G = C6.1452+ 1+4 order 192 = 26·3
54th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.1452+ 1+4, (C3×D4)⋊19D4, (C2×D4)⋊44D6, (C2×Q8)⋊36D6, (C22×C4)⋊33D6, D6⋊12(C4○D4), C23⋊2D6⋊31C2, C12⋊7D4⋊39C2, D4⋊10(C3⋊D4), C3⋊11(D4⋊5D4), D6⋊C4⋊37C22, D6⋊3Q8⋊44C2, (D4×Dic3)⋊42C2, C12.266(C2×D4), (C6×D4)⋊59C22, (C6×Q8)⋊40C22, C2.69(D4○D12), (C2×C6).315C24, C4⋊Dic3⋊46C22, C6.165(C22×D4), C23.14D6⋊44C2, C12.23D4⋊31C2, (C2×C12).652C23, Dic3⋊C4⋊40C22, (C22×C12)⋊24C22, (C4×Dic3)⋊44C22, (C2×D12).184C22, C23.28D6⋊31C2, C6.D4⋊41C22, (S3×C23).79C22, C22.324(S3×C23), C23.221(C22×S3), (C22×C6).241C23, (C22×S3).137C23, (C2×Dic3).162C23, (C22×Dic3)⋊36C22, (C2×S3×D4)⋊26C2, (C6×C4○D4)⋊7C2, (C2×D6⋊C4)⋊44C2, (C2×C4○D4)⋊11S3, (C4×C3⋊D4)⋊29C2, (C2×C6).81(C2×D4), C4.72(C2×C3⋊D4), C6.216(C2×C4○D4), C2.104(S3×C4○D4), C22.5(C2×C3⋊D4), (C2×C3⋊D4)⋊30C22, (S3×C2×C4).168C22, C2.38(C22×C3⋊D4), (C2×C4).250(C22×S3), SmallGroup(192,1388)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.1452+ 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=b-1, bd=db, ebe-1=a3b, cd=dc, ce=ec, ede-1=a3b2d >
Subgroups: 952 in 334 conjugacy classes, 113 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C22×S3, C22×C6, C22×C6, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C6.D4, S3×C2×C4, C2×D12, S3×D4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, S3×C23, D4⋊5D4, C2×D6⋊C4, C4×C3⋊D4, C23.28D6, C12⋊7D4, D4×Dic3, C23⋊2D6, C23.14D6, D6⋊3Q8, C12.23D4, C2×S3×D4, C6×C4○D4, C6.1452+ 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C4○D4, 2+ 1+4, C2×C3⋊D4, S3×C23, D4⋊5D4, S3×C4○D4, D4○D12, C22×C3⋊D4, C6.1452+ 1+4
►On 48 points
(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 12 18 46)(2 7 13 47)(3 8 14 48)(4 9 15 43)(5 10 16 44)(6 11 17 45)(19 41 29 31)(20 42 30 32)(21 37 25 33)(22 38 26 34)(23 39 27 35)(24 40 28 36)
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 10)(8 11)(9 12)(19 26)(20 27)(21 28)(22 29)(23 30)(24 25)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)
(1 22 18 26)(2 23 13 27)(3 24 14 28)(4 19 15 29)(5 20 16 30)(6 21 17 25)(7 39 47 35)(8 40 48 36)(9 41 43 31)(10 42 44 32)(11 37 45 33)(12 38 46 34)
(1 22 4 19)(2 21 5 24)(3 20 6 23)(7 40 10 37)(8 39 11 42)(9 38 12 41)(13 25 16 28)(14 30 17 27)(15 29 18 26)(31 43 34 46)(32 48 35 45)(33 47 36 44)
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), (1,12,18,46)(2,7,13,47)(3,8,14,48)(4,9,15,43)(5,10,16,44)(6,11,17,45)(19,41,29,31)(20,42,30,32)(21,37,25,33)(22,38,26,34)(23,39,27,35)(24,40,28,36), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,10)(8,11)(9,12)(19,26)(20,27)(21,28)(22,29)(23,30)(24,25)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48), (1,22,18,26)(2,23,13,27)(3,24,14,28)(4,19,15,29)(5,20,16,30)(6,21,17,25)(7,39,47,35)(8,40,48,36)(9,41,43,31)(10,42,44,32)(11,37,45,33)(12,38,46,34), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,40,10,37)(8,39,11,42)(9,38,12,41)(13,25,16,28)(14,30,17,27)(15,29,18,26)(31,43,34,46)(32,48,35,45)(33,47,36,44)>;
