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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×Dic6 — 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=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 280 in 122 conjugacy classes, 55 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D4.S3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×D4, C4×SD16, C4×C3⋊C8, C12.Q8, C6.SD16, D4⋊Dic3, C4×Dic6, C2×D4.S3, D4×C12, C4×D4.S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, SD16, C22×C4, C2×D4, C4○D4, C4×S3, C3⋊D4, C22×S3, C4×D4, C2×SD16, C4○D8, D4.S3, S3×C2×C4, C4○D12, C2×C3⋊D4, C4×SD16, 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 9)(2 10)(3 11)(4 12)(5 42)(6 43)(7 44)(8 41)(13 15)(14 16)(17 29)(18 30)(19 31)(20 32)(21 75)(22 76)(23 73)(24 74)(25 54)(26 55)(27 56)(28 53)(33 49)(34 50)(35 51)(36 52)(37 39)(38 40)(45 63)(46 64)(47 61)(48 62)(57 59)(58 60)(65 67)(66 68)(69 79)(70 80)(71 77)(72 78)(81 83)(82 84)(85 87)(86 88)(89 95)(90 96)(91 93)(92 94)
(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 86 60)(14 87 57)(15 88 58)(16 85 59)(17 33 63)(18 34 64)(19 35 61)(20 36 62)(25 80 96)(26 77 93)(27 78 94)(28 79 95)(29 49 45)(30 50 46)(31 51 47)(32 52 48)(37 82 66)(38 83 67)(39 84 68)(40 81 65)(53 69 89)(54 70 90)(55 71 91)(56 72 92)
(1 28 51 55)(2 25 52 56)(3 26 49 53)(4 27 50 54)(5 37 64 86)(6 38 61 87)(7 39 62 88)(8 40 63 85)(9 83 35 57)(10 84 36 58)(11 81 33 59)(12 82 34 60)(13 76 66 18)(14 73 67 19)(15 74 68 20)(16 75 65 17)(21 77 29 69)(22 78 30 70)(23 79 31 71)(24 80 32 72)(41 93 45 89)(42 94 46 90)(43 95 47 91)(44 96 48 92)

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

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

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

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 6D 6E 6F 6G 8A ··· 8H 12A 12B 12C 12D 12E ··· 12L 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 6 6 6 6 8 ··· 8 12 12 12 12 12 ··· 12 size 1 1 1 1 4 4 2 1 1 1 1 2 2 2 2 4 4 12 12 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 SD16 C4○D4 C3⋊D4 C4×S3 C4○D8 C4○D12 D4.S3 Q8.13D6 kernel C4×D4.S3 C4×C3⋊C8 C12.Q8 C6.SD16 D4⋊Dic3 C4×Dic6 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 GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 71 0 0 0 0 1 72 0 0 0 0 0 0 1 71 0 0 0 0 1 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 71 0 0 0 0 0 72 0 0 0 0 0 0 72 2 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 7 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 64 0 0 0 0 0 23 8
,
 0 61 0 0 0 0 67 0 0 0 0 0 0 0 0 12 0 0 0 0 6 0 0 0 0 0 0 0 3 53 0 0 0 0 15 70

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,1,0,0,0,0,71,72,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,71,72,0,0,0,0,0,0,72,0,0,0,0,0,2,1,0,0,0,0,0,0,72,7,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,23,0,0,0,0,0,8],[0,67,0,0,0,0,61,0,0,0,0,0,0,0,0,6,0,0,0,0,12,0,0,0,0,0,0,0,3,15,0,0,0,0,53,70] >;

C4×D4.S3 in GAP, Magma, Sage, TeX

C_4\times D_4.S_3
% in TeX

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

G:=SmallGroup(192,576);
// by ID

G=gap.SmallGroup(192,576);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,58,1684,851,102,6278]);
// Polycyclic

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

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