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## G = C4×S3×Q8order 192 = 26·3

### Direct product of C4, S3 and Q8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4×S3×Q8
G = < a,b,c,d,e | a4=b3=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 568 in 298 conjugacy classes, 169 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C2×C42, C2×C4⋊C4, C4×Q8, C4×Q8, C22×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, S3×Q8, C6×Q8, C2×C4×Q8, C4×Dic6, S3×C42, Dic6⋊C4, S3×C4⋊C4, Q8×Dic3, Q8×C12, C2×S3×Q8, C4×S3×Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, C24, C4×S3, C22×S3, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, S3×C2×C4, S3×Q8, S3×C23, C2×C4×Q8, S3×C22×C4, C2×S3×Q8, S3×C4○D4, C4×S3×Q8

Smallest permutation representation of C4×S3×Q8
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 37 23)(2 38 24)(3 39 21)(4 40 22)(5 84 94)(6 81 95)(7 82 96)(8 83 93)(9 29 27)(10 30 28)(11 31 25)(12 32 26)(13 91 17)(14 92 18)(15 89 19)(16 90 20)(33 47 61)(34 48 62)(35 45 63)(36 46 64)(41 54 52)(42 55 49)(43 56 50)(44 53 51)(57 71 85)(58 72 86)(59 69 87)(60 70 88)(65 78 76)(66 79 73)(67 80 74)(68 77 75)
(5 84)(6 81)(7 82)(8 83)(9 29)(10 30)(11 31)(12 32)(17 91)(18 92)(19 89)(20 90)(21 39)(22 40)(23 37)(24 38)(33 61)(34 62)(35 63)(36 64)(41 54)(42 55)(43 56)(44 53)(57 85)(58 86)(59 87)(60 88)(65 78)(66 79)(67 80)(68 77)
(1 49 27 45)(2 50 28 46)(3 51 25 47)(4 52 26 48)(5 65 91 86)(6 66 92 87)(7 67 89 88)(8 68 90 85)(9 63 37 42)(10 64 38 43)(11 61 39 44)(12 62 40 41)(13 72 94 76)(14 69 95 73)(15 70 96 74)(16 71 93 75)(17 58 84 78)(18 59 81 79)(19 60 82 80)(20 57 83 77)(21 53 31 33)(22 54 32 34)(23 55 29 35)(24 56 30 36)
(1 73 27 69)(2 74 28 70)(3 75 25 71)(4 76 26 72)(5 62 91 41)(6 63 92 42)(7 64 89 43)(8 61 90 44)(9 87 37 66)(10 88 38 67)(11 85 39 68)(12 86 40 65)(13 52 94 48)(14 49 95 45)(15 50 96 46)(16 51 93 47)(17 54 84 34)(18 55 81 35)(19 56 82 36)(20 53 83 33)(21 77 31 57)(22 78 32 58)(23 79 29 59)(24 80 30 60)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E ··· 4P 4Q 4R 4S 4T 4U ··· 4AF 6A 6B 6C 12A 12B 12C 12D 12E ··· 12P order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 4 4 4 4 ··· 4 6 6 6 12 12 12 12 12 ··· 12 size 1 1 1 1 3 3 3 3 2 1 1 1 1 2 ··· 2 3 3 3 3 6 ··· 6 2 2 2 2 2 2 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 Q8 D6 D6 D6 C4○D4 C4×S3 S3×Q8 S3×C4○D4 kernel C4×S3×Q8 C4×Dic6 S3×C42 Dic6⋊C4 S3×C4⋊C4 Q8×Dic3 Q8×C12 C2×S3×Q8 S3×Q8 C4×Q8 C4×S3 C42 C4⋊C4 C2×Q8 D6 Q8 C4 C2 # reps 1 3 3 3 3 1 1 1 16 1 4 3 3 1 4 8 2 2

Matrix representation of C4×S3×Q8 in GL4(𝔽13) generated by

 8 0 0 0 0 8 0 0 0 0 5 0 0 0 0 5
,
 0 12 0 0 1 12 0 0 0 0 1 0 0 0 0 1
,
 1 12 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 0 12 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 9 3 0 0 3 4
G:=sub<GL(4,GF(13))| [8,0,0,0,0,8,0,0,0,0,5,0,0,0,0,5],[0,1,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,9,3,0,0,3,4] >;

C4×S3×Q8 in GAP, Magma, Sage, TeX

C_4\times S_3\times Q_8
% in TeX

G:=Group("C4xS3xQ8");
// GroupNames label

G:=SmallGroup(192,1130);
// by ID

G=gap.SmallGroup(192,1130);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,80,6278]);
// Polycyclic

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

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