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## G = C2×Q8○D12order 192 = 26·3

### Direct product of C2 and Q8○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×Q8○D12
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C2×C4 — C2×S3×Q8 — C2×Q8○D12
 Lower central C3 — C6 — C2×Q8○D12
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C2×Q8○D12
G = < a,b,c,d,e | a2=b4=e2=1, c2=d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >

Subgroups: 1480 in 794 conjugacy classes, 447 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C22×Q8, C2×C4○D4, C2×C4○D4, 2- 1+4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, D42S3, S3×Q8, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C2×2- 1+4, C22×Dic6, C2×C4○D12, C2×D42S3, C2×S3×Q8, Q8○D12, C6×C4○D4, C2×Q8○D12
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2- 1+4, C25, S3×C23, C2×2- 1+4, Q8○D12, S3×C24, C2×Q8○D12

Smallest permutation representation of C2×Q8○D12
On 96 points
Generators in S96
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 68)(14 69)(15 70)(16 71)(17 72)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 82)(26 83)(27 84)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 81)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 85)(45 86)(46 87)(47 88)(48 89)
(1 69 7 63)(2 70 8 64)(3 71 9 65)(4 72 10 66)(5 61 11 67)(6 62 12 68)(13 60 19 54)(14 49 20 55)(15 50 21 56)(16 51 22 57)(17 52 23 58)(18 53 24 59)(25 96 31 90)(26 85 32 91)(27 86 33 92)(28 87 34 93)(29 88 35 94)(30 89 36 95)(37 82 43 76)(38 83 44 77)(39 84 45 78)(40 73 46 79)(41 74 47 80)(42 75 48 81)
(1 88 7 94)(2 89 8 95)(3 90 9 96)(4 91 10 85)(5 92 11 86)(6 93 12 87)(13 73 19 79)(14 74 20 80)(15 75 21 81)(16 76 22 82)(17 77 23 83)(18 78 24 84)(25 71 31 65)(26 72 32 66)(27 61 33 67)(28 62 34 68)(29 63 35 69)(30 64 36 70)(37 51 43 57)(38 52 44 58)(39 53 45 59)(40 54 46 60)(41 55 47 49)(42 56 48 50)
(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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 63)(14 62)(15 61)(16 72)(17 71)(18 70)(19 69)(20 68)(21 67)(22 66)(23 65)(24 64)(25 83)(26 82)(27 81)(28 80)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 84)(37 91)(38 90)(39 89)(40 88)(41 87)(42 86)(43 85)(44 96)(45 95)(46 94)(47 93)(48 92)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 3 4A ··· 4H 4I ··· 4T 6A 6B 6C 6D ··· 6I 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 ··· 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 6 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 ··· 2 6 6 6 6 2 2 ··· 2 6 ··· 6 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 2- 1+4 Q8○D12 kernel C2×Q8○D12 C22×Dic6 C2×C4○D12 C2×D4⋊2S3 C2×S3×Q8 Q8○D12 C6×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C6 C2 # reps 1 3 3 6 2 16 1 1 3 3 1 8 2 4

Matrix representation of C2×Q8○D12 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 3 7 2 0 0 0 6 10 0 2 0 0 0 0 10 6 0 0 0 0 7 3
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 2 9 10 0 0 0 4 11 0 10 0 0 5 0 11 4 0 0 0 5 9 2
,
 0 12 0 0 0 0 1 1 0 0 0 0 0 0 3 3 0 0 0 0 10 6 0 0 0 0 0 0 3 3 0 0 0 0 10 6
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 10 6 0 0 0 0 3 3 0 0 0 0 1 0 10 6 0 0 1 12 3 3

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,0,0,2,0,10,7,0,0,0,2,6,3],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,2,4,5,0,0,0,9,11,0,5,0,0,10,0,11,9,0,0,0,10,4,2],[0,1,0,0,0,0,12,1,0,0,0,0,0,0,3,10,0,0,0,0,3,6,0,0,0,0,0,0,3,10,0,0,0,0,3,6],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,3,1,1,0,0,6,3,0,12,0,0,0,0,10,3,0,0,0,0,6,3] >;

C2×Q8○D12 in GAP, Magma, Sage, TeX

C_2\times Q_8\circ D_{12}
% in TeX

G:=Group("C2xQ8oD12");
// GroupNames label

G:=SmallGroup(192,1522);
// by ID

G=gap.SmallGroup(192,1522);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,297,136,1684,102,6278]);
// Polycyclic

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

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