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G = C6×C4⋊1D4  order 192 = 26·3

Direct product of C6 and C4⋊1D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C6×C4⋊1D4, C4⋊1(C6×D4), (C2×C12)⋊33D4, C12⋊12(C2×D4), C42⋊22(C2×C6), (C2×C42)⋊14C6, (C22×D4)⋊8C6, (C4×C12)⋊59C22, (C6×D4)⋊63C22, C24.16(C2×C6), C22.63(C6×D4), (C2×C6).350C24, C6.186(C22×D4), C23.8(C22×C6), (C2×C12).961C23, (C22×C6).88C23, C22.24(C23×C6), (C23×C6).15C22, (C22×C12).596C22, (C2×C4×C12)⋊24C2, (D4×C2×C6)⋊20C2, (C2×C4)⋊7(C3×D4), C2.10(D4×C2×C6), (C2×D4)⋊11(C2×C6), (C2×C6).684(C2×D4), (C22×C4).131(C2×C6), (C2×C4).136(C22×C6), SmallGroup(192,1419)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C6×C4⋊1D4
Chief series
C1 — C2 — C22 — C2×C6 — C22×C6 — C6×D4 — C3×C4⋊1D4 — C6×C4⋊1D4
Lower central
C1 — C22 — C6×C4⋊1D4
Upper central
C1 — C22×C6 — C6×C4⋊1D4

Subgroups: 882 in 498 conjugacy classes, 210 normal (10 characteristic)
C1, C2 [×7], C2 [×8], C3, C4 [×12], C22, C22 [×6], C22 [×40], C6 [×7], C6 [×8], C2×C4 [×18], D4 [×48], C23, C23 [×8], C23 [×24], C12 [×12], C2×C6, C2×C6 [×6], C2×C6 [×40], C42 [×4], C22×C4 [×3], C2×D4 [×24], C2×D4 [×24], C24 [×4], C2×C12 [×18], C3×D4 [×48], C22×C6, C22×C6 [×8], C22×C6 [×24], C2×C42, C4⋊1D4 [×8], C22×D4 [×6], C4×C12 [×4], C22×C12 [×3], C6×D4 [×24], C6×D4 [×24], C23×C6 [×4], C2×C4⋊1D4, C2×C4×C12, C3×C4⋊1D4 [×8], D4×C2×C6 [×6], C6×C4⋊1D4

Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×12], C23 [×15], C2×C6 [×35], C2×D4 [×18], C24, C3×D4 [×12], C22×C6 [×15], C4⋊1D4 [×4], C22×D4 [×3], C6×D4 [×18], C23×C6, C2×C4⋊1D4, C3×C4⋊1D4 [×4], D4×C2×C6 [×3], C6×C4⋊1D4

Generators and relations
 G = < a,b,c,d | a6=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

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

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

400000
040000
009000
000900
000010
000001
,
1200000
0120000
001000
000100
0000012
000010
,
0120000
100000
000100
0012000
0000012
000010
,
010000
100000
0001200
0012000
0000120
000001

G:=sub<GL(6,GF(13))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1] >;
 

84 conjugacy classes

class 1 2A···2G2H···2O3A3B4A···4L6A···6N6O···6AD12A···12X
order12···22···2334···46···66···612···12
size11···14···4112···21···14···42···2

84 irreducible representations

dim1111111122
type+++++
imageC1C2C2C2C3C6C6C6D4C3×D4
kernelC6×C4⋊1D4C2×C4×C12C3×C4⋊1D4D4×C2×C6C2×C4⋊1D4C2×C42C4⋊1D4C22×D4C2×C12C2×C4
# reps11862216121224

In GAP, Magma, Sage, TeX 

C_6\times C_4\rtimes_1D_4
 
% in TeX
 
G:=Group("C6xC4:1D4");
 
// GroupNames label
 
G:=SmallGroup(192,1419);
 
// by ID
 
G=gap.SmallGroup(192,1419);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,344,2102,520]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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