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## G = C4×D6⋊C4order 192 = 26·3

### Direct product of C4 and D6⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C4×D6⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — S3×C22×C4 — C4×D6⋊C4
 Lower central C3 — C6 — C4×D6⋊C4
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C4×D6⋊C4
G = < a,b,c,d | a4=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 632 in 258 conjugacy classes, 107 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C4 [×10], C22 [×3], C22 [×4], C22 [×16], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×10], C2×C4 [×24], C23, C23 [×10], Dic3 [×6], C12 [×4], C12 [×4], D6 [×4], D6 [×12], C2×C6 [×3], C2×C6 [×4], C42 [×4], C22⋊C4 [×8], C22×C4, C22×C4 [×2], C22×C4 [×11], C24, C4×S3 [×8], C2×Dic3 [×6], C2×Dic3 [×6], C2×C12 [×10], C2×C12 [×4], C22×S3 [×6], C22×S3 [×4], C22×C6, C2.C42 [×2], C2×C42, C2×C42, C2×C22⋊C4 [×2], C23×C4, C4×Dic3 [×2], D6⋊C4 [×8], C4×C12 [×2], S3×C2×C4 [×4], S3×C2×C4 [×4], C22×Dic3, C22×Dic3 [×2], C22×C12, C22×C12 [×2], S3×C23, C4×C22⋊C4, C6.C42 [×2], C2×C4×Dic3, C2×D6⋊C4 [×2], C2×C4×C12, S3×C22×C4, C4×D6⋊C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], S3, C2×C4 [×18], D4 [×4], C23, D6 [×3], C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C4×S3 [×6], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], D6⋊C4 [×4], S3×C2×C4 [×3], C2×D12, C4○D12 [×2], C2×C3⋊D4, C4×C22⋊C4, S3×C42, C422S3, C4×D12 [×2], C2×D6⋊C4, C4×C3⋊D4 [×2], C4×D6⋊C4

Smallest permutation representation of C4×D6⋊C4
On 96 points
Generators in S96
(1 56 11 50)(2 57 12 51)(3 58 7 52)(4 59 8 53)(5 60 9 54)(6 55 10 49)(13 70 19 64)(14 71 20 65)(15 72 21 66)(16 67 22 61)(17 68 23 62)(18 69 24 63)(25 82 31 76)(26 83 32 77)(27 84 33 78)(28 79 34 73)(29 80 35 74)(30 81 36 75)(37 94 43 88)(38 95 44 89)(39 96 45 90)(40 91 46 85)(41 92 47 86)(42 93 48 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 22)(2 21)(3 20)(4 19)(5 24)(6 23)(7 14)(8 13)(9 18)(10 17)(11 16)(12 15)(25 47)(26 46)(27 45)(28 44)(29 43)(30 48)(31 41)(32 40)(33 39)(34 38)(35 37)(36 42)(49 68)(50 67)(51 72)(52 71)(53 70)(54 69)(55 62)(56 61)(57 66)(58 65)(59 64)(60 63)(73 95)(74 94)(75 93)(76 92)(77 91)(78 96)(79 89)(80 88)(81 87)(82 86)(83 85)(84 90)
(1 95 23 77)(2 96 24 78)(3 91 19 73)(4 92 20 74)(5 93 21 75)(6 94 22 76)(7 85 13 79)(8 86 14 80)(9 87 15 81)(10 88 16 82)(11 89 17 83)(12 90 18 84)(25 55 43 61)(26 56 44 62)(27 57 45 63)(28 58 46 64)(29 59 47 65)(30 60 48 66)(31 49 37 67)(32 50 38 68)(33 51 39 69)(34 52 40 70)(35 53 41 71)(36 54 42 72)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4H 4I ··· 4P 4Q ··· 4AB 6A ··· 6G 12A ··· 12X order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 6 6 6 6 2 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 S3 D4 D6 C4○D4 C4×S3 D12 C3⋊D4 C4○D12 kernel C4×D6⋊C4 C6.C42 C2×C4×Dic3 C2×D6⋊C4 C2×C4×C12 S3×C22×C4 D6⋊C4 S3×C2×C4 C2×C42 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C2×C4 C22 # reps 1 2 1 2 1 1 16 8 1 4 3 4 12 4 4 8

Matrix representation of C4×D6⋊C4 in GL4(𝔽13) generated by

 8 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 12 0 0 1 1
,
 12 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 12 0 0 0 0 8 0 0 0 0 3 6 0 0 7 10
G:=sub<GL(4,GF(13))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,1],[12,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[12,0,0,0,0,8,0,0,0,0,3,7,0,0,6,10] >;

C4×D6⋊C4 in GAP, Magma, Sage, TeX

C_4\times D_6\rtimes C_4
% in TeX

G:=Group("C4xD6:C4");
// GroupNames label

G:=SmallGroup(192,497);
// by ID

G=gap.SmallGroup(192,497);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,58,6278]);
// Polycyclic

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

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