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G = C4222D6order 192 = 26·3

20th semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4222D6, C6.1262+ 1+4, (C2×Q8)⋊11D6, (C4×D12)⋊45C2, C22⋊C434D6, Dic3⋊D442C2, C232D624C2, D6⋊D425C2, C423S38C2, (C4×C12)⋊24C22, D6⋊C469C22, D63Q830C2, D6.8(C4○D4), (C2×D4).110D6, C4.4D412S3, (C6×Q8)⋊14C22, C23.9D643C2, C2.50(D4○D12), (C2×C6).222C24, C4⋊Dic341C22, Dic34D431C2, C23.14D634C2, C2.75(D46D6), C12.23D422C2, (C2×C12).631C23, Dic3⋊C436C22, C38(C22.32C24), (C4×Dic3)⋊36C22, (C6×D4).210C22, C23.8D639C2, C23.54(C22×S3), (C22×C6).52C23, (C2×D12).224C22, C23.21D625C2, (C22×S3).96C23, (S3×C23).65C22, C22.243(S3×C23), (C2×Dic3).254C23, C6.D4.56C22, (C22×Dic3)⋊27C22, (S3×C2×C4)⋊52C22, C2.78(S3×C4○D4), (S3×C22⋊C4)⋊18C2, C6.189(C2×C4○D4), (C3×C4.4D4)⋊14C2, (C2×C3⋊D4)⋊24C22, (C3×C22⋊C4)⋊30C22, (C2×C4).197(C22×S3), SmallGroup(192,1237)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C4222D6
C1C3C6C2×C6C22×S3S3×C23S3×C22⋊C4 — C4222D6
C3C2×C6 — C4222D6
C1C22C4.4D4

Generators and relations for C4222D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >

Subgroups: 752 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×10], C22, C22 [×20], S3 [×4], C6 [×3], C6 [×2], C2×C4 [×5], C2×C4 [×9], D4 [×9], Q8, C23 [×2], C23 [×7], Dic3 [×5], C12 [×5], D6 [×2], D6 [×12], C2×C6, C2×C6 [×6], C42, C42, C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×6], C2×Q8, C24, C4×S3 [×3], D12 [×3], C2×Dic3 [×5], C2×Dic3, C3⋊D4 [×5], C2×C12 [×5], C3×D4, C3×Q8, C22×S3 [×3], C22×S3 [×4], C22×C6 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C4.4D4, C422C2 [×2], C4×Dic3, Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×8], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×4], S3×C2×C4 [×3], C2×D12 [×2], C22×Dic3, C2×C3⋊D4 [×4], C6×D4, C6×Q8, S3×C23, C22.32C24, C4×D12, C423S3, C23.8D6, S3×C22⋊C4, Dic34D4, D6⋊D4, C23.9D6, Dic3⋊D4 [×2], C23.21D6, C232D6, C23.14D6, D63Q8, C12.23D4, C3×C4.4D4, C4222D6
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ 1+4 [×2], S3×C23, C22.32C24, D46D6, S3×C4○D4, D4○D12, C4222D6

Smallest permutation representation of C4222D6
On 48 points
Generators in S48
(1 19 12 45)(2 23 10 43)(3 21 11 47)(4 46 9 20)(5 44 7 24)(6 48 8 22)(13 29 16 37)(14 41 17 27)(15 25 18 39)(26 33 40 36)(28 35 42 32)(30 31 38 34)
(1 31 8 17)(2 35 9 15)(3 33 7 13)(4 18 10 32)(5 16 11 36)(6 14 12 34)(19 38 22 27)(20 25 23 42)(21 40 24 29)(26 44 37 47)(28 46 39 43)(30 48 41 45)
(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)
(1 7)(2 9)(3 8)(4 10)(5 12)(6 11)(13 14)(15 18)(16 17)(19 21)(22 24)(25 28)(26 27)(29 30)(31 36)(32 35)(33 34)(37 38)(39 42)(40 41)(44 48)(45 47)

