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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, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22⋊C8, D4⋊C4, C2.D8, C2×C4⋊C4, C4⋊D4, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C3×C22⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C6×D4, C6×D4, C22.D8, C6.Q16, C12.55D4, D4⋊Dic3, C2×C4⋊Dic3, C3×C4⋊D4, (C2×C6).D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, C2×D8, C8.C22, D4⋊S3, D42S3, 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 22)(2 87)(3 24)(4 81)(5 18)(6 83)(7 20)(8 85)(9 25)(10 51)(11 27)(12 53)(13 29)(14 55)(15 31)(16 49)(17 33)(19 35)(21 37)(23 39)(26 73)(28 75)(30 77)(32 79)(34 82)(36 84)(38 86)(40 88)(41 62)(42 89)(43 64)(44 91)(45 58)(46 93)(47 60)(48 95)(50 80)(52 74)(54 76)(56 78)(57 69)(59 71)(61 65)(63 67)(66 96)(68 90)(70 92)(72 94)
(1 79 95 38 16 61)(2 62 9 39 96 80)(3 73 89 40 10 63)(4 64 11 33 90 74)(5 75 91 34 12 57)(6 58 13 35 92 76)(7 77 93 36 14 59)(8 60 15 37 94 78)(17 68 52 81 43 27)(18 28 44 82 53 69)(19 70 54 83 45 29)(20 30 46 84 55 71)(21 72 56 85 47 31)(22 32 48 86 49 65)(23 66 50 87 41 25)(24 26 42 88 51 67)
(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 14 80 77)(10 76 73 13)(11 12 74 75)(15 16 78 79)(17 82 81 18)(19 88 83 24)(20 23 84 87)(21 86 85 22)(25 55 50 30)(26 29 51 54)(27 53 52 28)(31 49 56 32)(41 71 66 46)(42 45 67 70)(43 69 68 44)(47 65 72 48)(57 90 91 64)(58 63 92 89)(59 96 93 62)(60 61 94 95)

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

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

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

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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