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## G = C2×S3×C4⋊C4order 192 = 26·3

### Direct product of C2, S3 and C4⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×S3×C4⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22×C4 — C2×S3×C4⋊C4
 Lower central C3 — C6 — C2×S3×C4⋊C4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×S3×C4⋊C4
G = < a,b,c,d,e | a2=b3=c2=d4=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 888 in 418 conjugacy classes, 207 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C2×C4⋊C4, C2×C4⋊C4, C23×C4, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, S3×C23, C22×C4⋊C4, S3×C4⋊C4, C2×Dic3⋊C4, C2×C4⋊Dic3, C6×C4⋊C4, S3×C22×C4, S3×C22×C4, C2×S3×C4⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C4×S3, C22×S3, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, S3×C2×C4, S3×D4, S3×Q8, S3×C23, C22×C4⋊C4, S3×C4⋊C4, S3×C22×C4, C2×S3×D4, C2×S3×Q8, C2×S3×C4⋊C4

Smallest permutation representation of C2×S3×C4⋊C4
On 96 points
Generators in S96
(1 72)(2 69)(3 70)(4 71)(5 53)(6 54)(7 55)(8 56)(9 35)(10 36)(11 33)(12 34)(13 46)(14 47)(15 48)(16 45)(17 61)(18 62)(19 63)(20 64)(21 31)(22 32)(23 29)(24 30)(25 88)(26 85)(27 86)(28 87)(37 82)(38 83)(39 84)(40 81)(41 80)(42 77)(43 78)(44 79)(49 75)(50 76)(51 73)(52 74)(57 91)(58 92)(59 89)(60 90)(65 95)(66 96)(67 93)(68 94)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 44 39)(6 41 40)(7 42 37)(8 43 38)(9 73 58)(10 74 59)(11 75 60)(12 76 57)(21 85 66)(22 86 67)(23 87 68)(24 88 65)(25 95 30)(26 96 31)(27 93 32)(28 94 29)(33 49 90)(34 50 91)(35 51 92)(36 52 89)(45 61 72)(46 62 69)(47 63 70)(48 64 71)(53 79 84)(54 80 81)(55 77 82)(56 78 83)
(9 73)(10 74)(11 75)(12 76)(13 18)(14 19)(15 20)(16 17)(21 85)(22 86)(23 87)(24 88)(25 30)(26 31)(27 32)(28 29)(33 49)(34 50)(35 51)(36 52)(37 42)(38 43)(39 44)(40 41)(45 61)(46 62)(47 63)(48 64)(77 82)(78 83)(79 84)(80 81)
(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 95 91 54)(2 94 92 53)(3 93 89 56)(4 96 90 55)(5 69 68 58)(6 72 65 57)(7 71 66 60)(8 70 67 59)(9 44 46 23)(10 43 47 22)(11 42 48 21)(12 41 45 24)(13 29 35 79)(14 32 36 78)(15 31 33 77)(16 30 34 80)(17 25 50 81)(18 28 51 84)(19 27 52 83)(20 26 49 82)(37 64 85 75)(38 63 86 74)(39 62 87 73)(40 61 88 76)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 4A ··· 4L 4M ··· 4X 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 3 ··· 3 2 2 ··· 2 6 ··· 6 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 Q8 D6 D6 C4×S3 S3×D4 S3×Q8 kernel C2×S3×C4⋊C4 S3×C4⋊C4 C2×Dic3⋊C4 C2×C4⋊Dic3 C6×C4⋊C4 S3×C22×C4 S3×C2×C4 C2×C4⋊C4 C22×S3 C22×S3 C4⋊C4 C22×C4 C2×C4 C22 C22 # reps 1 8 2 1 1 3 16 1 4 4 4 3 8 2 2

Matrix representation of C2×S3×C4⋊C4 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
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 12 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 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 10 9 0 0 0 0 9 3
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 0

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],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,12,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,9,0,0,0,0,9,3],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;

C2×S3×C4⋊C4 in GAP, Magma, Sage, TeX

C_2\times S_3\times C_4\rtimes C_4
% in TeX

G:=Group("C2xS3xC4:C4");
// GroupNames label

G:=SmallGroup(192,1060);
// by ID

G=gap.SmallGroup(192,1060);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,297,80,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^4=e^4=1,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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