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G = C4.(C2×D12)  order 192 = 26·3

14th non-split extension by C4 of C2×D12 acting via C2×D12/C22×S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4○D124C4, C4⋊C4.235D6, C4.63(C2×D12), C4.52(D6⋊C4), C42⋊C22S3, D12.26(C2×C4), C6.85(C4○D8), (C2×C4).146D12, C12.143(C2×D4), C6.D843C2, (C2×C12).496D4, (C22×C6).75D4, C6.SD1643C2, C12.65(C22×C4), C22.1(D6⋊C4), Dic6.26(C2×C4), (C22×C4).355D6, C12.64(C22⋊C4), (C2×C12).329C23, C2.2(Q8.13D6), C23.45(C3⋊D4), C33(C23.24D4), (C2×D12).235C22, (C22×C12).151C22, (C2×Dic6).262C22, C4.52(S3×C2×C4), (C22×C3⋊C8)⋊2C2, (C2×C4).81(C4×S3), C2.18(C2×D6⋊C4), (C2×C12).92(C2×C4), (C2×C4○D12).7C2, (C2×C6).458(C2×D4), C6.45(C2×C22⋊C4), (C3×C42⋊C2)⋊2C2, (C2×C3⋊C8).243C22, C22.73(C2×C3⋊D4), (C2×C4).274(C3⋊D4), (C3×C4⋊C4).266C22, (C2×C6).15(C22⋊C4), (C2×C4).429(C22×S3), SmallGroup(192,561)

Series: Derived Chief Lower central Upper central

C1C12 — C4.(C2×D12)
C1C3C6C2×C6C2×C12C2×D12C2×C4○D12 — C4.(C2×D12)
C3C6C12 — C4.(C2×D12)
C1C2×C4C22×C4C42⋊C2

Generators and relations for C4.(C2×D12)
 G = < a,b,c,d | a4=b2=c12=1, d2=a, ab=ba, cac-1=a-1, ad=da, cbc-1=a2b, bd=db, dcd-1=ac-1 >

Subgroups: 424 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], S3 [×2], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×7], D4 [×7], Q8 [×3], C23, C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×4], C22×C4, C22×C4, C2×D4 [×2], C2×Q8, C4○D4 [×6], C3⋊C8 [×2], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C22×S3, C22×C6, D4⋊C4 [×2], Q8⋊C4 [×2], C42⋊C2, C22×C8, C2×C4○D4, C2×C3⋊C8 [×2], C2×C3⋊C8 [×2], C4×C12, C3×C22⋊C4, C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C2×C3⋊D4, C22×C12, C23.24D4, C6.D8 [×2], C6.SD16 [×2], C22×C3⋊C8, C3×C42⋊C2, C2×C4○D12, C4.(C2×D12)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C4○D8 [×2], D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C23.24D4, C2×D6⋊C4, Q8.13D6 [×2], C4.(C2×D12)

Smallest permutation representation of C4.(C2×D12)
On 96 points
Generators in S96
(1 21 44 59)(2 60 45 22)(3 23 46 49)(4 50 47 24)(5 13 48 51)(6 52 37 14)(7 15 38 53)(8 54 39 16)(9 17 40 55)(10 56 41 18)(11 19 42 57)(12 58 43 20)(25 91 64 75)(26 76 65 92)(27 93 66 77)(28 78 67 94)(29 95 68 79)(30 80 69 96)(31 85 70 81)(32 82 71 86)(33 87 72 83)(34 84 61 88)(35 89 62 73)(36 74 63 90)
(1 36)(2 64)(3 26)(4 66)(5 28)(6 68)(7 30)(8 70)(9 32)(10 72)(11 34)(12 62)(13 78)(14 95)(15 80)(16 85)(17 82)(18 87)(19 84)(20 89)(21 74)(22 91)(23 76)(24 93)(25 45)(27 47)(29 37)(31 39)(33 41)(35 43)(38 69)(40 71)(42 61)(44 63)(46 65)(48 67)(49 92)(50 77)(51 94)(52 79)(53 96)(54 81)(55 86)(56 83)(57 88)(58 73)(59 90)(60 75)
(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 58 21 43 44 20 59 12)(2 11 60 19 45 42 22 57)(3 56 23 41 46 18 49 10)(4 9 50 17 47 40 24 55)(5 54 13 39 48 16 51 8)(6 7 52 15 37 38 14 53)(25 61 91 88 64 34 75 84)(26 83 76 33 65 87 92 72)(27 71 93 86 66 32 77 82)(28 81 78 31 67 85 94 70)(29 69 95 96 68 30 79 80)(35 63 89 90 62 36 73 74)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C6D6E8A···8H12A12B12C12D12E···12N
order122222223444444444444666668···81212121212···12
size1111221212211112244441212222446···622224···4

48 irreducible representations

dim111111122222222224
type++++++++++++
imageC1C2C2C2C2C2C4S3D4D4D6D6C4×S3D12C3⋊D4C3⋊D4C4○D8Q8.13D6
kernelC4.(C2×D12)C6.D8C6.SD16C22×C3⋊C8C3×C42⋊C2C2×C4○D12C4○D12C42⋊C2C2×C12C22×C6C4⋊C4C22×C4C2×C4C2×C4C2×C4C23C6C2
# reps122111813121442284

Matrix representation of C4.(C2×D12) in GL4(𝔽73) generated by

1000
0100
00125
007072
,
1000
0100
002718
006546
,
666600
75900
00035
00480
,
75900
666600
004138
00480
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,70,0,0,25,72],[1,0,0,0,0,1,0,0,0,0,27,65,0,0,18,46],[66,7,0,0,66,59,0,0,0,0,0,48,0,0,35,0],[7,66,0,0,59,66,0,0,0,0,41,48,0,0,38,0] >;

C4.(C2×D12) in GAP, Magma, Sage, TeX

C_4.(C_2\times D_{12})
% in TeX

G:=Group("C4.(C2xD12)");
// GroupNames label

G:=SmallGroup(192,561);
// by ID

G=gap.SmallGroup(192,561);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,422,58,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^12=1,d^2=a,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations

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