metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊14D6, C4⋊C4⋊45D6, (C4×D4)⋊9S3, (C4×S3)⋊14D4, D6⋊2(C4○D4), (C4×D12)⋊25C2, (D4×C12)⋊11C2, C22⋊C4⋊44D6, D6.38(C2×D4), C4.219(S3×D4), (C22×C4)⋊13D6, C23⋊2D6⋊33C2, D6⋊D4⋊30C2, Dic3⋊D4⋊47C2, (C4×C12)⋊17C22, D6⋊C4⋊65C22, D6⋊Q8⋊51C2, (C2×D4).211D6, C12.378(C2×D4), (C2×C6).91C24, C6.47(C22×D4), C42⋊2S3⋊13C2, C23.9D6⋊53C2, D6.D4⋊49C2, C22⋊3(C4○D12), C4⋊Dic3⋊57C22, Dic3.43(C2×D4), C23.14D6⋊46C2, (C2×C12).157C23, Dic3⋊C4⋊70C22, (C22×C12)⋊15C22, C3⋊2(C22.19C24), (C4×Dic3)⋊51C22, (C2×Dic6)⋊52C22, (C6×D4).304C22, Dic3.D4⋊50C2, (C2×D12).209C22, C6.D4⋊49C22, C23.180(C22×S3), C22.116(S3×C23), (C22×C6).161C23, (C2×Dic3).38C23, (S3×C23).106C22, (C22×S3).170C23, (C22×Dic3).220C22, C2.19(C2×S3×D4), (C2×C4○D12)⋊5C2, (C2×C6)⋊1(C4○D4), (C4×C3⋊D4)⋊42C2, (S3×C2×C4)⋊47C22, (S3×C22×C4)⋊22C2, C2.20(S3×C4○D4), C6.39(C2×C4○D4), (C3×C4⋊C4)⋊57C22, C2.43(C2×C4○D12), (C2×C3⋊D4)⋊37C22, (C3×C22⋊C4)⋊55C22, (C2×C4).156(C22×S3), SmallGroup(192,1106)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 872 in 330 conjugacy classes, 109 normal (91 characteristic)
C1, C2 [×3], C2 [×8], C3, C4 [×2], C4 [×10], C22, C22 [×2], C22 [×24], S3 [×5], C6 [×3], C6 [×3], C2×C4 [×5], C2×C4 [×23], D4 [×14], Q8 [×2], C23 [×2], C23 [×9], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×4], D6 [×4], D6 [×15], C2×C6, C2×C6 [×2], C2×C6 [×5], C42, C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×10], C2×D4, C2×D4 [×6], C2×Q8, C4○D4 [×4], C24, Dic6 [×2], C4×S3 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×5], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×5], C2×C12 [×4], C3×D4 [×2], C22×S3 [×3], C22×S3 [×6], C22×C6 [×2], C42⋊C2, C4×D4, C4×D4 [×3], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×4], C4⋊Dic3, D6⋊C4 [×6], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×5], S3×C2×C4 [×4], C2×D12 [×2], C4○D12 [×4], C22×Dic3, C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, S3×C23, C22.19C24, C42⋊2S3, C4×D12, Dic3.D4, D6⋊D4, C23.9D6, Dic3⋊D4, D6.D4, D6⋊Q8, C4×C3⋊D4 [×2], C23⋊2D6, C23.14D6, D4×C12, S3×C22×C4, C2×C4○D12, C42⋊14D6
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], C4○D12 [×2], S3×D4 [×2], S3×C23, C22.19C24, C2×C4○D12, C2×S3×D4, S3×C4○D4, C42⋊14D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, bd=db, dcd=c-1 >
►On 48 points
(1 19 14 26)(2 27 15 20)(3 21 16 28)(4 29 17 22)(5 23 18 30)(6 25 13 24)(7 39 44 31)(8 32 45 40)(9 41 46 33)(10 34 47 42)(11 37 48 35)(12 36 43 38)
(1 33 4 36)(2 34 5 31)(3 35 6 32)(7 27 47 23)(8 28 48 24)(9 29 43 19)(10 30 44 20)(11 25 45 21)(12 26 46 22)(13 40 16 37)(14 41 17 38)(15 42 18 39)
