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G = C6.512+ 1+4order 192 = 26·3

51st non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.512+ 1+4, C4⋊C411D6, (C2×Q8)⋊8D6, D6.2(C2×Q8), C22⋊Q811S3, (C6×Q8)⋊8C22, (C22×S3)⋊3Q8, D6⋊Q821C2, D63Q816C2, C4.D1227C2, C22.7(S3×Q8), C22⋊C4.59D6, C32(C232Q8), C2.35(D4○D12), C6.36(C22×Q8), (C2×C12).57C23, (C2×C6).178C24, C4⋊Dic336C22, (C2×Dic6)⋊9C22, (C22×C4).256D6, C12.48D446C2, C2.53(D46D6), Dic3⋊C418C22, D6⋊C4.147C22, Dic3.D424C2, (S3×C23).53C22, (C22×C6).206C23, C23.201(C22×S3), C22.199(S3×C23), (C2×Dic3).89C23, (C22×S3).200C23, (C22×C12).315C22, C6.D4.34C22, (C22×Dic3).119C22, C2.19(C2×S3×Q8), (C2×C6).7(C2×Q8), (C3×C4⋊C4)⋊20C22, (C2×D6⋊C4).20C2, (S3×C22⋊C4).2C2, (S3×C2×C4).98C22, (C3×C22⋊Q8)⋊14C2, (C2×C4).183(C22×S3), (C3×C22⋊C4).33C22, SmallGroup(192,1193)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C6.512+ 1+4
C1C3C6C2×C6C22×S3S3×C23S3×C22⋊C4 — C6.512+ 1+4
C3C2×C6 — C6.512+ 1+4
C1C22C22⋊Q8

Generators and relations for C6.512+ 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=a3b-1, dbd-1=ebe-1=a3b, cd=dc, ce=ec, ede-1=a3b2d >

Subgroups: 672 in 242 conjugacy classes, 103 normal (31 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×12], C22, C22 [×2], C22 [×14], S3 [×4], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], Q8 [×4], C23, C23 [×8], Dic3 [×6], C12 [×6], D6 [×4], D6 [×8], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4, C22×C4 [×5], C2×Q8, C2×Q8 [×3], C24, Dic6 [×3], C4×S3 [×4], C2×Dic3 [×6], C2×Dic3, C2×C12 [×2], C2×C12 [×4], C2×C12, C3×Q8, C22×S3 [×6], C22×S3 [×2], C22×C6, C2×C22⋊C4 [×3], C22⋊Q8, C22⋊Q8 [×11], Dic3⋊C4 [×6], C4⋊Dic3, C4⋊Dic3 [×2], D6⋊C4 [×8], C6.D4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C2×Dic6, C2×Dic6 [×2], S3×C2×C4 [×4], C22×Dic3, C22×C12, C6×Q8, S3×C23, C232Q8, Dic3.D4 [×2], S3×C22⋊C4 [×2], D6⋊Q8 [×4], C4.D12 [×2], C12.48D4, C2×D6⋊C4, D63Q8 [×2], C3×C22⋊Q8, C6.512+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C24, C22×S3 [×7], C22×Q8, 2+ 1+4 [×2], S3×Q8 [×2], S3×C23, C232Q8, D46D6, C2×S3×Q8, D4○D12, C6.512+ 1+4

Smallest permutation representation of C6.512+ 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 25 10 34)(2 26 11 35)(3 27 12 36)(4 28 7 31)(5 29 8 32)(6 30 9 33)(13 37 22 46)(14 38 23 47)(15 39 24 48)(16 40 19 43)(17 41 20 44)(18 42 21 45)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)
(1 19 10 16)(2 24 11 15)(3 23 12 14)(4 22 7 13)(5 21 8 18)(6 20 9 17)(25 46 34 37)(26 45 35 42)(27 44 36 41)(28 43 31 40)(29 48 32 39)(30 47 33 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), (1,25,10,34)(2,26,11,35)(3,27,12,36)(4,28,7,31)(5,29,8,32)(6,30,9,33)(13,37,22,46)(14,38,23,47)(15,39,24,48)(16,40,19,43)(17,41,20,44)(18,42,21,45), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45), (1,19,10,16)(2,24,11,15)(3,23,12,14)(4,22,7,13)(5,21,8,18)(6,20,9,17)(25,46,34,37)(26,45,35,42)(27,44,36,41)(28,43,31,40)(29,48,32,39)(30,47,33,38), (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), (1,25,10,34)(2,26,11,35)(3,27,12,36)(4,28,7,31)(5,29,8,32)(6,30,9,33)(13,37,22,46)(14,38,23,47)(15,39,24,48)(16,40,19,43)(17,41,20,44)(18,42,21,45), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45), (1,19,10,16)(2,24,11,15)(3,23,12,14)(4,22,7,13)(5,21,8,18)(6,20,9,17)(25,46,34,37)(26,45,35,42)(27,44,36,41)(28,43,31,40)(29,48,32,39)(30,47,33,38), (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)], [(1,25,10,34),(2,26,11,35),(3,27,12,36),(4,28,7,31),(5,29,8,32),(6,30,9,33),(13,37,22,46),(14,38,23,47),(15,39,24,48),(16,40,19,43),(17,41,20,44),(18,42,21,45)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,34),(26,35),(27,36),(28,31),(29,32),(30,33),(37,46),(38,47),(39,48),(40,43),(41,44),(42,45)], [(1,19,10,16),(2,24,11,15),(3,23,12,14),(4,22,7,13),(5,21,8,18),(6,20,9,17),(25,46,34,37),(26,45,35,42),(27,44,36,41),(28,43,31,40),(29,48,32,39),(30,47,33,38)], [(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)])

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A···4F4G···4L6A6B6C6D6E12A12B12C12D12E12F12G12H
order122222222234···44···4666661212121212121212
size111122666624···412···122224444448888

36 irreducible representations

dim1111111112222224444
type++++++++++-+++++-+
imageC1C2C2C2C2C2C2C2C2S3Q8D6D6D6D62+ 1+4S3×Q8D46D6D4○D12
kernelC6.512+ 1+4Dic3.D4S3×C22⋊C4D6⋊Q8C4.D12C12.48D4C2×D6⋊C4D63Q8C3×C22⋊Q8C22⋊Q8C22×S3C22⋊C4C4⋊C4C22×C4C2×Q8C6C22C2C2
# reps1224211211423112222

Matrix representation of C6.512+ 1+4 in GL6(𝔽13)

1200000
0120000
00121200
001000
00001212
000010
,
4100000
1090000
0000120
0000012
001000
000100
,
100000
010000
001000
000100
0000120
0000012
,
0120000
100000
0091100
002400
0000911
000024
,
0120000
100000
002400
0091100
000024
0000911

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[4,10,0,0,0,0,10,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,12,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,9,2,0,0,0,0,11,4,0,0,0,0,0,0,9,2,0,0,0,0,11,4],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,0,0,2,9,0,0,0,0,4,11] >;

C6.512+ 1+4 in GAP, Magma, Sage, TeX

C_6._{51}2_+^{1+4}
% in TeX

G:=Group("C6.51ES+(2,2)");
// GroupNames label

G:=SmallGroup(192,1193);
// by ID

G=gap.SmallGroup(192,1193);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,675,297,136,6278]);
// 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=a^3*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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