G:=sub<Sym(48)| (1,19,12,45)(2,23,10,43)(3,21,11,47)(4,46,9,20)(5,44,7,24)(6,48,8,22)(13,29,16,37)(14,41,17,27)(15,25,18,39)(26,33,40,36)(28,35,42,32)(30,31,38,34), (1,31,8,17)(2,35,9,15)(3,33,7,13)(4,18,10,32)(5,16,11,36)(6,14,12,34)(19,38,22,27)(20,25,23,42)(21,40,24,29)(26,44,37,47)(28,46,39,43)(30,48,41,45), (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), (1,7)(2,9)(3,8)(4,10)(5,12)(6,11)(13,14)(15,18)(16,17)(19,21)(22,24)(25,28)(26,27)(29,30)(31,36)(32,35)(33,34)(37,38)(39,42)(40,41)(44,48)(45,47)>;

G:=Group( (1,19,12,45)(2,23,10,43)(3,21,11,47)(4,46,9,20)(5,44,7,24)(6,48,8,22)(13,29,16,37)(14,41,17,27)(15,25,18,39)(26,33,40,36)(28,35,42,32)(30,31,38,34), (1,31,8,17)(2,35,9,15)(3,33,7,13)(4,18,10,32)(5,16,11,36)(6,14,12,34)(19,38,22,27)(20,25,23,42)(21,40,24,29)(26,44,37,47)(28,46,39,43)(30,48,41,45), (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), (1,7)(2,9)(3,8)(4,10)(5,12)(6,11)(13,14)(15,18)(16,17)(19,21)(22,24)(25,28)(26,27)(29,30)(31,36)(32,35)(33,34)(37,38)(39,42)(40,41)(44,48)(45,47) );

G=PermutationGroup([(1,19,12,45),(2,23,10,43),(3,21,11,47),(4,46,9,20),(5,44,7,24),(6,48,8,22),(13,29,16,37),(14,41,17,27),(15,25,18,39),(26,33,40,36),(28,35,42,32),(30,31,38,34)], [(1,31,8,17),(2,35,9,15),(3,33,7,13),(4,18,10,32),(5,16,11,36),(6,14,12,34),(19,38,22,27),(20,25,23,42),(21,40,24,29),(26,44,37,47),(28,46,39,43),(30,48,41,45)], [(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)], [(1,7),(2,9),(3,8),(4,10),(5,12),(6,11),(13,14),(15,18),(16,17),(19,21),(22,24),(25,28),(26,27),(29,30),(31,36),(32,35),(33,34),(37,38),(39,42),(40,41),(44,48),(45,47)])

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C6D6E12A···12F12G12H
order122222222234444444444446666612···121212
size11114466121222244446612121212222884···488

36 irreducible representations

dim1111111111111112222224444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2S3D6D6D6D6C4○D42+ 1+4D46D6S3×C4○D4D4○D12
kernelC4222D6C4×D12C423S3C23.8D6S3×C22⋊C4Dic34D4D6⋊D4C23.9D6Dic3⋊D4C23.21D6C232D6C23.14D6D63Q8C12.23D4C3×C4.4D4C4.4D4C42C22⋊C4C2×D4C2×Q8D6C6C2C2C2
# reps1111111121111111141142222

Matrix representation of C4222D6 in GL8(𝔽13)

120000000
012000000
00800000
00080000
0000120110
000001258
00001010
00001301
,
10000000
01000000
00130000
000120000
00001300
000081200
00000001
000000120
,
11000000
120000000
001200000
00510000
000012000
00005100
00001010
00001210012
,
11000000
012000000
00100000
008120000
000012000
000001200
00001010
00001301

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,1,1,0,0,0,0,0,12,0,3,0,0,0,0,11,5,1,0,0,0,0,0,0,8,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,12,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,3,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,5,1,12,0,0,0,0,0,1,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,1,1,0,0,0,0,0,12,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C4222D6 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{22}D_6
% in TeX

G:=Group("C4^2:22D6");
// GroupNames label

G:=SmallGroup(192,1237);
// by ID

G=gap.SmallGroup(192,1237);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,675,570,80,6278]);
// Polycyclic

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

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