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), (1,12,18,46)(2,7,13,47)(3,8,14,48)(4,9,15,43)(5,10,16,44)(6,11,17,45)(19,41,29,31)(20,42,30,32)(21,37,25,33)(22,38,26,34)(23,39,27,35)(24,40,28,36), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,10)(8,11)(9,12)(19,26)(20,27)(21,28)(22,29)(23,30)(24,25)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48), (1,22,18,26)(2,23,13,27)(3,24,14,28)(4,19,15,29)(5,20,16,30)(6,21,17,25)(7,39,47,35)(8,40,48,36)(9,41,43,31)(10,42,44,32)(11,37,45,33)(12,38,46,34), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,40,10,37)(8,39,11,42)(9,38,12,41)(13,25,16,28)(14,30,17,27)(15,29,18,26)(31,43,34,46)(32,48,35,45)(33,47,36,44) );
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)], [(1,12,18,46),(2,7,13,47),(3,8,14,48),(4,9,15,43),(5,10,16,44),(6,11,17,45),(19,41,29,31),(20,42,30,32),(21,37,25,33),(22,38,26,34),(23,39,27,35),(24,40,28,36)], [(1,15),(2,16),(3,17),(4,18),(5,13),(6,14),(7,10),(8,11),(9,12),(19,26),(20,27),(21,28),(22,29),(23,30),(24,25),(31,34),(32,35),(33,36),(37,40),(38,41),(39,42),(43,46),(44,47),(45,48)], [(1,22,18,26),(2,23,13,27),(3,24,14,28),(4,19,15,29),(5,20,16,30),(6,21,17,25),(7,39,47,35),(8,40,48,36),(9,41,43,31),(10,42,44,32),(11,37,45,33),(12,38,46,34)], [(1,22,4,19),(2,21,5,24),(3,20,6,23),(7,40,10,37),(8,39,11,42),(9,38,12,41),(13,25,16,28),(14,30,17,27),(15,29,18,26),(31,43,34,46),(32,48,35,45),(33,47,36,44)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | ··· | 6I | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | 2+ 1+4 | S3×C4○D4 | D4○D12 |
| kernel | C6.1452+ 1+4 | C2×D6⋊C4 | C4×C3⋊D4 | C23.28D6 | C12⋊7D4 | D4×Dic3 | C23⋊2D6 | C23.14D6 | D6⋊3Q8 | C12.23D4 | C2×S3×D4 | C6×C4○D4 | C2×C4○D4 | C3×D4 | C22×C4 | C2×D4 | C2×Q8 | D6 | D4 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 1 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C6.1452+ 1+4 ►in GL4(𝔽13) generated by
| 1 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 2 | 4 | 0 | 0 |
| 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 0 | 5 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 11 | 9 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 2 | 4 | 0 | 0 |
| 2 | 11 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [1,12,0,0,1,0,0,0,0,0,12,0,0,0,0,12],[2,9,0,0,4,11,0,0,0,0,0,5,0,0,5,0],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[11,4,0,0,9,2,0,0,0,0,8,0,0,0,0,8],[2,2,0,0,4,11,0,0,0,0,8,0,0,0,0,5] >;
C6.1452+ 1+4 in GAP, Magma, Sage, TeX
C_6._{145}2_+^{1+4} % in TeX
G:=Group("C6.145ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1388);
// by ID
G=gap.SmallGroup(192,1388);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,136,6278]);
// 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,a*d=d*a,e*a*e^-1=a^-1,c*b*c=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