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G = C6.1202+ 1+4order 192 = 26·3

29th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.1202+ 1+4, C4⋊C414D6, D6.5(C2×D4), C22⋊C414D6, (C22×S3)⋊8D4, (C22×C4)⋊25D6, C232D615C2, C12⋊D429C2, Dic3⋊D428C2, D6⋊D418C2, D6⋊C421C22, (C2×D4).162D6, C34(C233D4), C22.44(S3×D4), C6.82(C22×D4), D6.D426C2, C2.40(D4○D12), (C22×D12)⋊10C2, (C2×D12)⋊47C22, (C2×C6).197C24, (C2×C12).70C23, Dic3⋊C44C22, C22.D43S3, (S3×C23)⋊11C22, (C22×C12)⋊12C22, (C6×D4).135C22, C23.28D67C2, (C22×C6).32C23, C6.D429C22, C22.218(S3×C23), C23.210(C22×S3), (C22×S3).205C23, (C2×Dic3).101C23, (C2×S3×D4)⋊14C2, C2.55(C2×S3×D4), (S3×C2×C4)⋊20C22, (C2×C6).58(C2×D4), (C3×C4⋊C4)⋊24C22, (S3×C22⋊C4)⋊11C2, (C2×C3⋊D4)⋊18C22, (C3×C22⋊C4)⋊20C22, (C2×C4).189(C22×S3), (C3×C22.D4)⋊5C2, SmallGroup(192,1212)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C6.1202+ 1+4
C1C3C6C2×C6C22×S3S3×C23C2×S3×D4 — C6.1202+ 1+4
C3C2×C6 — C6.1202+ 1+4
C1C22C22.D4

Generators and relations for C6.1202+ 1+4
 G = < a,b,c,d,e | a6=b4=c2=e2=1, d2=a3b2, ab=ba, ac=ca, ad=da, eae=a-1, cbc=a3b-1, dbd-1=ebe=a3b, cd=dc, ce=ec, ede=a3b2d >

Subgroups: 1168 in 346 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×8], C22, C22 [×2], C22 [×34], S3 [×7], C6, C6 [×2], C6 [×3], C2×C4, C2×C4 [×4], C2×C4 [×9], D4 [×20], C23 [×2], C23 [×19], Dic3 [×3], C12 [×5], D6 [×4], D6 [×25], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×19], C24 [×3], C4×S3 [×4], D12 [×12], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×6], C2×C12, C2×C12 [×4], C2×C12 [×2], C3×D4 [×2], C22×S3 [×3], C22×S3 [×6], C22×S3 [×10], C22×C6 [×2], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4 [×4], C22.D4, C22.D4 [×3], C22×D4 [×2], Dic3⋊C4 [×2], D6⋊C4 [×8], C6.D4, C3×C22⋊C4, C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], S3×C2×C4, S3×C2×C4 [×2], C2×D12, C2×D12 [×6], C2×D12 [×4], S3×D4 [×4], C2×C3⋊D4 [×2], C2×C3⋊D4 [×2], C22×C12, C6×D4, S3×C23 [×3], C233D4, S3×C22⋊C4, D6⋊D4, D6⋊D4 [×2], Dic3⋊D4 [×2], D6.D4 [×2], C12⋊D4 [×2], C23.28D6, C232D6, C3×C22.D4, C22×D12, C2×S3×D4, C6.1202+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, 2+ 1+4 [×2], S3×D4 [×2], S3×C23, C233D4, C2×S3×D4, D4○D12 [×2], C6.1202+ 1+4

Smallest permutation representation of C6.1202+ 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 40 10 43)(2 41 11 44)(3 42 12 45)(4 37 7 46)(5 38 8 47)(6 39 9 48)(13 34 22 25)(14 35 23 26)(15 36 24 27)(16 31 19 28)(17 32 20 29)(18 33 21 30)
(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 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 43 34 37)(29 44 35 38)(30 45 36 39)
(1 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)(25 43)(26 48)(27 47)(28 46)(29 45)(30 44)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M 3 4A···4E4F4G4H6A6B6C6D6E6F12A12B12C12D12E12F12G
order1222222222222234···444466666612121212121212
size1111224666612121224···41212122224484444888

36 irreducible representations

dim11111111111222222444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2S3D4D6D6D6D62+ 1+4S3×D4D4○D12
kernelC6.1202+ 1+4S3×C22⋊C4D6⋊D4Dic3⋊D4D6.D4C12⋊D4C23.28D6C232D6C3×C22.D4C22×D12C2×S3×D4C22.D4C22×S3C22⋊C4C4⋊C4C22×C4C2×D4C6C22C2
# reps11322211111143211224

Matrix representation of C6.1202+ 1+4 in GL6(𝔽13)

1200000
0120000
0001200
0011200
00001212
000010
,
0120000
100000
0010699
007309
0000107
000063
,
100000
010000
001000
000100
0088120
00108012
,
100000
0120000
003700
0061000
0000107
000063
,
100000
0120000
0061000
003700
007636
0060310

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,9,0,10,6,0,0,9,9,7,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,8,10,0,0,0,1,8,8,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,0,0,0,0,10,6,0,0,0,0,7,3],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,6,3,7,6,0,0,10,7,6,0,0,0,0,0,3,3,0,0,0,0,6,10] >;

C6.1202+ 1+4 in GAP, Magma, Sage, TeX

C_6._{120}2_+^{1+4}
% in TeX

G:=Group("C6.120ES+(2,2)");
// GroupNames label

G:=SmallGroup(192,1212);
// by ID

G=gap.SmallGroup(192,1212);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,675,297,80,6278]);
// Polycyclic

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

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