metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊7S3, (C4×S3)⋊2C4, C4.14(C4×S3), D6⋊C4.4C2, D6.4(C2×C4), (C2×C4).44D6, C12.11(C2×C4), Dic3○2(C4⋊C4), C4⋊Dic3⋊12C2, C3⋊3(C42⋊C2), (C4×Dic3)⋊13C2, C6.26(C4○D4), (C2×C6).33C23, C6.10(C22×C4), Dic3.9(C2×C4), C2.4(D4⋊2S3), (C2×C12).56C22, C2.1(Q8⋊3S3), C22.17(C22×S3), (C22×S3).19C22, (C2×Dic3).30C22, (C3×C4⋊C4)⋊3C2, (S3×C2×C4).2C2, C2.12(S3×C2×C4), SmallGroup(96,99)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊7S3
G = < a,b,c,d | a4=b4=c3=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 154 in 76 conjugacy classes, 41 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C42⋊C2, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C4⋊C4⋊7S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, S3×C2×C4, D4⋊2S3, Q8⋊3S3, C4⋊C4⋊7S3
Character table of C4⋊C4⋊7S3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -i | -i | i | i | 1 | 1 | 1 | -1 | -1 | i | i | -i | -i | -1 | -1 | 1 | i | -i | -i | 1 | i | -1 | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | i | -i | i | -i | -1 | 1 | 1 | -1 | -1 | -i | i | i | -i | -1 | -1 | 1 | -i | i | -i | -1 | i | 1 | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -i | -i | i | i | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -1 | -1 | 1 | i | -i | -i | 1 | i | -1 | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | -1 | -1 | 1 | -i | i | -i | -1 | i | 1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -1 | -1 | 1 | i | -i | i | -1 | -i | 1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | i | i | -i | -i | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | -1 | -1 | 1 | -i | i | i | 1 | -i | -1 | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -i | i | -i | i | -1 | 1 | 1 | -1 | -1 | i | -i | -i | i | -1 | -1 | 1 | i | -i | i | -1 | -i | 1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | i | i | -i | -i | 1 | 1 | 1 | -1 | -1 | -i | -i | i | i | -1 | -1 | 1 | -i | i | i | 1 | -i | -1 | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | -2i | -2i | 2i | 2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -i | i | i | -1 | -i | 1 | complex lifted from C4×S3 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2i | 2i | -2i | -2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | i | -i | -i | -1 | i | 1 | complex lifted from C4×S3 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2i | 2i | -2i | 2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -i | i | -i | 1 | i | -1 | complex lifted from C4×S3 |
| ρ25 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | 2i | -2i | 2i | -2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | i | -i | i | 1 | -i | -1 | complex lifted from C4×S3 |
| ρ27 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ29 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
| ρ30 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
►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 47 43 8)(2 46 44 7)(3 45 41 6)(4 48 42 5)(9 29 15 23)(10 32 16 22)(11 31 13 21)(12 30 14 24)(17 38 26 33)(18 37 27 36)(19 40 28 35)(20 39 25 34)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 29 34)(6 30 35)(7 31 36)(8 32 33)(9 25 42)(10 26 43)(11 27 44)(12 28 41)(21 37 46)(22 38 47)(23 39 48)(24 40 45)
