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## G = C2×Q8.15D6order 192 = 26·3

### Direct product of C2 and Q8.15D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×Q8.15D6
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C2×C4 — C2×S3×Q8 — C2×Q8.15D6
 Lower central C3 — C6 — C2×Q8.15D6
 Upper central C1 — C22 — C22×Q8

Generators and relations for C2×Q8.15D6
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, dcd-1=ece-1=b2c, ede-1=d5 >

Subgroups: 1512 in 794 conjugacy classes, 447 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×Q8, C22×S3, C22×C6, C22×Q8, C22×Q8, C2×C4○D4, 2- 1+4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, S3×Q8, Q83S3, C2×C3⋊D4, C22×C12, C6×Q8, C2×2- 1+4, C2×C4○D12, C2×S3×Q8, C2×Q83S3, Q8.15D6, Q8×C2×C6, C2×Q8.15D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2- 1+4, C25, S3×C23, C2×2- 1+4, Q8.15D6, S3×C24, C2×Q8.15D6

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

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F ··· 2M 3 4A ··· 4L 4M ··· 4T 6A ··· 6G 12A ··· 12L order 1 2 2 2 2 2 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 6 ··· 6 2 2 ··· 2 6 ··· 6 2 ··· 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 S3 D6 D6 2- 1+4 Q8.15D6 kernel C2×Q8.15D6 C2×C4○D12 C2×S3×Q8 C2×Q8⋊3S3 Q8.15D6 Q8×C2×C6 C22×Q8 C22×C4 C2×Q8 C6 C2 # reps 1 6 4 4 16 1 1 3 12 2 4

Matrix representation of C2×Q8.15D6 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 12 4 0 0 0 1 6 0 4 0 0 4 0 6 1 0 0 0 4 12 7
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 12 0 0 0 0 1 0 0 0 0 0 0 0 3 10 8 8 0 0 3 0 5 3 0 0 8 8 10 3 0 0 5 3 10 0
,
 1 0 0 0 0 0 1 12 0 0 0 0 0 0 2 9 0 0 0 0 11 11 0 0 0 0 0 0 11 4 0 0 0 0 2 2

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,1,4,0,0,0,12,6,0,4,0,0,4,0,6,12,0,0,0,4,1,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,12,0,0],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,3,3,8,5,0,0,10,0,8,3,0,0,8,5,10,10,0,0,8,3,3,0],[1,1,0,0,0,0,0,12,0,0,0,0,0,0,2,11,0,0,0,0,9,11,0,0,0,0,0,0,11,2,0,0,0,0,4,2] >;

C2×Q8.15D6 in GAP, Magma, Sage, TeX

C_2\times Q_8._{15}D_6
% in TeX

G:=Group("C2xQ8.15D6");
// GroupNames label

G:=SmallGroup(192,1519);
// by ID

G=gap.SmallGroup(192,1519);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,297,136,1684,6278]);
// 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,d*c*d^-1=e*c*e^-1=b^2*c,e*d*e^-1=d^5>;
// generators/relations

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