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## G = C4×D4⋊S3order 192 = 26·3

### Direct product of C4 and D4⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4×D4⋊S3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — C2×D4⋊S3 — C4×D4⋊S3
 Lower central C3 — C6 — C12 — C4×D4⋊S3
 Upper central C1 — C2×C4 — C42 — C4×D4

Generators and relations for C4×D4⋊S3
G = < a,b,c,d,e | a4=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Subgroups: 376 in 134 conjugacy classes, 55 normal (39 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×3], C22, C22 [×8], S3 [×2], C6 [×3], C6 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×2], D4 [×4], C23 [×2], Dic3, C12 [×2], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×4], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4, C2×D4, C3⋊C8 [×2], C3⋊C8, C4×S3 [×2], D12 [×2], D12, C2×Dic3, C2×C12 [×3], C2×C12 [×3], C3×D4 [×2], C3×D4, C22×S3, C22×C6, C4×C8, D4⋊C4 [×2], C2.D8, C4×D4, C4×D4, C2×D8, C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, D4⋊S3 [×4], C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×C12, C6×D4, C4×D8, C4×C3⋊C8, C6.Q16, C6.D8, D4⋊Dic3, C4×D12, C2×D4⋊S3, D4×C12, C4×D4⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], D8 [×2], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4×D4, C2×D8, C4○D8, D4⋊S3 [×2], S3×C2×C4, C4○D12, C2×C3⋊D4, C4×D8, C4×C3⋊D4, C2×D4⋊S3, Q8.13D6, C4×D4⋊S3

Smallest permutation representation of C4×D4⋊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 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 83)(2 84)(3 81)(4 82)(5 90)(6 91)(7 92)(8 89)(9 55)(10 56)(11 53)(12 54)(13 30)(14 31)(15 32)(16 29)(17 77)(18 78)(19 79)(20 80)(21 65)(22 66)(23 67)(24 68)(25 36)(26 33)(27 34)(28 35)(37 42)(38 43)(39 44)(40 41)(45 85)(46 86)(47 87)(48 88)(49 59)(50 60)(51 57)(52 58)(61 95)(62 96)(63 93)(64 94)(69 75)(70 76)(71 73)(72 74)
(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)
(1 3)(2 4)(5 20)(6 17)(7 18)(8 19)(9 33)(10 34)(11 35)(12 36)(13 90)(14 91)(15 92)(16 89)(21 43)(22 44)(23 41)(24 42)(25 84)(26 81)(27 82)(28 83)(29 47)(30 48)(31 45)(32 46)(37 78)(38 79)(39 80)(40 77)(49 51)(50 52)(53 59)(54 60)(55 57)(56 58)(61 75)(62 76)(63 73)(64 74)(65 93)(66 94)(67 95)(68 96)(69 85)(70 86)(71 87)(72 88)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 6D 6E 6F 6G 8A ··· 8H 12A 12B 12C 12D 12E ··· 12L 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 6 6 6 6 8 ··· 8 12 12 12 12 12 ··· 12 size 1 1 1 1 4 4 12 12 2 1 1 1 1 2 2 2 2 4 4 12 12 2 2 2 4 4 4 4 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 D8 C4○D4 C3⋊D4 C4×S3 C4○D8 C4○D12 D4⋊S3 Q8.13D6 kernel C4×D4⋊S3 C4×C3⋊C8 C6.Q16 C6.D8 D4⋊Dic3 C4×D12 C2×D4⋊S3 D4×C12 D4⋊S3 C4×D4 C2×C12 C42 C4⋊C4 C2×D4 C12 C12 C2×C4 D4 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×D4⋊S3 in GL5(𝔽73)

 27 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 46 0 0 0 0 0 46
,
 1 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 72 0
,
 72 0 0 0 0 0 43 13 0 0 0 60 30 0 0 0 0 0 57 16 0 0 0 16 16
,
 1 0 0 0 0 0 0 1 0 0 0 72 72 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 72 0 0 0 0 0 1

G:=sub<GL(5,GF(73))| [27,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,46,0,0,0,0,0,46],[1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,1,0],[72,0,0,0,0,0,43,60,0,0,0,13,30,0,0,0,0,0,57,16,0,0,0,16,16],[1,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,72,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1] >;

C4×D4⋊S3 in GAP, Magma, Sage, TeX

C_4\times D_4\rtimes S_3
% in TeX

G:=Group("C4xD4:S3");
// GroupNames label

G:=SmallGroup(192,572);
// by ID

G=gap.SmallGroup(192,572);
# by ID

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

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

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