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## G = Q8⋊7(C4×S3)  order 192 = 26·3

### 2nd semidirect product of Q8 and C4×S3 acting via C4×S3/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Q8⋊7(C4×S3)
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — S3×C2×C4 — C2×Q8⋊3S3 — Q8⋊7(C4×S3)
 Lower central C3 — C6 — C12 — Q8⋊7(C4×S3)
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for Q87(C4×S3)
G = < a,b,c,d,e | a4=c4=d3=e2=1, b2=a2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=ab, bd=db, ebe=a2b, cd=dc, ce=ec, ede=d-1 >

Subgroups: 488 in 162 conjugacy classes, 55 normal (37 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×8], S3 [×4], C6 [×3], C8 [×2], C2×C4, C2×C4 [×14], D4 [×7], Q8 [×2], Q8, C23 [×2], Dic3 [×2], Dic3, C12 [×2], C12 [×3], D6 [×2], D6 [×6], C2×C6, C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, M4(2) [×2], C22×C4 [×3], C2×D4 [×2], C2×Q8, C4○D4 [×6], C3⋊C8, C24, C4×S3 [×4], C4×S3 [×6], D12 [×2], D12 [×5], C2×Dic3, C2×Dic3, C2×C12, C2×C12 [×2], C3×Q8 [×2], C3×Q8, C22×S3, C22×S3, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C8⋊S3 [×2], C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4 [×2], C2×D12, C2×D12, Q83S3 [×4], Q83S3 [×2], C6×Q8, C23.36D4, C6.D8, C2.D24, Q82Dic3, C3×Q8⋊C4, S3×C4⋊C4, C2×C8⋊S3, C2×Q83S3, Q87(C4×S3)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C22×S3, C2×C22⋊C4, C8⋊C22, C8.C22, S3×C2×C4, S3×D4 [×2], C23.36D4, S3×C22⋊C4, Q83D6, Q16⋊S3, Q87(C4×S3)

Smallest permutation representation of Q87(C4×S3)
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 60 3 58)(2 59 4 57)(5 56 7 54)(6 55 8 53)(9 52 11 50)(10 51 12 49)(13 66 15 68)(14 65 16 67)(17 62 19 64)(18 61 20 63)(21 69 23 71)(22 72 24 70)(25 78 27 80)(26 77 28 79)(29 74 31 76)(30 73 32 75)(33 81 35 83)(34 84 36 82)(37 90 39 92)(38 89 40 91)(41 86 43 88)(42 85 44 87)(45 93 47 95)(46 96 48 94)
(1 46 22 34)(2 45 23 33)(3 48 24 36)(4 47 21 35)(5 76 85 67)(6 75 86 66)(7 74 87 65)(8 73 88 68)(9 78 89 63)(10 77 90 62)(11 80 91 61)(12 79 92 64)(13 56 32 44)(14 55 29 43)(15 54 30 42)(16 53 31 41)(17 52 26 40)(18 51 27 39)(19 50 28 38)(20 49 25 37)(57 94 69 82)(58 93 70 81)(59 96 71 84)(60 95 72 83)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 95 10)(6 96 11)(7 93 12)(8 94 9)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 39 44)(34 40 41)(35 37 42)(36 38 43)(45 51 56)(46 52 53)(47 49 54)(48 50 55)(57 63 68)(58 64 65)(59 61 66)(60 62 67)(69 78 73)(70 79 74)(71 80 75)(72 77 76)(81 92 87)(82 89 88)(83 90 85)(84 91 86)
(1 58)(2 59)(3 60)(4 57)(5 50)(6 51)(7 52)(8 49)(9 54)(10 55)(11 56)(12 53)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 84)(34 81)(35 82)(36 83)(37 88)(38 85)(39 86)(40 87)(41 92)(42 89)(43 90)(44 91)(45 96)(46 93)(47 94)(48 95)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D4 D4 D6 D6 D6 C4×S3 C8⋊C22 C8.C22 S3×D4 S3×D4 Q8⋊3D6 Q16⋊S3 kernel Q8⋊7(C4×S3) C6.D8 C2.D24 Q8⋊2Dic3 C3×Q8⋊C4 S3×C4⋊C4 C2×C8⋊S3 C2×Q8⋊3S3 Q8⋊3S3 Q8⋊C4 C4×S3 C2×Dic3 C22×S3 C4⋊C4 C2×C8 C2×Q8 Q8 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 1 1 4 1 1 1 1 2 2

Matrix representation of Q87(C4×S3) in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 1 0
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 67 37 36 0 0 67 0 37 37 0 0 36 36 6 0 0 0 37 36 0 67
,
 27 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 72 0 0 0 0 72 0 0 0 0 0 0 0 0 67 37 36 0 0 6 0 36 36 0 0 37 36 0 67 0 0 36 36 6 0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,67,36,37,0,0,67,0,36,36,0,0,37,37,6,0,0,0,36,37,0,67],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,6,37,36,0,0,67,0,36,36,0,0,37,36,0,6,0,0,36,36,67,0] >;

Q87(C4×S3) in GAP, Magma, Sage, TeX

Q_8\rtimes_7(C_4\times S_3)
% in TeX

G:=Group("Q8:7(C4xS3)");
// GroupNames label

G:=SmallGroup(192,362);
// by ID

G=gap.SmallGroup(192,362);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,219,58,851,438,102,6278]);
// Polycyclic

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

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