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G = C6.1222+ 1+4order 192 = 26·3

31st non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.1222+ 1+4, C4⋊C430D6, C22⋊C417D6, C4.D1232C2, (D4×Dic3)⋊28C2, (C2×D4).166D6, C232D6.2C2, D6.21(C4○D4), C23.9D631C2, C2.43(D4○D12), (C2×C12).74C23, (C2×C6).202C24, D6⋊C4.33C22, C4⋊Dic339C22, (C22×C4).274D6, C22.D47S3, Dic34D421C2, Dic3⋊C424C22, (C4×Dic3)⋊55C22, (C2×Dic6)⋊30C22, (C6×D4).140C22, Dic3.D432C2, C23.11D633C2, C36(C22.45C24), C23.21D621C2, C6.D431C22, C23.26D612C2, (C22×S3).86C23, (S3×C23).59C22, C23.212(C22×S3), C22.223(S3×C23), (C22×C6).222C23, C22.19(D42S3), (C22×C12).114C22, (C2×Dic3).245C23, (C22×Dic3)⋊26C22, (C2×D6⋊C4)⋊24C2, C4⋊C4⋊S328C2, C4⋊C47S332C2, C2.64(S3×C4○D4), (S3×C22⋊C4)⋊14C2, (C3×C4⋊C4)⋊28C22, C6.176(C2×C4○D4), (C2×C6).47(C4○D4), C2.53(C2×D42S3), (S3×C2×C4).111C22, (C3×C22⋊C4)⋊24C22, (C2×C4).297(C22×S3), (C2×C3⋊D4).47C22, (C3×C22.D4)⋊10C2, SmallGroup(192,1217)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C6.1222+ 1+4
C1C3C6C2×C6C22×S3S3×C23C2×D6⋊C4 — C6.1222+ 1+4
C3C2×C6 — C6.1222+ 1+4
C1C22C22.D4

Generators and relations for C6.1222+ 1+4
 G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3, bab-1=eae-1=a-1, ac=ca, ad=da, cbc=b-1, dbd-1=a3b, be=eb, dcd-1=a3c, ce=ec, ede-1=a3b2d >

Subgroups: 672 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×11], C22, C22 [×2], C22 [×16], S3 [×3], C6 [×3], C6 [×3], C2×C4 [×5], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], Dic3 [×6], C12 [×5], D6 [×2], D6 [×9], C2×C6, C2×C6 [×2], C2×C6 [×5], C42 [×3], C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic6, C4×S3 [×3], C2×Dic3 [×6], C2×Dic3 [×2], C3⋊D4 [×3], C2×C12 [×5], C2×C12 [×2], C3×D4 [×2], C22×S3 [×2], C22×S3 [×5], C22×C6 [×2], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C22.D4 [×2], C4.4D4, C422C2 [×2], C4×Dic3 [×3], Dic3⋊C4 [×2], C4⋊Dic3 [×4], D6⋊C4 [×8], C6.D4 [×3], C3×C22⋊C4 [×3], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×2], C22×Dic3 [×2], C2×C3⋊D4 [×2], C22×C12, C6×D4, S3×C23, C22.45C24, Dic3.D4, S3×C22⋊C4, Dic34D4, C23.9D6, C23.11D6, C23.21D6, C4⋊C47S3, C4.D12, C4⋊C4⋊S3 [×2], C23.26D6, C2×D6⋊C4, D4×Dic3, C232D6, C3×C22.D4, C6.1222+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2+ 1+4, D42S3 [×2], S3×C23, C22.45C24, C2×D42S3, S3×C4○D4, D4○D12, C6.1222+ 1+4

Smallest permutation representation of C6.1222+ 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 43 16 9)(2 48 17 8)(3 47 18 7)(4 46 13 12)(5 45 14 11)(6 44 15 10)(19 41 29 31)(20 40 30 36)(21 39 25 35)(22 38 26 34)(23 37 27 33)(24 42 28 32)
(1 46)(2 47)(3 48)(4 43)(5 44)(6 45)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)(19 38)(20 39)(21 40)(22 41)(23 42)(24 37)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 22 16 26)(2 23 17 27)(3 24 18 28)(4 19 13 29)(5 20 14 30)(6 21 15 25)(7 35 47 39)(8 36 48 40)(9 31 43 41)(10 32 44 42)(11 33 45 37)(12 34 46 38)
(1 29 4 26)(2 28 5 25)(3 27 6 30)(7 37 10 40)(8 42 11 39)(9 41 12 38)(13 22 16 19)(14 21 17 24)(15 20 18 23)(31 46 34 43)(32 45 35 48)(33 44 36 47)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G···4L4M4N4O6A6B6C6D6E6F12A12B12C12D12E12F12G
order122222222234444444···444466666612121212121212
size1111224661222244446···61212122224484444888

39 irreducible representations

dim11111111111111122222224444
type+++++++++++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2S3D6D6D6D6C4○D4C4○D42+ 1+4D42S3S3×C4○D4D4○D12
kernelC6.1222+ 1+4Dic3.D4S3×C22⋊C4Dic34D4C23.9D6C23.11D6C23.21D6C4⋊C47S3C4.D12C4⋊C4⋊S3C23.26D6C2×D6⋊C4D4×Dic3C232D6C3×C22.D4C22.D4C22⋊C4C4⋊C4C22×C4C2×D4D6C2×C6C6C22C2C2
# reps11111111121111113211441222

Matrix representation of C6.1222+ 1+4 in GL6(𝔽13)

1200000
0120000
00121200
001000
0000120
0000012
,
080000
500000
001000
00121200
000049
000019
,
050000
800000
001000
000100
000049
000079
,
010000
100000
0012000
0001200
000080
000035
,
010000
1200000
0012000
001100
000080
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],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,4,1,0,0,0,0,9,9],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,7,0,0,0,0,9,9],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,3,0,0,0,0,0,5],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;

C6.1222+ 1+4 in GAP, Magma, Sage, TeX

C_6._{122}2_+^{1+4}
% in TeX

G:=Group("C6.122ES+(2,2)");
// GroupNames label

G:=SmallGroup(192,1217);
// by ID

G=gap.SmallGroup(192,1217);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,100,346,297,136,6278]);
// Polycyclic

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

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