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G = C6.402+ 1+4order 192 = 26·3

40th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.402+ 1+4, C4⋊C46D6, C3⋊D42D4, (C2×D4)⋊24D6, D68(C4○D4), C4⋊D413S3, C35(D45D4), C22⋊C427D6, D6.17(C2×D4), (C22×C4)⋊21D6, C232D623C2, Dic3⋊D421C2, C22.6(S3×D4), D6⋊C467C22, D6⋊Q814C2, (C6×D4)⋊30C22, C6.69(C22×D4), Dic34D49C2, D6.D412C2, (C2×C6).154C24, (C2×C12).41C23, Dic3.22(C2×D4), C23.14D630C2, C2.42(D46D6), Dic3⋊C416C22, (C22×C12)⋊40C22, (C2×Dic6)⋊25C22, (C4×Dic3)⋊54C22, C23.23D69C2, (C22×C6).21C23, C23.11D619C2, (C2×D12).143C22, C6.D452C22, (S3×C23).47C22, C22.175(S3×C23), C23.192(C22×S3), (C2×Dic3).74C23, (C22×S3).188C23, (C22×Dic3)⋊20C22, (C2×S3×D4)⋊12C2, C2.42(C2×S3×D4), (C2×C6).6(C2×D4), (C2×D6⋊C4)⋊36C2, (C4×C3⋊D4)⋊54C2, (S3×C22⋊C4)⋊5C2, (S3×C2×C4)⋊50C22, C2.39(S3×C4○D4), (C3×C4⋊D4)⋊16C2, (C3×C4⋊C4)⋊12C22, C6.152(C2×C4○D4), (C2×D42S3)⋊14C2, (C2×C3⋊D4)⋊16C22, (C3×C22⋊C4)⋊14C22, (C2×C4).177(C22×S3), SmallGroup(192,1169)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C6.402+ 1+4
C1C3C6C2×C6C22×S3S3×C23C2×S3×D4 — C6.402+ 1+4
C3C2×C6 — C6.402+ 1+4
C1C22C4⋊D4

Generators and relations for C6.402+ 1+4
 G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=ebe-1=a3b, cd=dc, ce=ec, ede-1=a3b2d >

Subgroups: 992 in 334 conjugacy classes, 105 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×10], C22, C22 [×2], C22 [×27], S3 [×5], C6 [×3], C6 [×4], C2×C4 [×4], C2×C4 [×15], D4 [×18], Q8 [×2], C23 [×3], C23 [×13], Dic3 [×2], Dic3 [×4], C12 [×4], D6 [×4], D6 [×15], C2×C6, C2×C6 [×2], C2×C6 [×8], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4 [×3], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×5], D12 [×2], C2×Dic3 [×5], C2×Dic3 [×4], C3⋊D4 [×4], C3⋊D4 [×7], C2×C12 [×4], C2×C12, C3×D4 [×5], C22×S3 [×3], C22×S3 [×10], C22×C6 [×3], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×3], D6⋊C4 [×7], C6.D4 [×3], C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×3], C2×D12, S3×D4 [×4], D42S3 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×5], C22×C12, C6×D4 [×3], S3×C23 [×2], D45D4, S3×C22⋊C4, Dic34D4, Dic3⋊D4, C23.11D6, D6.D4, D6⋊Q8, C2×D6⋊C4, C4×C3⋊D4, C23.23D6, C232D6 [×2], C23.14D6, C3×C4⋊D4, C2×S3×D4, C2×D42S3, C6.402+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2+ 1+4, S3×D4 [×2], S3×C23, D45D4, C2×S3×D4, D46D6, S3×C4○D4, C6.402+ 1+4

Smallest permutation representation of C6.402+ 1+4
On 48 points
Generators in S48
(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 28 7 34)(2 29 8 35)(3 30 9 36)(4 25 10 31)(5 26 11 32)(6 27 12 33)(13 40 19 46)(14 41 20 47)(15 42 21 48)(16 37 22 43)(17 38 23 44)(18 39 24 45)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 22 7 16)(2 21 8 15)(3 20 9 14)(4 19 10 13)(5 24 11 18)(6 23 12 17)(25 43 31 37)(26 48 32 42)(27 47 33 41)(28 46 34 40)(29 45 35 39)(30 44 36 38)
(1 16 4 13)(2 17 5 14)(3 18 6 15)(7 22 10 19)(8 23 11 20)(9 24 12 21)(25 37 28 40)(26 38 29 41)(27 39 30 42)(31 43 34 46)(32 44 35 47)(33 45 36 48)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L 3 4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C6D6E6F6G12A12B12C12D12E12F
order122222222222234444444444446666666121212121212
size1111224466661222244466661212122224488444488

39 irreducible representations

dim11111111111111122222224444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2S3D4D6D6D6D6C4○D42+ 1+4S3×D4D46D6S3×C4○D4
kernelC6.402+ 1+4S3×C22⋊C4Dic34D4Dic3⋊D4C23.11D6D6.D4D6⋊Q8C2×D6⋊C4C4×C3⋊D4C23.23D6C232D6C23.14D6C3×C4⋊D4C2×S3×D4C2×D42S3C4⋊D4C3⋊D4C22⋊C4C4⋊C4C22×C4C2×D4D6C6C22C2C2
# reps11111111112111114211341222

Matrix representation of C6.402+ 1+4 in GL6(𝔽13)

1200000
0120000
00121200
001000
0000120
0000012
,
010000
1200000
0012000
0001200
0000120
000031
,
100000
0120000
001000
000100
000010
000001
,
800000
050000
001000
00121200
000081
000025
,
500000
080000
001000
000100
0000512
000008

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,3,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,8,2,0,0,0,0,1,5],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,12,8] >;

C6.402+ 1+4 in GAP, Magma, Sage, TeX

C_6._{40}2_+^{1+4}
% in TeX

G:=Group("C6.40ES+(2,2)");
// GroupNames label

G:=SmallGroup(192,1169);
// by ID

G=gap.SmallGroup(192,1169);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,297,136,6278]);
// Polycyclic

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

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