(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 6)(2 5)(3 4)(7 44)(8 43)(9 48)(10 47)(11 46)(12 45)(13 14)(15 18)(16 17)(19 28)(20 27)(21 26)(22 25)(23 30)(24 29)(31 34)(32 33)(35 36)(37 38)(39 42)(40 41)
G:=sub<Sym(48)| (1,19,14,26)(2,27,15,20)(3,21,16,28)(4,29,17,22)(5,23,18,30)(6,25,13,24)(7,39,44,31)(8,32,45,40)(9,41,46,33)(10,34,47,42)(11,37,48,35)(12,36,43,38), (1,33,4,36)(2,34,5,31)(3,35,6,32)(7,27,47,23)(8,28,48,24)(9,29,43,19)(10,30,44,20)(11,25,45,21)(12,26,46,22)(13,40,16,37)(14,41,17,38)(15,42,18,39), (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), (1,6)(2,5)(3,4)(7,44)(8,43)(9,48)(10,47)(11,46)(12,45)(13,14)(15,18)(16,17)(19,28)(20,27)(21,26)(22,25)(23,30)(24,29)(31,34)(32,33)(35,36)(37,38)(39,42)(40,41)>;
G:=Group( (1,19,14,26)(2,27,15,20)(3,21,16,28)(4,29,17,22)(5,23,18,30)(6,25,13,24)(7,39,44,31)(8,32,45,40)(9,41,46,33)(10,34,47,42)(11,37,48,35)(12,36,43,38), (1,33,4,36)(2,34,5,31)(3,35,6,32)(7,27,47,23)(8,28,48,24)(9,29,43,19)(10,30,44,20)(11,25,45,21)(12,26,46,22)(13,40,16,37)(14,41,17,38)(15,42,18,39), (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), (1,6)(2,5)(3,4)(7,44)(8,43)(9,48)(10,47)(11,46)(12,45)(13,14)(15,18)(16,17)(19,28)(20,27)(21,26)(22,25)(23,30)(24,29)(31,34)(32,33)(35,36)(37,38)(39,42)(40,41) );
G=PermutationGroup([(1,19,14,26),(2,27,15,20),(3,21,16,28),(4,29,17,22),(5,23,18,30),(6,25,13,24),(7,39,44,31),(8,32,45,40),(9,41,46,33),(10,34,47,42),(11,37,48,35),(12,36,43,38)], [(1,33,4,36),(2,34,5,31),(3,35,6,32),(7,27,47,23),(8,28,48,24),(9,29,43,19),(10,30,44,20),(11,25,45,21),(12,26,46,22),(13,40,16,37),(14,41,17,38),(15,42,18,39)], [(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)], [(1,6),(2,5),(3,4),(7,44),(8,43),(9,48),(10,47),(11,46),(12,45),(13,14),(15,18),(16,17),(19,28),(20,27),(21,26),(22,25),(23,30),(24,29),(31,34),(32,33),(35,36),(37,38),(39,42),(40,41)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 12 |
G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
48 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 | ··· | 12L |
| 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 | 6 | 6 | 6 | 6 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | S3×D4 | S3×C4○D4 |
| kernel | C42⋊14D6 | C42⋊2S3 | C4×D12 | Dic3.D4 | D6⋊D4 | C23.9D6 | Dic3⋊D4 | D6.D4 | D6⋊Q8 | C4×C3⋊D4 | C23⋊2D6 | C23.14D6 | D4×C12 | S3×C22×C4 | C2×C4○D12 | C4×D4 | C4×S3 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D6 | C2×C6 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{14}D_6 % in TeX
G:=Group("C4^2:14D6"); // GroupNames label
G:=SmallGroup(192,1106);
// by ID
G=gap.SmallGroup(192,1106);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,675,570,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations