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G = (C2×C6).D8order 192 = 26·3

8th non-split extension by C2×C6 of D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C2×C6).D8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C2×C4⋊Dic3 — (C2×C6).D8
 Lower central C3 — C6 — C2×C12 — (C2×C6).D8
 Upper central C1 — C22 — C22×C4 — C4⋊D4

Generators and relations for (C2×C6).D8
G = < a,b,c,d | a2=b6=c8=1, d2=b3, ab=ba, cac-1=ab3, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 304 in 114 conjugacy classes, 43 normal (27 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C6 [×3], C6 [×3], C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4, C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], C22×C4, C22×C4, C2×D4, C2×D4, C3⋊C8 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C22×C6, C22×C6, C22⋊C8, D4⋊C4 [×2], C2.D8 [×2], C2×C4⋊C4, C4⋊D4, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C3×C22⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C6×D4, C6×D4, C22.D8, C6.Q16 [×2], C12.55D4, D4⋊Dic3 [×2], C2×C4⋊Dic3, C3×C4⋊D4, (C2×C6).D8
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], D42S3 [×2], C2×C3⋊D4, C22.D8, C2×D4⋊S3, C23.23D6, Q8.14D6, (C2×C6).D8

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - - + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C4○D4 D8 C3⋊D4 C3⋊D4 C8.C22 D4⋊2S3 D4⋊S3 Q8.14D6 kernel (C2×C6).D8 C6.Q16 C12.55D4 D4⋊Dic3 C2×C4⋊Dic3 C3×C4⋊D4 C4⋊D4 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×D4 C12 C2×C6 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 1 1 1 1 1 1 1 1 4 4 2 2 1 2 2 2

Matrix representation of (C2×C6).D8 in GL6(𝔽73)

 1 3 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 27 0 0 0 0 0 55 46 0 0 0 0 0 0 57 16 0 0 0 0 57 57 0 0 0 0 0 0 70 31 0 0 0 0 28 3
,
 27 0 0 0 0 0 0 27 0 0 0 0 0 0 57 16 0 0 0 0 16 16 0 0 0 0 0 0 70 31 0 0 0 0 28 3

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,3,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[27,55,0,0,0,0,0,46,0,0,0,0,0,0,57,57,0,0,0,0,16,57,0,0,0,0,0,0,70,28,0,0,0,0,31,3],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,57,16,0,0,0,0,16,16,0,0,0,0,0,0,70,28,0,0,0,0,31,3] >;

(C2×C6).D8 in GAP, Magma, Sage, TeX

(C_2\times C_6).D_8
% in TeX

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

G:=SmallGroup(192,592);
// by ID

G=gap.SmallGroup(192,592);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,254,219,1123,297,136,6278]);
// Polycyclic

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

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