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## G = C2×Q8.14D6order 192 = 26·3

### Direct product of C2 and Q8.14D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×Q8.14D6
 Chief series C1 — C3 — C6 — C12 — Dic6 — C2×Dic6 — C22×Dic6 — C2×Q8.14D6
 Lower central C3 — C6 — C12 — C2×Q8.14D6
 Upper central C1 — C22 — C22×C4 — C2×C4○D4

Generators and relations for C2×Q8.14D6
G = < a,b,c,d,e | a2=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d-1 >

Subgroups: 552 in 258 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C6, C6 [×2], C6 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×2], Q8 [×11], C23, C23, Dic3 [×4], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×6], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×9], C4○D4 [×4], C4○D4 [×2], C3⋊C8 [×4], Dic6 [×4], Dic6 [×6], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×4], C2×C12 [×5], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8, C22×C6, C22×C6, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C2×C3⋊C8 [×2], C4.Dic3 [×4], D4.S3 [×8], C3⋊Q16 [×8], C2×Dic6 [×6], C2×Dic6 [×3], C22×Dic3, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4 [×4], C3×C4○D4 [×2], C2×C8.C22, C2×C4.Dic3, C2×D4.S3 [×2], C2×C3⋊Q16 [×2], Q8.14D6 [×8], C22×Dic6, C6×C4○D4, C2×Q8.14D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C8.C22 [×2], C22×D4, C2×C3⋊D4 [×6], S3×C23, C2×C8.C22, Q8.14D6 [×2], C22×C3⋊D4, C2×Q8.14D6

Smallest permutation representation of C2×Q8.14D6
On 96 points
Generators in S96
(1 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 66)(20 61)(21 62)(22 63)(23 64)(24 65)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 53)(38 54)(39 49)(40 50)(41 51)(42 52)(55 70)(56 71)(57 72)(58 67)(59 68)(60 69)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 13 47 63)(2 14 48 64)(3 15 43 65)(4 16 44 66)(5 17 45 61)(6 18 46 62)(7 35 22 26)(8 36 23 27)(9 31 24 28)(10 32 19 29)(11 33 20 30)(12 34 21 25)(37 94 68 77)(38 95 69 78)(39 96 70 73)(40 91 71 74)(41 92 72 75)(42 93 67 76)(49 90 55 79)(50 85 56 80)(51 86 57 81)(52 87 58 82)(53 88 59 83)(54 89 60 84)
(1 32 47 29)(2 30 48 33)(3 34 43 25)(4 26 44 35)(5 36 45 27)(6 28 46 31)(7 16 22 66)(8 61 23 17)(9 18 24 62)(10 63 19 13)(11 14 20 64)(12 65 21 15)(37 85 68 80)(38 81 69 86)(39 87 70 82)(40 83 71 88)(41 89 72 84)(42 79 67 90)(49 93 55 76)(50 77 56 94)(51 95 57 78)(52 73 58 96)(53 91 59 74)(54 75 60 92)
(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 96 47 73)(2 95 48 78)(3 94 43 77)(4 93 44 76)(5 92 45 75)(6 91 46 74)(7 49 22 55)(8 54 23 60)(9 53 24 59)(10 52 19 58)(11 51 20 57)(12 50 21 56)(13 39 63 70)(14 38 64 69)(15 37 65 68)(16 42 66 67)(17 41 61 72)(18 40 62 71)(25 85 34 80)(26 90 35 79)(27 89 36 84)(28 88 31 83)(29 87 32 82)(30 86 33 81)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D6 C3⋊D4 C3⋊D4 C8.C22 Q8.14D6 kernel C2×Q8.14D6 C2×C4.Dic3 C2×D4.S3 C2×C3⋊Q16 Q8.14D6 C22×Dic6 C6×C4○D4 C2×C4○D4 C2×C12 C22×C6 C22×C4 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C6 C2 # reps 1 1 2 2 8 1 1 1 3 1 1 1 1 4 6 2 2 4

Matrix representation of C2×Q8.14D6 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 1 0
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 72 0 0 0 0 0 0 1 0 0
,
 0 1 0 0 0 0 72 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 72 0 0 0 0 0 0 72 0 0
,
 45 3 0 0 0 0 31 28 0 0 0 0 0 0 18 55 71 71 0 0 55 55 71 2 0 0 71 71 18 55 0 0 71 2 55 55

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,72,0,0],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,72,0,0,0,0,0,0,72,0,0],[45,31,0,0,0,0,3,28,0,0,0,0,0,0,18,55,71,71,0,0,55,55,71,2,0,0,71,71,18,55,0,0,71,2,55,55] >;

C2×Q8.14D6 in GAP, Magma, Sage, TeX

C_2\times Q_8._{14}D_6
% in TeX

G:=Group("C2xQ8.14D6");
// GroupNames label

G:=SmallGroup(192,1382);
// by ID

G=gap.SmallGroup(192,1382);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,675,297,1684,235,102,6278]);
// Polycyclic

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

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