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## G = (C3×Q8)⋊13D4order 192 = 26·3

### 1st semidirect product of C3×Q8 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C3×Q8)⋊13D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C12⋊7D4 — (C3×Q8)⋊13D4
 Lower central C3 — C6 — C2×C12 — (C3×Q8)⋊13D4
 Upper central C1 — C22 — C22×C4 — C22×Q8

Generators and relations for (C3×Q8)⋊13D4
G = < a,b,c,d,e | a3=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, dad-1=eae=a-1, cbc-1=dbd-1=ebe=b-1, dcd-1=ece=b-1c, ede=d-1 >

Subgroups: 424 in 158 conjugacy classes, 51 normal (25 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×6], C22, C22 [×2], C22 [×5], S3, C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×9], D4 [×4], Q8 [×4], Q8 [×6], C23, C23, Dic3, C12 [×2], C12 [×5], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×4], C22×C4, C22×C4, C2×D4 [×2], C2×Q8 [×2], C2×Q8 [×5], C3⋊C8 [×2], D12 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×8], C3×Q8 [×4], C3×Q8 [×6], C22×S3, C22×C6, C22⋊C8, Q8⋊C4 [×2], C4⋊D4, C2×SD16 [×2], C22×Q8, C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, Q82S3 [×4], C2×D12, C2×C3⋊D4, C22×C12, C22×C12, C6×Q8 [×2], C6×Q8 [×5], Q8⋊D4, C12.55D4, Q82Dic3 [×2], C127D4, C2×Q82S3 [×2], Q8×C2×C6, (C3×Q8)⋊13D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], SD16 [×2], C2×D4 [×3], C3⋊D4 [×6], C22×S3, C22≀C2, C2×SD16, C8.C22, Q82S3 [×2], C2×C3⋊D4 [×3], Q8⋊D4, C2×Q82S3, Q8.11D6, C244S3, (C3×Q8)⋊13D4

Smallest permutation representation of (C3×Q8)⋊13D4
On 96 points
Generators in S96
(1 35 39)(2 36 40)(3 33 37)(4 34 38)(5 10 14)(6 11 15)(7 12 16)(8 9 13)(17 22 89)(18 23 90)(19 24 91)(20 21 92)(25 45 30)(26 46 31)(27 47 32)(28 48 29)(41 60 64)(42 57 61)(43 58 62)(44 59 63)(49 54 69)(50 55 70)(51 56 71)(52 53 72)(65 82 86)(66 83 87)(67 84 88)(68 81 85)(73 93 78)(74 94 79)(75 95 80)(76 96 77)
(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 92 3 90)(2 91 4 89)(5 95 7 93)(6 94 8 96)(9 77 11 79)(10 80 12 78)(13 76 15 74)(14 75 16 73)(17 36 19 34)(18 35 20 33)(21 37 23 39)(22 40 24 38)(25 56 27 54)(26 55 28 53)(29 52 31 50)(30 51 32 49)(41 67 43 65)(42 66 44 68)(45 71 47 69)(46 70 48 72)(57 83 59 81)(58 82 60 84)(61 87 63 85)(62 86 64 88)
(1 66 5 71)(2 65 6 70)(3 68 7 69)(4 67 8 72)(9 53 34 88)(10 56 35 87)(11 55 36 86)(12 54 33 85)(13 52 38 84)(14 51 39 83)(15 50 40 82)(16 49 37 81)(17 63 77 27)(18 62 78 26)(19 61 79 25)(20 64 80 28)(21 60 75 29)(22 59 76 32)(23 58 73 31)(24 57 74 30)(41 95 48 92)(42 94 45 91)(43 93 46 90)(44 96 47 89)
(2 4)(6 8)(9 15)(10 14)(11 13)(12 16)(17 21)(18 24)(19 23)(20 22)(25 58)(26 57)(27 60)(28 59)(29 63)(30 62)(31 61)(32 64)(33 37)(34 40)(35 39)(36 38)(41 47)(42 46)(43 45)(44 48)(49 85)(50 88)(51 87)(52 86)(53 82)(54 81)(55 84)(56 83)(65 72)(66 71)(67 70)(68 69)(73 79)(74 78)(75 77)(76 80)(89 92)(90 91)(93 94)(95 96)

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C ··· 4G 4H 6A ··· 6G 8A 8B 8C 8D 12A ··· 12L order 1 2 2 2 2 2 2 3 4 4 4 ··· 4 4 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 1 1 2 2 24 2 2 2 4 ··· 4 24 2 ··· 2 12 12 12 12 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 SD16 C3⋊D4 C3⋊D4 C3⋊D4 C8.C22 Q8⋊2S3 Q8.11D6 kernel (C3×Q8)⋊13D4 C12.55D4 Q8⋊2Dic3 C12⋊7D4 C2×Q8⋊2S3 Q8×C2×C6 C22×Q8 C2×C12 C3×Q8 C22×C6 C22×C4 C2×Q8 C2×C6 C2×C4 Q8 C23 C6 C22 C2 # reps 1 1 2 1 2 1 1 1 4 1 1 2 4 2 8 2 1 2 2

Matrix representation of (C3×Q8)⋊13D4 in GL4(𝔽73) generated by

 72 72 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 0 72 0 0 0 0 1 71 0 0 1 72
,
 30 60 0 0 13 43 0 0 0 0 12 61 0 0 6 61
,
 30 60 0 0 30 43 0 0 0 0 72 0 0 0 72 1
,
 1 0 0 0 72 72 0 0 0 0 1 0 0 0 1 72
G:=sub<GL(4,GF(73))| [72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,1,1,0,0,71,72],[30,13,0,0,60,43,0,0,0,0,12,6,0,0,61,61],[30,30,0,0,60,43,0,0,0,0,72,72,0,0,0,1],[1,72,0,0,0,72,0,0,0,0,1,1,0,0,0,72] >;

(C3×Q8)⋊13D4 in GAP, Magma, Sage, TeX

(C_3\times Q_8)\rtimes_{13}D_4
% in TeX

G:=Group("(C3xQ8):13D4");
// GroupNames label

G:=SmallGroup(192,786);
// by ID

G=gap.SmallGroup(192,786);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,184,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=d^4=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

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