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## G = (C2×Q8).51D6order 192 = 26·3

### 27th non-split extension by C2×Q8 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C2×Q8).51D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C23.26D6 — (C2×Q8).51D6
 Lower central C3 — C6 — C2×C12 — (C2×Q8).51D6
 Upper central C1 — C22 — C22×C4 — C22⋊Q8

Generators and relations for (C2×Q8).51D6
G = < a,b,c,d,e | a2=b4=1, c2=d6=b2, e2=a, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ab2c, ece-1=b-1c, ede-1=d5 >

Subgroups: 224 in 96 conjugacy classes, 39 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×2], C4 [×5], C22, C22 [×3], C6 [×3], C6, C8 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], C22×C4, C2×Q8, C3⋊C8 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], C22×C6, C22⋊C8, Q8⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C2×C3⋊C8 [×2], C4×Dic3, C4⋊Dic3 [×2], C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C6×Q8, C23.20D4, C6.Q16, C12.Q8, C12.55D4, Q82Dic3 [×2], C23.26D6, C3×C22⋊Q8, (C2×Q8).51D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C22.D4, C4○D8, C8.C22, D42S3 [×2], C2×C3⋊D4, C23.20D4, C23.23D6, Q8.11D6, Q8.13D6, (C2×Q8).51D6

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 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 C2 S3 D4 D4 D6 D6 D6 C4○D4 C3⋊D4 C3⋊D4 C4○D8 C8.C22 D4⋊2S3 Q8.11D6 Q8.13D6 kernel (C2×Q8).51D6 C6.Q16 C12.Q8 C12.55D4 Q8⋊2Dic3 C23.26D6 C3×C22⋊Q8 C22⋊Q8 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×Q8 C12 C2×C4 C23 C6 C6 C4 C2 C2 # reps 1 1 1 1 2 1 1 1 1 1 1 1 1 4 2 2 4 1 2 2 2

Matrix representation of (C2×Q8).51D6 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 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 46 0 0 0 0 0 32 27 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 17 71 0 0 0 0 71 56 0 0 0 0 0 0 12 2 0 0 0 0 37 61 0 0 0 0 0 0 30 13 0 0 0 0 60 43
,
 72 0 0 0 0 0 56 1 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 0 1 0 0 0 0 72 1
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 45 44 0 0 0 0 17 28 0 0 0 0 0 0 60 2 0 0 0 0 62 13

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,32,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[17,71,0,0,0,0,71,56,0,0,0,0,0,0,12,37,0,0,0,0,2,61,0,0,0,0,0,0,30,60,0,0,0,0,13,43],[72,56,0,0,0,0,0,1,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,45,17,0,0,0,0,44,28,0,0,0,0,0,0,60,62,0,0,0,0,2,13] >;

(C2×Q8).51D6 in GAP, Magma, Sage, TeX

(C_2\times Q_8)._{51}D_6
% in TeX

G:=Group("(C2xQ8).51D6");
// GroupNames label

G:=SmallGroup(192,604);
// by ID

G=gap.SmallGroup(192,604);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,254,219,184,1123,297,136,6278]);
// Polycyclic

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

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