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G = C2×C23.6D6order 192 = 26·3

Direct product of C2 and C23.6D6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C23.6D6, C24.22D6, C23.20D12, C61(C23⋊C4), (S3×C23)⋊4C4, C22⋊C436D6, C23.22(C4×S3), (C22×C6).66D4, (C22×Dic3)⋊5C4, C22.14(C2×D12), C22.44(D6⋊C4), C23.81(C3⋊D4), (C23×C6).37C22, C23.82(C22×S3), C6.D442C22, (C22×C6).111C23, C32(C2×C23⋊C4), (C2×C3⋊D4)⋊3C4, C2.8(C2×D6⋊C4), (C2×C22⋊C4)⋊1S3, (C6×C22⋊C4)⋊1C2, C22.18(S3×C2×C4), (C22×S3)⋊2(C2×C4), (C2×Dic3)⋊2(C2×C4), (C2×C6).433(C2×D4), C6.35(C2×C22⋊C4), (C2×C6.D4)⋊1C2, (C2×C6).12(C22×C4), (C22×C6).52(C2×C4), (C22×C3⋊D4).1C2, C22.26(C2×C3⋊D4), (C2×C6).56(C22⋊C4), (C3×C22⋊C4)⋊44C22, (C2×C3⋊D4).83C22, SmallGroup(192,513)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C2×C23.6D6
C1C3C6C2×C6C22×C6C2×C3⋊D4C22×C3⋊D4 — C2×C23.6D6
C3C6C2×C6 — C2×C23.6D6
C1C22C24C2×C22⋊C4

Generators and relations for C2×C23.6D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=cde5 >

Subgroups: 680 in 210 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×6], C22 [×7], C22 [×18], S3 [×2], C6, C6 [×2], C6 [×6], C2×C4 [×12], D4 [×8], C23 [×7], C23 [×8], Dic3 [×4], C12 [×2], D6 [×8], C2×C6 [×7], C2×C6 [×10], C22⋊C4 [×2], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×8], C24, C24, C2×Dic3 [×2], C2×Dic3 [×6], C3⋊D4 [×8], C2×C12 [×4], C22×S3 [×2], C22×S3 [×4], C22×C6 [×7], C22×C6 [×2], C23⋊C4 [×4], C2×C22⋊C4, C2×C22⋊C4, C22×D4, C6.D4 [×2], C6.D4, C3×C22⋊C4 [×2], C3×C22⋊C4, C22×Dic3, C22×Dic3, C2×C3⋊D4 [×4], C2×C3⋊D4 [×4], C22×C12, S3×C23, C23×C6, C2×C23⋊C4, C23.6D6 [×4], C2×C6.D4, C6×C22⋊C4, C22×C3⋊D4, C2×C23.6D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C23⋊C4 [×2], C2×C22⋊C4, D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C2×C23⋊C4, C23.6D6 [×2], C2×D6⋊C4, C2×C23.6D6

Smallest permutation representation of C2×C23.6D6
On 48 points
Generators in S48
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)(25 46)(26 47)(27 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(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 16)(2 36)(3 18)(4 26)(5 20)(6 28)(7 22)(8 30)(9 24)(10 32)(11 14)(12 34)(13 41)(15 43)(17 45)(19 47)(21 37)(23 39)(25 46)(27 48)(29 38)(31 40)(33 42)(35 44)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 32)(14 33)(15 34)(16 35)(17 36)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)
(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 21 29 12)(2 42 23 20)(3 26 31 41)(4 9 13 25)(5 17 33 8)(6 38 15 16)(7 34 35 37)(10 46 19 24)(11 30 27 45)(14 39 48 36)(18 47 40 32)(22 43 44 28)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D···2I2J2K 3 4A4B4C4D4E···4J6A···6G6H6I6J6K12A···12H
order12222···222344444···46···6666612···12
size11112···212122444412···122···244444···4

42 irreducible representations

dim11111111222222244
type+++++++++++
imageC1C2C2C2C2C4C4C4S3D4D6D6C4×S3D12C3⋊D4C23⋊C4C23.6D6
kernelC2×C23.6D6C23.6D6C2×C6.D4C6×C22⋊C4C22×C3⋊D4C22×Dic3C2×C3⋊D4S3×C23C2×C22⋊C4C22×C6C22⋊C4C24C23C23C23C6C2
# reps14111242142144424

Matrix representation of C2×C23.6D6 in GL6(𝔽13)

1200000
0120000
001000
000100
000010
000001
,
100000
010000
002400
0091100
000024
0000911
,
100000
010000
001000
000100
0000120
0000012
,
100000
010000
0012000
0001200
0000120
0000012
,
1120000
100000
000001
00001212
0091100
0021100
,
0120000
1200000
00001212
000001
0011200
004200

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

C2×C23.6D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3._6D_6
% in TeX

G:=Group("C2xC2^3.6D6");
// GroupNames label

G:=SmallGroup(192,513);
// by ID

G=gap.SmallGroup(192,513);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,58,1123,438,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=b,f^2=b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^5>;
// generators/relations

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