(5 7)(6 8)(9 25)(10 26)(11 27)(12 28)(13 18)(14 19)(15 20)(16 17)(21 39)(22 40)(23 37)(24 38)(29 36)(30 33)(31 34)(32 35)(45 47)(46 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), (1,47,43,8)(2,46,44,7)(3,45,41,6)(4,48,42,5)(9,29,15,23)(10,32,16,22)(11,31,13,21)(12,30,14,24)(17,38,26,33)(18,37,27,36)(19,40,28,35)(20,39,25,34), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,29,34)(6,30,35)(7,31,36)(8,32,33)(9,25,42)(10,26,43)(11,27,44)(12,28,41)(21,37,46)(22,38,47)(23,39,48)(24,40,45), (5,7)(6,8)(9,25)(10,26)(11,27)(12,28)(13,18)(14,19)(15,20)(16,17)(21,39)(22,40)(23,37)(24,38)(29,36)(30,33)(31,34)(32,35)(45,47)(46,48)>;
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), (1,47,43,8)(2,46,44,7)(3,45,41,6)(4,48,42,5)(9,29,15,23)(10,32,16,22)(11,31,13,21)(12,30,14,24)(17,38,26,33)(18,37,27,36)(19,40,28,35)(20,39,25,34), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,29,34)(6,30,35)(7,31,36)(8,32,33)(9,25,42)(10,26,43)(11,27,44)(12,28,41)(21,37,46)(22,38,47)(23,39,48)(24,40,45), (5,7)(6,8)(9,25)(10,26)(11,27)(12,28)(13,18)(14,19)(15,20)(16,17)(21,39)(22,40)(23,37)(24,38)(29,36)(30,33)(31,34)(32,35)(45,47)(46,48) );
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)], [(1,47,43,8),(2,46,44,7),(3,45,41,6),(4,48,42,5),(9,29,15,23),(10,32,16,22),(11,31,13,21),(12,30,14,24),(17,38,26,33),(18,37,27,36),(19,40,28,35),(20,39,25,34)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,29,34),(6,30,35),(7,31,36),(8,32,33),(9,25,42),(10,26,43),(11,27,44),(12,28,41),(21,37,46),(22,38,47),(23,39,48),(24,40,45)], [(5,7),(6,8),(9,25),(10,26),(11,27),(12,28),(13,18),(14,19),(15,20),(16,17),(21,39),(22,40),(23,37),(24,38),(29,36),(30,33),(31,34),(32,35),(45,47),(46,48)]])
C4⋊C4⋊7S3 is a maximal subgroup of
C4⋊C4⋊19D6 D4⋊2S3⋊C4 D6⋊C8⋊11C2 C24⋊1C4⋊C2 (S3×Q8)⋊C4 C4⋊C4.150D6 D6⋊C8.C2 C8⋊Dic3⋊C2 (S3×C8)⋊C4 C8⋊(C4×S3) C4.Q8⋊S3 C6.(C4○D8) C8.27(C4×S3) C8⋊S3⋊C4 C2.D8⋊S3 C2.D8⋊7S3 C6.82+ 1+4 C6.52- 1+4 C6.112+ 1+4 S3×C42⋊C2 C42.91D6 C42.97D6 C42.98D6 C4×D4⋊2S3 C42⋊13D6 C42.113D6 C42.117D6 C42.125D6 C4×Q8⋊3S3 C42.132D6 C42.135D6 C4⋊C4⋊21D6 C6.382+ 1+4 C6.452+ 1+4 C6.1152+ 1+4 C4⋊C4.187D6 C4⋊C4⋊26D6 C6.162- 1+4 C6.532+ 1+4 C6.212- 1+4 C6.222- 1+4 C6.232- 1+4 C6.772- 1+4 C4⋊C4.197D6 C6.1222+ 1+4 C6.622+ 1+4 C6.662+ 1+4 C42.236D6 C42.148D6 C42.237D6 C42.151D6 C42.152D6 C42.153D6 C42.154D6 C42⋊26D6 C42.189D6 C42.162D6 C42.164D6 C42.241D6 C42.174D6 C42.177D6 C42.179D6 C4⋊C4⋊7D9 C62.6C23 C62.11C23 C62.19C23 C62.48C23 C62.236C23 (S3×C20)⋊5C4 (C4×D15)⋊8C4 (S3×Dic5)⋊C4 D30.23(C2×C4) C4⋊C4⋊7D15 C4⋊F5⋊3S3
C4⋊C4⋊7S3 is a maximal quotient of
Dic3.5C42 C3⋊(C42⋊5C4) Dic3⋊C4⋊C4 C22.58(S3×D4) D6⋊C42 D6⋊C4⋊3C4 C42.200D6 C42.202D6 C42.31D6 Dic3×C4⋊C4 (C4×Dic3)⋊9C4 C6.67(C4×D4) C4⋊(D6⋊C4) D6⋊C4⋊7C4 C4⋊C4⋊7D9 C62.6C23 C62.11C23 C62.19C23 C62.48C23 C62.236C23 (S3×C20)⋊5C4 (C4×D15)⋊8C4 (S3×Dic5)⋊C4 D30.23(C2×C4) C4⋊C4⋊7D15 C4⋊F5⋊3S3
Matrix representation of C4⋊C4⋊7S3 ►in GL4(𝔽13) generated by
| 5 | 0 | 0 | 0 |
| 3 | 8 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 8 | 8 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 12 |
| 1 | 0 | 0 | 0 |
| 11 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [5,3,0,0,0,8,0,0,0,0,12,0,0,0,0,12],[8,0,0,0,8,5,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,12],[1,11,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;
C4⋊C4⋊7S3 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_7S_3
% in TeX
G:=Group("C4:C4:7S3"); // GroupNames label
G:=SmallGroup(96,99);
// by ID
G=gap.SmallGroup(96,99);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,362,188,50,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^3=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
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