direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C42⋊4S3, C42⋊39D6, C6⋊1C4≀C2, C4○D12⋊1C4, (C2×D12)⋊6C4, (C2×C42)⋊6S3, D12⋊16(C2×C4), (C2×Dic6)⋊6C4, C4.81(C2×D12), (C2×C4).88D12, C4.21(D6⋊C4), (C4×C12)⋊52C22, Dic6⋊15(C2×C4), (C2×C12).480D4, C12.301(C2×D4), (C22×C4).433D6, (C22×C6).179D4, C12.45(C22⋊C4), C12.109(C22×C4), (C2×C12).793C23, C22.43(D6⋊C4), C4○D12.37C22, C23.80(C3⋊D4), C4.Dic3⋊19C22, (C22×C12).536C22, C3⋊2(C2×C4≀C2), (C2×C4×C12)⋊10C2, C4.67(S3×C2×C4), C2.4(C2×D6⋊C4), (C2×C4).107(C4×S3), (C2×C4○D12).3C2, (C2×C6).422(C2×D4), C6.30(C2×C22⋊C4), (C2×C4.Dic3)⋊3C2, (C2×C12).222(C2×C4), C22.25(C2×C3⋊D4), (C2×C4).236(C3⋊D4), (C2×C6).55(C22⋊C4), (C2×C4).707(C22×S3), SmallGroup(192,486)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C42⋊4S3
G = < a,b,c,d,e | a2=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, ebe=bc=cb, bd=db, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 440 in 170 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3, C4.Dic3, C4×C12, C4×C12, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, C22×C12, C2×C4≀C2, C42⋊4S3, C2×C4.Dic3, C2×C4×C12, C2×C4○D12, C2×C42⋊4S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C4≀C2, C2×C22⋊C4, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C2×C4≀C2, C42⋊4S3, C2×D6⋊C4, C2×C42⋊4S3
►On 48 points
(1 22)(2 21)(3 12)(4 11)(5 19)(6 20)(7 14)(8 13)(9 16)(10 15)(17 23)(18 24)(25 30)(26 31)(27 32)(28 29)(33 47)(34 48)(35 45)(36 46)(37 43)(38 44)(39 41)(40 42)
(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 3 21 11)(2 4 22 12)(5 9 20 15)(6 10 19 16)(7 17 13 24)(8 18 14 23)(25 31 27 29)(26 32 28 30)(33 48 35 46)(34 45 36 47)(37 44 39 42)(38 41 40 43)
(1 19 14)(2 20 13)(3 16 23)(4 15 24)(5 7 22)(6 8 21)(9 17 12)(10 18 11)(25 38 48)(26 39 45)(27 40 46)(28 37 47)(29 43 33)(30 44 34)(31 41 35)(32 42 36)
(1 30)(2 27)(3 28)(4 31)(5 48)(6 36)(7 38)(8 42)(9 33)(10 45)(11 26)(12 29)(13 40)(14 44)(15 35)(16 47)(17 43)(18 39)(19 34)(20 46)(21 32)(22 25)(23 37)(24 41)
G:=sub<Sym(48)| (1,22)(2,21)(3,12)(4,11)(5,19)(6,20)(7,14)(8,13)(9,16)(10,15)(17,23)(18,24)(25,30)(26,31)(27,32)(28,29)(33,47)(34,48)(35,45)(36,46)(37,43)(38,44)(39,41)(40,42), (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), (1,3,21,11)(2,4,22,12)(5,9,20,15)(6,10,19,16)(7,17,13,24)(8,18,14,23)(25,31,27,29)(26,32,28,30)(33,48,35,46)(34,45,36,47)(37,44,39,42)(38,41,40,43), (1,19,14)(2,20,13)(3,16,23)(4,15,24)(5,7,22)(6,8,21)(9,17,12)(10,18,11)(25,38,48)(26,39,45)(27,40,46)(28,37,47)(29,43,33)(30,44,34)(31,41,35)(32,42,36), (1,30)(2,27)(3,28)(4,31)(5,48)(6,36)(7,38)(8,42)(9,33)(10,45)(11,26)(12,29)(13,40)(14,44)(15,35)(16,47)(17,43)(18,39)(19,34)(20,46)(21,32)(22,25)(23,37)(24,41)>;
