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G = C6.1212+ (1+4)  order 192 = 26·3

30th non-split extension by C6 of 2+ (1+4) acting via 2+ (1+4)/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.1212+ (1+4), C4⋊C4⋊15D6, C3⋊D4⋊6D4, C3⋊8(D4⋊5D4), C22⋊C4⋊15D6, D6.22(C2×D4), (C22×C4)⋊26D6, D6⋊10(C4○D4), C12⋊D4⋊30C2, Dic3⋊D4⋊29C2, D6⋊D4⋊19C2, D6⋊C4⋊22C22, D6⋊Q8⋊27C2, (C2×D4).163D6, Dic3⋊5D4⋊31C2, C6.83(C22×D4), C22.12(S3×D4), D6.D4⋊27C2, C2.41(D4○D12), (C2×D12)⋊26C22, (C2×C12).71C23, (C2×C6).198C24, Dic3.26(C2×D4), C22.D4⋊4S3, C23.14D6⋊20C2, Dic3⋊4D4⋊18C2, Dic3⋊C4⋊22C22, (C22×C12)⋊17C22, (C2×Dic6)⋊55C22, (C4×Dic3)⋊31C22, (C6×D4).136C22, C23.36(C22×S3), (C22×C6).33C23, C23.11D6⋊31C2, C6.D4⋊30C22, (S3×C23).57C22, C22.219(S3×C23), (C22×S3).206C23, (C2×Dic3).102C23, (C22×Dic3)⋊24C22, (C2×S3×D4)⋊15C2, C2.56(C2×S3×D4), (C2×D6⋊C4)⋊23C2, (S3×C2×C4)⋊21C22, C2.60(S3×C4○D4), (C2×C6).59(C2×D4), (C2×C4○D12)⋊11C2, (S3×C22⋊C4)⋊12C2, (C3×C4⋊C4)⋊25C22, C6.172(C2×C4○D4), (C2×C3⋊D4)⋊41C22, (C2×C4).61(C22×S3), (C3×C22⋊C4)⋊21C22, (C3×C22.D4)⋊6C2, SmallGroup(192,1213)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C6.1212+ (1+4)
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×D6⋊C4 — C6.1212+ (1+4)
Lower central
C3 — C2×C6 — C6.1212+ (1+4)

Subgroups: 1024 in 334 conjugacy classes, 105 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×10], C22, C22 [×2], C22 [×27], S3 [×6], C6 [×3], C6 [×3], C2×C4 [×5], C2×C4 [×14], D4 [×18], Q8 [×2], C23 [×2], C23 [×14], Dic3 [×2], Dic3 [×3], C12 [×5], D6 [×4], D6 [×18], C2×C6, C2×C6 [×2], C2×C6 [×5], C42, C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×7], D12 [×7], C2×Dic3 [×4], C2×Dic3, C3⋊D4 [×4], C3⋊D4 [×5], C2×C12 [×5], C2×C12 [×2], C3×D4 [×2], C22×S3 [×4], C22×S3 [×10], C22×C6 [×2], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×2], D6⋊C4 [×8], C6.D4, C3×C22⋊C4 [×3], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×4], C2×D12 [×4], C4○D12 [×4], S3×D4 [×4], C22×Dic3, C2×C3⋊D4 [×4], C22×C12, C6×D4, S3×C23 [×2], D4⋊5D4, S3×C22⋊C4, Dic3⋊4D4, D6⋊D4 [×2], Dic3⋊D4, C23.11D6, Dic3⋊5D4, D6.D4, C12⋊D4, D6⋊Q8, C2×D6⋊C4, C23.14D6, C3×C22.D4, C2×C4○D12, C2×S3×D4, C6.1212+ (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, D4⋊5D4, C2×S3×D4, S3×C4○D4, D4○D12, C6.1212+ (1+4)

Generators and relations
 G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a3b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

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

(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 24 16 21)(14 23 17 20)(15 22 18 19)(31 46 34 43)(32 45 35 48)(33 44 36 47)

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

1200000
0120000
0012000
0001200
0000012
0000112
,
0120000
1200000
005000
000500
000010
000001
,
010000
1200000
005000
000500
000001
000010
,
1200000
0120000
0012500
0010100
0000012
0000120
,
1200000
0120000
0012500
000100
0000120
0000012

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,10,0,0,0,0,5,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,5,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
 

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L 3 4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C6D6E6F12A12B12C12D12E12F12G
order1222222222222344444444444466666612121212121212
size1111224666612122224444666612122224484444888

39 irreducible representations

dim11111111111111122222224444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2S3D4D6D6D6D6C4○D42+ (1+4)S3×D4S3×C4○D4D4○D12
kernelC6.1212+ (1+4)S3×C22⋊C4Dic3⋊4D4D6⋊D4Dic3⋊D4C23.11D6Dic3⋊5D4D6.D4C12⋊D4D6⋊Q8C2×D6⋊C4C23.14D6C3×C22.D4C2×C4○D12C2×S3×D4C22.D4C3⋊D4C22⋊C4C4⋊C4C22×C4C2×D4D6C6C22C2C2
# reps11121111111111114321141222

In GAP, Magma, Sage, TeX 

C_6._{121}2_+^{(1+4)}
 
% in TeX
 
G:=Group("C6.121ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(192,1213);
 
// by ID
 
G=gap.SmallGroup(192,1213);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,100,346,297,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^4=e^2=1,c^2=a^3,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^3*b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
 
// generators/relations
 

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