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## G = C2×Q8.13D6order 192 = 26·3

### Direct product of C2 and Q8.13D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×Q8.13D6
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — C2×C4○D12 — C2×Q8.13D6
 Lower central C3 — C6 — C12 — C2×Q8.13D6
 Upper central C1 — C2×C4 — C22×C4 — C2×C4○D4

Generators and relations for C2×Q8.13D6
G = < a,b,c,d,e | a2=b4=1, c2=d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d5 >

Subgroups: 616 in 266 conjugacy classes, 111 normal (35 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], S3 [×2], C6, C6 [×2], C6 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23, C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×6], C2×C8 [×6], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×4], C4○D4 [×8], C3⋊C8 [×4], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×5], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8, C22×S3, C22×C6, C22×C6, C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4, C2×C4○D4, C2×C3⋊C8 [×2], C2×C3⋊C8 [×4], D4⋊S3 [×4], D4.S3 [×4], Q82S3 [×4], C3⋊Q16 [×4], C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4 [×4], C3×C4○D4 [×2], C2×C4○D8, C22×C3⋊C8, C2×D4⋊S3, C2×D4.S3, C2×Q82S3, C2×C3⋊Q16, Q8.13D6 [×8], C2×C4○D12, C6×C4○D4, C2×Q8.13D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C4○D8 [×2], C22×D4, C2×C3⋊D4 [×6], S3×C23, C2×C4○D8, Q8.13D6 [×2], C22×C3⋊D4, C2×Q8.13D6

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D6 C3⋊D4 C3⋊D4 C4○D8 Q8.13D6 kernel C2×Q8.13D6 C22×C3⋊C8 C2×D4⋊S3 C2×D4.S3 C2×Q8⋊2S3 C2×C3⋊Q16 Q8.13D6 C2×C4○D12 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 1 1 1 1 8 1 1 1 3 1 1 1 1 4 6 2 8 4

Matrix representation of C2×Q8.13D6 in GL5(𝔽73)

 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 72
,
 72 0 0 0 0 0 67 67 0 0 0 67 6 0 0 0 0 0 30 60 0 0 0 13 43
,
 1 0 0 0 0 0 46 0 0 0 0 0 46 0 0 0 0 0 0 72 0 0 0 1 1
,
 72 0 0 0 0 0 46 0 0 0 0 0 27 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,67,67,0,0,0,67,6,0,0,0,0,0,30,13,0,0,0,60,43],[1,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,72,1],[72,0,0,0,0,0,46,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,1,0] >;

C2×Q8.13D6 in GAP, Magma, Sage, TeX

C_2\times Q_8._{13}D_6
% in TeX

G:=Group("C2xQ8.13D6");
// GroupNames label

G:=SmallGroup(192,1380);
// by ID

G=gap.SmallGroup(192,1380);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,1684,235,102,6278]);
// Polycyclic

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

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