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## G = C22×C4.Dic3order 192 = 26·3

### Direct product of C22 and C4.Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C22×C4.Dic3
 Chief series C1 — C3 — C6 — C12 — C3⋊C8 — C2×C3⋊C8 — C22×C3⋊C8 — C22×C4.Dic3
 Lower central C3 — C6 — C22×C4.Dic3
 Upper central C1 — C22×C4 — C23×C4

Generators and relations for C22×C4.Dic3
G = < a,b,c,d,e | a2=b2=c4=1, d6=c2, e2=c2d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d5 >

Subgroups: 440 in 298 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C22×C4, C24, C3⋊C8, C2×C12, C22×C6, C22×C6, C22×C6, C22×C8, C2×M4(2), C23×C4, C2×C3⋊C8, C4.Dic3, C22×C12, C22×C12, C23×C6, C22×M4(2), C22×C3⋊C8, C2×C4.Dic3, C23×C12, C22×C4.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, M4(2), C22×C4, C24, C2×Dic3, C22×S3, C2×M4(2), C23×C4, C4.Dic3, C22×Dic3, S3×C23, C22×M4(2), C2×C4.Dic3, C23×Dic3, C22×C4.Dic3

Smallest permutation representation of C22×C4.Dic3
On 96 points
Generators in S96
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(1 22 7 16)(2 23 8 17)(3 24 9 18)(4 13 10 19)(5 14 11 20)(6 15 12 21)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 37 34 43)(29 38 35 44)(30 39 36 45)(49 64 55 70)(50 65 56 71)(51 66 57 72)(52 67 58 61)(53 68 59 62)(54 69 60 63)(73 88 79 94)(74 89 80 95)(75 90 81 96)(76 91 82 85)(77 92 83 86)(78 93 84 87)
(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 50 10 59 7 56 4 53)(2 55 11 52 8 49 5 58)(3 60 12 57 9 54 6 51)(13 62 22 71 19 68 16 65)(14 67 23 64 20 61 17 70)(15 72 24 69 21 66 18 63)(25 74 34 83 31 80 28 77)(26 79 35 76 32 73 29 82)(27 84 36 81 33 78 30 75)(37 86 46 95 43 92 40 89)(38 91 47 88 44 85 41 94)(39 96 48 93 45 90 42 87)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4H 4I 4J 4K 4L 6A ··· 6O 8A ··· 8P 12A ··· 12P order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 1 ··· 1 2 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C2 C4 C4 S3 Dic3 D6 Dic3 M4(2) C4.Dic3 kernel C22×C4.Dic3 C22×C3⋊C8 C2×C4.Dic3 C23×C12 C22×C12 C23×C6 C23×C4 C22×C4 C22×C4 C24 C2×C6 C22 # reps 1 2 12 1 14 2 1 7 7 1 8 16

Matrix representation of C22×C4.Dic3 in GL4(𝔽73) generated by

 1 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 1 0 0 0 0 27 0 0 0 30 46
,
 1 0 0 0 0 1 0 0 0 0 3 0 0 0 37 24
,
 1 0 0 0 0 1 0 0 0 0 66 71 0 0 38 7
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,27,30,0,0,0,46],[1,0,0,0,0,1,0,0,0,0,3,37,0,0,0,24],[1,0,0,0,0,1,0,0,0,0,66,38,0,0,71,7] >;

C22×C4.Dic3 in GAP, Magma, Sage, TeX

C_2^2\times C_4.{\rm Dic}_3
% in TeX

G:=Group("C2^2xC4.Dic3");
// GroupNames label

G:=SmallGroup(192,1340);
// by ID

G=gap.SmallGroup(192,1340);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,1123,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=1,d^6=c^2,e^2=c^2*d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^5>;
// generators/relations

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