metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊21D6, (C2×D4)⋊22D6, (C4×S3)⋊12D4, D6⋊3(C4○D4), C23⋊2D6⋊8C2, C4⋊D4⋊26S3, C22⋊C4⋊26D6, D6.40(C2×D4), C4.182(S3×D4), D6⋊3D4⋊17C2, C4.D12⋊20C2, (D4×Dic3)⋊20C2, C12.226(C2×D4), (C6×D4)⋊11C22, C6.65(C22×D4), Dic3⋊4D4⋊8C2, C23.9D6⋊18C2, (C2×C6).150C24, D6⋊C4.14C22, C4⋊Dic3⋊30C22, Dic3.63(C2×D4), (C22×C4).384D6, C12.48D4⋊33C2, C22⋊3(D4⋊2S3), (C2×C12).594C23, Dic3⋊C4⋊28C22, C3⋊4(C22.19C24), (C2×Dic6)⋊24C22, (C4×Dic3)⋊20C22, C23.25(C22×S3), (C22×C6).19C23, C6.D4⋊22C22, C22.171(S3×C23), (C2×Dic3).71C23, (C22×S3).185C23, (S3×C23).107C22, (C22×C12).240C22, (C22×Dic3)⋊19C22, C2.38(C2×S3×D4), (S3×C22×C4)⋊4C2, (C2×C6)⋊5(C4○D4), (C3×C4⋊C4)⋊9C22, C4⋊C4⋊7S3⋊19C2, C2.38(S3×C4○D4), (C3×C4⋊D4)⋊12C2, C6.151(C2×C4○D4), (C2×D4⋊2S3)⋊12C2, C2.36(C2×D4⋊2S3), (S3×C2×C4).247C22, (C3×C22⋊C4)⋊11C22, (C2×C4).294(C22×S3), (C2×C3⋊D4).27C22, SmallGroup(192,1165)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 848 in 330 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×8], C3, C4 [×2], C4 [×10], C22, C22 [×2], C22 [×24], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×24], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×8], Dic3 [×2], Dic3 [×5], C12 [×2], C12 [×3], D6 [×4], D6 [×12], C2×C6, C2×C6 [×2], C2×C6 [×8], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×5], C22×C4, C22×C4 [×11], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24, Dic6 [×2], C4×S3 [×4], C4×S3 [×6], C2×Dic3 [×2], C2×Dic3 [×4], C2×Dic3 [×6], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×6], C22×S3 [×2], C22×S3 [×6], C22×C6, C22×C6 [×2], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4, C4⋊D4, C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4 [×4], C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×4], S3×C2×C4 [×4], D4⋊2S3 [×4], C22×Dic3, C22×Dic3 [×2], C2×C3⋊D4 [×4], C22×C12, C6×D4, C6×D4 [×2], S3×C23, C22.19C24, Dic3⋊4D4 [×2], C23.9D6 [×2], C4⋊C4⋊7S3, C4.D12, C12.48D4, D4×Dic3 [×2], C23⋊2D6 [×2], D6⋊3D4, C3×C4⋊D4, S3×C22×C4, C2×D4⋊2S3, C4⋊C4⋊21D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C22×S3 [×7], C22×D4, C2×C4○D4 [×2], S3×D4 [×2], D4⋊2S3 [×2], S3×C23, C22.19C24, C2×S3×D4, C2×D4⋊2S3, S3×C4○D4, C4⋊C4⋊21D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=b-1, dbd=a2b, dcd=c-1 >
►On 48 points
(1 28 7 34)(2 29 8 35)(3 30 9 36)(4 25 10 31)(5 26 11 32)(6 27 12 33)(13 40 16 37)(14 41 17 38)(15 42 18 39)(19 46 22 43)(20 47 23 44)(21 48 24 45)
(1 46 10 40)(2 41 11 47)(3 48 12 42)(4 37 7 43)(5 44 8 38)(6 39 9 45)(13 28 19 31)(14 32 20 29)(15 30 21 33)(16 34 22 25)(17 26 23 35)(18 36 24 27)
(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 9)(2 8)(3 7)(4 12)(5 11)(6 10)(13 15)(16 18)(19 21)(22 24)(25 33)(26 32)(27 31)(28 36)(29 35)(30 34)(37 39)(40 42)(43 45)(46 48)
G:=sub<Sym(48)| (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,25,10,31)(5,26,11,32)(6,27,12,33)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45), (1,46,10,40)(2,41,11,47)(3,48,12,42)(4,37,7,43)(5,44,8,38)(6,39,9,45)(13,28,19,31)(14,32,20,29)(15,30,21,33)(16,34,22,25)(17,26,23,35)(18,36,24,27), (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), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,15)(16,18)(19,21)(22,24)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34)(37,39)(40,42)(43,45)(46,48)>;
G:=Group( (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,25,10,31)(5,26,11,32)(6,27,12,33)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45), (1,46,10,40)(2,41,11,47)(3,48,12,42)(4,37,7,43)(5,44,8,38)(6,39,9,45)(13,28,19,31)(14,32,20,29)(15,30,21,33)(16,34,22,25)(17,26,23,35)(18,36,24,27), (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), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,15)(16,18)(19,21)(22,24)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34)(37,39)(40,42)(43,45)(46,48) );
G=PermutationGroup([(1,28,7,34),(2,29,8,35),(3,30,9,36),(4,25,10,31),(5,26,11,32),(6,27,12,33),(13,40,16,37),(14,41,17,38),(15,42,18,39),(19,46,22,43),(20,47,23,44),(21,48,24,45)], [(1,46,10,40),(2,41,11,47),(3,48,12,42),(4,37,7,43),(5,44,8,38),(6,39,9,45),(13,28,19,31),(14,32,20,29),(15,30,21,33),(16,34,22,25),(17,26,23,35),(18,36,24,27)], [(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)], [(1,9),(2,8),(3,7),(4,12),(5,11),(6,10),(13,15),(16,18),(19,21),(22,24),(25,33),(26,32),(27,31),(28,36),(29,35),(30,34),(37,39),(40,42),(43,45),(46,48)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,8,0],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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 | D6 | C4○D4 | C4○D4 | S3×D4 | D4⋊2S3 | S3×C4○D4 |
| kernel | C4⋊C4⋊21D6 | Dic3⋊4D4 | C23.9D6 | C4⋊C4⋊7S3 | C4.D12 | C12.48D4 | D4×Dic3 | C23⋊2D6 | D6⋊3D4 | C3×C4⋊D4 | S3×C22×C4 | C2×D4⋊2S3 | C4⋊D4 | C4×S3 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D6 | C2×C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 1 | 3 | 4 | 4 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_{21}D_6 % in TeX
G:=Group("C4:C4:21D6"); // GroupNames label
G:=SmallGroup(192,1165);
// by ID
G=gap.SmallGroup(192,1165);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,1123,794,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations