×C4 — C2×C4⋊C4

Generators and relations for (C2×C6).40D8
G = < a,b,c,d | a6=b2=c8=1, d2=a3, ab=ba, cac-1=a-1, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 344 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], S3, C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, Dic3, C12 [×2], C12 [×3], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×2], C3⋊C8 [×2], D12 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×6], C22×S3, C22×C6, C22⋊C8, D4⋊C4 [×2], C2.D8 [×2], C2×C4⋊C4, C4⋊D4, C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, C3×C4⋊C4 [×2], C3×C4⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C22×C12, C22.D8, C6.Q16 [×2], C6.D8 [×2], C12.55D4, C127D4, C6×C4⋊C4, (C2×C6).40D8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], D8 [×2], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C22.D4, C2×D8, C8.C22, D4⋊S3 [×2], C4○D12 [×2], C2×C3⋊D4, C22.D8, C23.28D6, C2×D4⋊S3, Q8.11D6, (C2×C6).40D8

Smallest permutation representation of (C2×C6).40D8
On 96 points
Generators in S96
(1 10 38 68 51 81)(2 82 52 69 39 11)(3 12 40 70 53 83)(4 84 54 71 33 13)(5 14 34 72 55 85)(6 86 56 65 35 15)(7 16 36 66 49 87)(8 88 50 67 37 9)(17 62 74 95 30 48)(18 41 31 96 75 63)(19 64 76 89 32 42)(20 43 25 90 77 57)(21 58 78 91 26 44)(22 45 27 92 79 59)(23 60 80 93 28 46)(24 47 29 94 73 61)
(1 32)(2 57)(3 26)(4 59)(5 28)(6 61)(7 30)(8 63)(9 75)(10 42)(11 77)(12 44)(13 79)(14 46)(15 73)(16 48)(17 36)(18 88)(19 38)(20 82)(21 40)(22 84)(23 34)(24 86)(25 69)(27 71)(29 65)(31 67)(33 92)(35 94)(37 96)(39 90)(41 50)(43 52)(45 54)(47 56)(49 74)(51 76)(53 78)(55 80)(58 70)(60 72)(62 66)(64 68)(81 89)(83 91)(85 93)(87 95)
(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 27 68 59)(2 26 69 58)(3 25 70 57)(4 32 71 64)(5 31 72 63)(6 30 65 62)(7 29 66 61)(8 28 67 60)(9 93 50 23)(10 92 51 22)(11 91 52 21)(12 90 53 20)(13 89 54 19)(14 96 55 18)(15 95 56 17)(16 94 49 24)(33 76 84 42)(34 75 85 41)(35 74 86 48)(36 73 87 47)(37 80 88 46)(38 79 81 45)(39 78 82 44)(40 77 83 43)

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 D8 C3⋊D4 C3⋊D4 C4○D12 C8.C22 D4⋊S3 Q8.11D6 kernel (C2×C6).40D8 C6.Q16 C6.D8 C12.55D4 C12⋊7D4 C6×C4⋊C4 C2×C4⋊C4 C2×C12 C22×C6 C4⋊C4 C22×C4 C12 C2×C6 C2×C4 C23 C4 C6 C22 C2 # reps 1 2 2 1 1 1 1 1 1 2 1 4 4 2 2 8 1 2 2

Matrix representation of (C2×C6).40D8 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 1
,
 72 0 0 0 0 72 0 0 0 0 43 60 0 0 13 30
,
 57 57 0 0 16 57 0 0 0 0 14 66 0 0 7 59
,
 57 57 0 0 57 16 0 0 0 0 46 0 0 0 0 46
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,1],[72,0,0,0,0,72,0,0,0,0,43,13,0,0,60,30],[57,16,0,0,57,57,0,0,0,0,14,7,0,0,66,59],[57,57,0,0,57,16,0,0,0,0,46,0,0,0,0,46] >;

(C2×C6).40D8 in GAP, Magma, Sage, TeX

(C_2\times C_6)._{40}D_8
% in TeX

G:=Group("(C2xC6).40D8");
// GroupNames label

G:=SmallGroup(192,526);
// by ID

G=gap.SmallGroup(192,526);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,100,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^8=1,d^2=a^3,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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