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## G = C4×Q8⋊2S3order 192 = 26·3

### Direct product of C4 and Q8⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4×Q8⋊2S3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — C2×Q8⋊2S3 — C4×Q8⋊2S3
 Lower central C3 — C6 — C12 — C4×Q8⋊2S3
 Upper central C1 — C2×C4 — C42 — C4×Q8

Generators and relations for C4×Q82S3
G = < a,b,c,d,e | a4=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

Subgroups: 328 in 122 conjugacy classes, 55 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×5], C22, C22 [×4], S3 [×2], C6 [×3], C8 [×3], C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×2], Q8, C23, Dic3, C12 [×2], C12 [×2], C12 [×4], D6 [×4], C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8, C3⋊C8 [×2], C3⋊C8, C4×S3 [×2], D12 [×2], D12, C2×Dic3, C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C3×Q8, C22×S3, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, Q82S3 [×4], C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C6×Q8, C4×SD16, C4×C3⋊C8, C12.Q8, C6.D8, Q82Dic3, C4×D12, C2×Q82S3, Q8×C12, C4×Q82S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], SD16 [×2], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4×D4, C2×SD16, C4○D8, Q82S3 [×2], S3×C2×C4, C4○D12, C2×C3⋊D4, C4×SD16, C4×C3⋊D4, C2×Q82S3, Q8.13D6, C4×Q82S3

Smallest permutation representation of C4×Q82S3
On 96 points
Generators in S96
(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 33 51 11)(2 34 52 12)(3 35 49 9)(4 36 50 10)(5 44 64 48)(6 41 61 45)(7 42 62 46)(8 43 63 47)(13 80 66 72)(14 77 67 69)(15 78 68 70)(16 79 65 71)(17 31 75 23)(18 32 76 24)(19 29 73 21)(20 30 74 22)(25 82 56 60)(26 83 53 57)(27 84 54 58)(28 81 55 59)(37 92 86 96)(38 89 87 93)(39 90 88 94)(40 91 85 95)
(1 26 51 53)(2 27 52 54)(3 28 49 55)(4 25 50 56)(5 39 64 88)(6 40 61 85)(7 37 62 86)(8 38 63 87)(9 81 35 59)(10 82 36 60)(11 83 33 57)(12 84 34 58)(13 74 66 20)(14 75 67 17)(15 76 68 18)(16 73 65 19)(21 79 29 71)(22 80 30 72)(23 77 31 69)(24 78 32 70)(41 95 45 91)(42 96 46 92)(43 93 47 89)(44 94 48 90)
(1 43 23)(2 44 24)(3 41 21)(4 42 22)(5 76 12)(6 73 9)(7 74 10)(8 75 11)(13 60 86)(14 57 87)(15 58 88)(16 59 85)(17 33 63)(18 34 64)(19 35 61)(20 36 62)(25 96 80)(26 93 77)(27 94 78)(28 95 79)(29 49 45)(30 50 46)(31 51 47)(32 52 48)(37 66 82)(38 67 83)(39 68 84)(40 65 81)(53 89 69)(54 90 70)(55 91 71)(56 92 72)
(5 18)(6 19)(7 20)(8 17)(9 35)(10 36)(11 33)(12 34)(13 92)(14 89)(15 90)(16 91)(21 41)(22 42)(23 43)(24 44)(25 82)(26 83)(27 84)(28 81)(29 45)(30 46)(31 47)(32 48)(37 80)(38 77)(39 78)(40 79)(53 57)(54 58)(55 59)(56 60)(61 73)(62 74)(63 75)(64 76)(65 95)(66 96)(67 93)(68 94)(69 87)(70 88)(71 85)(72 86)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 6A 6B 6C 8A ··· 8H 12A 12B 12C 12D 12E ··· 12P order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 8 ··· 8 12 12 12 12 12 ··· 12 size 1 1 1 1 12 12 2 1 1 1 1 2 2 2 2 4 4 4 4 12 12 2 2 2 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 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 D6 SD16 C4○D4 C3⋊D4 C4×S3 C4○D8 C4○D12 Q8⋊2S3 Q8.13D6 kernel C4×Q8⋊2S3 C4×C3⋊C8 C12.Q8 C6.D8 Q8⋊2Dic3 C4×D12 C2×Q8⋊2S3 Q8×C12 Q8⋊2S3 C4×Q8 C2×C12 C42 C4⋊C4 C2×Q8 C12 C12 C2×C4 Q8 C6 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 4 2 4 4 4 4 2 2

Matrix representation of C4×Q82S3 in GL4(𝔽73) generated by

 27 0 0 0 0 27 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 3 0 0 48 72
,
 1 0 0 0 0 1 0 0 0 0 61 55 0 0 4 12
,
 72 1 0 0 72 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 48 72
G:=sub<GL(4,GF(73))| [27,0,0,0,0,27,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,48,0,0,3,72],[1,0,0,0,0,1,0,0,0,0,61,4,0,0,55,12],[72,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,48,0,0,0,72] >;

C4×Q82S3 in GAP, Magma, Sage, TeX

C_4\times Q_8\rtimes_2S_3
% in TeX

G:=Group("C4xQ8:2S3");
// GroupNames label

G:=SmallGroup(192,584);
// by ID

G=gap.SmallGroup(192,584);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,58,1684,851,102,6278]);
// Polycyclic

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

׿
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