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## G = C2×C4×C3⋊D4order 192 = 26·3

### Direct product of C2×C4 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C4×C3⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×C3⋊D4 — C2×C4×C3⋊D4
 Lower central C3 — C6 — C2×C4×C3⋊D4
 Upper central C1 — C22×C4 — C23×C4

Generators and relations for C2×C4×C3⋊D4
G = < a,b,c,d,e | a2=b4=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 920 in 426 conjugacy classes, 183 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C3, C4 [×4], C4 [×10], C22, C22 [×10], C22 [×28], S3 [×4], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×8], C2×C4 [×32], D4 [×16], C23, C23 [×6], C23 [×14], Dic3 [×4], Dic3 [×4], C12 [×4], C12 [×2], D6 [×4], D6 [×12], C2×C6, C2×C6 [×10], C2×C6 [×12], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×15], C2×D4 [×12], C24, C24, C4×S3 [×8], C2×Dic3 [×10], C2×Dic3 [×4], C3⋊D4 [×16], C2×C12 [×8], C2×C12 [×10], C22×S3 [×6], C22×S3 [×4], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4, C23×C4, C22×D4, C4×Dic3 [×4], Dic3⋊C4 [×4], D6⋊C4 [×4], C6.D4 [×4], S3×C2×C4 [×4], S3×C2×C4 [×4], C22×Dic3 [×3], C2×C3⋊D4 [×12], C22×C12 [×2], C22×C12 [×4], C22×C12 [×4], S3×C23, C23×C6, C2×C4×D4, C2×C4×Dic3, C2×Dic3⋊C4, C2×D6⋊C4, C4×C3⋊D4 [×8], C2×C6.D4, S3×C22×C4, C22×C3⋊D4, C23×C12, C2×C4×C3⋊D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], D4 [×4], C23 [×15], D6 [×7], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×S3 [×4], C3⋊D4 [×4], C22×S3 [×7], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, S3×C2×C4 [×6], C4○D12 [×2], C2×C3⋊D4 [×6], S3×C23, C2×C4×D4, C4×C3⋊D4 [×4], S3×C22×C4, C2×C4○D12, C22×C3⋊D4, C2×C4×C3⋊D4

Smallest permutation representation of C2×C4×C3⋊D4
On 96 points
Generators in S96
(1 51)(2 52)(3 49)(4 50)(5 28)(6 25)(7 26)(8 27)(9 91)(10 92)(11 89)(12 90)(13 33)(14 34)(15 35)(16 36)(17 85)(18 86)(19 87)(20 88)(21 46)(22 47)(23 48)(24 45)(29 60)(30 57)(31 58)(32 59)(37 80)(38 77)(39 78)(40 79)(41 70)(42 71)(43 72)(44 69)(53 84)(54 81)(55 82)(56 83)(61 73)(62 74)(63 75)(64 76)(65 94)(66 95)(67 96)(68 93)
(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 41)(2 38 42)(3 39 43)(4 40 44)(5 84 22)(6 81 23)(7 82 24)(8 83 21)(9 17 30)(10 18 31)(11 19 32)(12 20 29)(13 96 73)(14 93 74)(15 94 75)(16 95 76)(25 54 48)(26 55 45)(27 56 46)(28 53 47)(33 67 61)(34 68 62)(35 65 63)(36 66 64)(49 78 72)(50 79 69)(51 80 70)(52 77 71)(57 91 85)(58 92 86)(59 89 87)(60 90 88)
(1 8 19 75)(2 5 20 76)(3 6 17 73)(4 7 18 74)(9 13 43 81)(10 14 44 82)(11 15 41 83)(12 16 42 84)(21 32 94 37)(22 29 95 38)(23 30 96 39)(24 31 93 40)(25 85 61 49)(26 86 62 50)(27 87 63 51)(28 88 64 52)(33 72 54 91)(34 69 55 92)(35 70 56 89)(36 71 53 90)(45 58 68 79)(46 59 65 80)(47 60 66 77)(48 57 67 78)
(5 76)(6 73)(7 74)(8 75)(9 30)(10 31)(11 32)(12 29)(13 23)(14 24)(15 21)(16 22)(25 61)(26 62)(27 63)(28 64)(33 48)(34 45)(35 46)(36 47)(37 41)(38 42)(39 43)(40 44)(53 66)(54 67)(55 68)(56 65)(57 91)(58 92)(59 89)(60 90)(69 79)(70 80)(71 77)(72 78)(81 96)(82 93)(83 94)(84 95)

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 kernel C2×C4×C3⋊D4 C2×C4×Dic3 C2×Dic3⋊C4 C2×D6⋊C4 C4×C3⋊D4 C2×C6.D4 S3×C22×C4 C22×C3⋊D4 C23×C12 C2×C3⋊D4 C23×C4 C2×C12 C22×C4 C24 C2×C6 C2×C4 C23 C22 # reps 1 1 1 1 8 1 1 1 1 16 1 4 6 1 4 8 8 8

Matrix representation of C2×C4×C3⋊D4 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 5 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 1 12
,
 1 0 0 0 0 0 0 12 0 0 0 1 0 0 0 0 0 0 2 2 0 0 0 4 11
,
 12 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1 12 0 0 0 0 12

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,2,4,0,0,0,2,11],[12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,12,12] >;

C2×C4×C3⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_3\rtimes D_4
% in TeX

G:=Group("C2xC4xC3:D4");
// GroupNames label

G:=SmallGroup(192,1347);
// by ID

G=gap.SmallGroup(192,1347);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,80,6278]);
// Polycyclic

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

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