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G = C6.562+ 1+4order 192 = 26·3

56th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.562+ 1+4, C4⋊C413D6, (C2×Q8)⋊10D6, C22⋊Q821S3, D6⋊D417C2, C12⋊D427C2, C127D446C2, D6⋊C433C22, (C2×D12)⋊8C22, C22⋊C4.64D6, (C6×Q8)⋊10C22, D6.D422C2, C2.38(D4○D12), (C2×C6).188C24, (C2×C12).64C23, C4⋊Dic314C22, (C22×C4).266D6, Dic34D415C2, C12.23D417C2, C2.58(D46D6), Dic3⋊C420C22, C36(C22.32C24), (C4×Dic3)⋊30C22, (S3×C23).55C22, (C22×S3).79C23, C22.209(S3×C23), (C22×C6).216C23, C23.206(C22×S3), C22.4(Q83S3), (C22×C12).316C22, (C2×Dic3).241C23, (C22×Dic3).124C22, (C2×D6⋊C4)⋊27C2, (S3×C2×C4)⋊19C22, C4⋊C4⋊S323C2, (C3×C4⋊C4)⋊22C22, C6.116(C2×C4○D4), (C3×C22⋊Q8)⋊24C2, (C2×C6).28(C4○D4), C2.20(C2×Q83S3), (C2×C4).185(C22×S3), (C2×C3⋊D4).40C22, (C3×C22⋊C4).43C22, SmallGroup(192,1203)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C6.562+ 1+4
C1C3C6C2×C6C22×S3S3×C23D6⋊D4 — C6.562+ 1+4
C3C2×C6 — C6.562+ 1+4
C1C22C22⋊Q8

Generators and relations for C6.562+ 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, bd=db, ebe-1=a3b, cd=dc, ce=ec, ede-1=a3b2d >

Subgroups: 768 in 250 conjugacy classes, 95 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C422C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, C6×Q8, S3×C23, C22.32C24, Dic34D4, D6⋊D4, D6.D4, C12⋊D4, C4⋊C4⋊S3, C2×D6⋊C4, C127D4, C12.23D4, C3×C22⋊Q8, C6.562+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, Q83S3, S3×C23, C22.32C24, D46D6, C2×Q83S3, D4○D12, C6.562+ 1+4

Smallest permutation representation of C6.562+ 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 10 31)(2 29 11 32)(3 30 12 33)(4 25 7 34)(5 26 8 35)(6 27 9 36)(13 40 22 43)(14 41 23 44)(15 42 24 45)(16 37 19 46)(17 38 20 47)(18 39 21 48)
(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 10 13)(2 21 11 18)(3 20 12 17)(4 19 7 16)(5 24 8 15)(6 23 9 14)(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), (1,28,10,31)(2,29,11,32)(3,30,12,33)(4,25,7,34)(5,26,8,35)(6,27,9,36)(13,40,22,43)(14,41,23,44)(15,42,24,45)(16,37,19,46)(17,38,20,47)(18,39,21,48), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,22,10,13)(2,21,11,18)(3,20,12,17)(4,19,7,16)(5,24,8,15)(6,23,9,14)(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), (1,28,10,31)(2,29,11,32)(3,30,12,33)(4,25,7,34)(5,26,8,35)(6,27,9,36)(13,40,22,43)(14,41,23,44)(15,42,24,45)(16,37,19,46)(17,38,20,47)(18,39,21,48), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,22,10,13)(2,21,11,18)(3,20,12,17)(4,19,7,16)(5,24,8,15)(6,23,9,14)(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)], [(1,28,10,31),(2,29,11,32),(3,30,12,33),(4,25,7,34),(5,26,8,35),(6,27,9,36),(13,40,22,43),(14,41,23,44),(15,42,24,45),(16,37,19,46),(17,38,20,47),(18,39,21,48)], [(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,22,10,13),(2,21,11,18),(3,20,12,17),(4,19,7,16),(5,24,8,15),(6,23,9,14),(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···4F4G4H4I4J4K4L6A6B6C6D6E12A12B12C12D12E12F12G12H
order122222222234···4444444666661212121212121212
size1111221212121224···4666612122224444448888

36 irreducible representations

dim11111111112222224444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2S3D6D6D6D6C4○D42+ 1+4Q83S3D46D6D4○D12
kernelC6.562+ 1+4Dic34D4D6⋊D4D6.D4C12⋊D4C4⋊C4⋊S3C2×D6⋊C4C127D4C12.23D4C3×C22⋊Q8C22⋊Q8C22⋊C4C4⋊C4C22×C4C2×Q8C2×C6C6C22C2C2
# reps12222211211231142222

Matrix representation of C6.562+ 1+4 in GL6(𝔽13)

1200000
0120000
0001200
0011200
0000012
0000112
,
010000
1200000
0000120
0000012
001000
000100
,
1200000
0120000
001000
000100
0000120
0000012
,
800000
080000
0041100
002900
0000411
000029
,
500000
080000
002900
0041100
000029
0000411

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,12,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,4,2,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0,11,9],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,2,4,0,0,0,0,9,11,0,0,0,0,0,0,2,4,0,0,0,0,9,11] >;

C6.562+ 1+4 in GAP, Magma, Sage, TeX

C_6._{56}2_+^{1+4}
% in TeX

G:=Group("C6.56ES+(2,2)");
// GroupNames label

G:=SmallGroup(192,1203);
// by ID

G=gap.SmallGroup(192,1203);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,184,675,297,136,6278]);
// 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,b*d=d*b,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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