G:=Group( (1,22)(2,21)(3,12)(4,11)(5,19)(6,20)(7,14)(8,13)(9,16)(10,15)(17,23)(18,24)(25,30)(26,31)(27,32)(28,29)(33,47)(34,48)(35,45)(36,46)(37,43)(38,44)(39,41)(40,42), (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), (1,3,21,11)(2,4,22,12)(5,9,20,15)(6,10,19,16)(7,17,13,24)(8,18,14,23)(25,31,27,29)(26,32,28,30)(33,48,35,46)(34,45,36,47)(37,44,39,42)(38,41,40,43), (1,19,14)(2,20,13)(3,16,23)(4,15,24)(5,7,22)(6,8,21)(9,17,12)(10,18,11)(25,38,48)(26,39,45)(27,40,46)(28,37,47)(29,43,33)(30,44,34)(31,41,35)(32,42,36), (1,30)(2,27)(3,28)(4,31)(5,48)(6,36)(7,38)(8,42)(9,33)(10,45)(11,26)(12,29)(13,40)(14,44)(15,35)(16,47)(17,43)(18,39)(19,34)(20,46)(21,32)(22,25)(23,37)(24,41) );
G=PermutationGroup([[(1,22),(2,21),(3,12),(4,11),(5,19),(6,20),(7,14),(8,13),(9,16),(10,15),(17,23),(18,24),(25,30),(26,31),(27,32),(28,29),(33,47),(34,48),(35,45),(36,46),(37,43),(38,44),(39,41),(40,42)], [(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)], [(1,3,21,11),(2,4,22,12),(5,9,20,15),(6,10,19,16),(7,17,13,24),(8,18,14,23),(25,31,27,29),(26,32,28,30),(33,48,35,46),(34,45,36,47),(37,44,39,42),(38,41,40,43)], [(1,19,14),(2,20,13),(3,16,23),(4,15,24),(5,7,22),(6,8,21),(9,17,12),(10,18,11),(25,38,48),(26,39,45),(27,40,46),(28,37,47),(29,43,33),(30,44,34),(31,41,35),(32,42,36)], [(1,30),(2,27),(3,28),(4,31),(5,48),(6,36),(7,38),(8,42),(9,33),(10,45),(11,26),(12,29),(13,40),(14,44),(15,35),(16,47),(17,43),(18,39),(19,34),(20,46),(21,32),(22,25),(23,37),(24,41)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 6A | ··· | 6G | 8A | 8B | 8C | 8D | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 12 | 12 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | C3⋊D4 | C4≀C2 | C42⋊4S3 |
| kernel | C2×C42⋊4S3 | C42⋊4S3 | C2×C4.Dic3 | C2×C4×C12 | C2×C4○D12 | C2×Dic6 | C2×D12 | C4○D12 | C2×C42 | C2×C12 | C22×C6 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 3 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 8 | 16 |
Matrix representation of C2×C42⋊4S3 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 46 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 27 |
| 1 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 46 |
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 64 |
| 72 | 0 | 0 |
| 0 | 0 | 72 |
| 0 | 72 | 0 |
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[46,0,0,0,1,0,0,0,27],[1,0,0,0,27,0,0,0,46],[1,0,0,0,8,0,0,0,64],[72,0,0,0,0,72,0,72,0] >;
C2×C42⋊4S3 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes_4S_3
% in TeX
G:=Group("C2xC4^2:4S3"); // GroupNames label
G:=SmallGroup(192,486);
// by ID
G=gap.SmallGroup(192,486);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,58,1123,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e=b*c=c*b,b*d=d*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations