metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊25D6, C6.1372+ (1+4), C4⋊C4⋊33D6, (C4×D12)⋊13C2, (C4×C12)⋊7C22, D6⋊C4⋊7C22, Dic3⋊D4⋊44C2, D6⋊D4⋊27C2, C12⋊D4⋊36C2, C42⋊7S3⋊8C2, C42⋊2C2⋊2S3, D6⋊Q8⋊40C2, C22⋊C4.40D6, Dic3⋊5D4⋊39C2, D6.12(C4○D4), D6.D4⋊38C2, C23.9D6⋊48C2, C2.62(D4○D12), (C2×D12)⋊29C22, (C2×C6).248C24, C4⋊Dic3⋊61C22, (C2×C12).193C23, Dic3⋊C4⋊27C22, C3⋊9(C22.32C24), (C4×Dic3)⋊38C22, (C2×Dic6)⋊11C22, (C22×C6).62C23, C23.64(C22×S3), C23.11D6⋊44C2, (S3×C23).68C22, C22.269(S3×C23), (C22×S3).111C23, (C2×Dic3).264C23, C6.D4.64C22, (S3×C2×C4)⋊27C22, C4⋊C4⋊S3⋊41C2, C2.95(S3×C4○D4), (S3×C22⋊C4)⋊20C2, (C3×C4⋊C4)⋊32C22, C6.206(C2×C4○D4), (C3×C42⋊2C2)⋊3C2, (C2×C4).85(C22×S3), (C2×C3⋊D4).68C22, (C3×C22⋊C4).73C22, SmallGroup(192,1263)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 784 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×10], C22, C22 [×20], S3 [×5], C6 [×3], C6, C2×C4 [×6], C2×C4 [×8], D4 [×9], Q8, C23, C23 [×8], Dic3 [×4], C12 [×6], D6 [×2], D6 [×15], C2×C6, C2×C6 [×3], C42, C42, C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×4], C2×D4 [×7], C2×Q8, C24, Dic6, C4×S3 [×4], D12 [×7], C2×Dic3 [×4], C3⋊D4 [×2], C2×C12 [×6], C22×S3 [×4], C22×S3 [×4], C22×C6, C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C42⋊2C2, C42⋊2C2, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×10], C6.D4, C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C2×Dic6, S3×C2×C4 [×4], C2×D12 [×5], C2×C3⋊D4 [×2], S3×C23, C22.32C24, C4×D12, C42⋊7S3, S3×C22⋊C4, D6⋊D4 [×2], C23.9D6, Dic3⋊D4, C23.11D6, Dic3⋊5D4, D6.D4, C12⋊D4 [×2], D6⋊Q8, C4⋊C4⋊S3, C3×C42⋊2C2, C42⋊25D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ (1+4) [×2], S3×C23, C22.32C24, S3×C4○D4, D4○D12 [×2], C42⋊25D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=b-1, dcd=c-1 >
►On 48 points
(1 31 4 26)(2 35 5 30)(3 33 6 28)(7 34 10 29)(8 32 11 27)(9 36 12 25)(13 41 16 43)(14 47 17 39)(15 37 18 45)(19 46 22 38)(20 42 23 44)(21 48 24 40)
(1 22 10 13)(2 20 11 17)(3 24 12 15)(4 19 7 16)(5 23 8 14)(6 21 9 18)(25 37 33 40)(26 46 34 43)(27 39 35 42)(28 48 36 45)(29 41 31 38)(30 44 32 47)
(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 12)(8 11)(9 10)(13 21)(14 20)(15 19)(16 24)(17 23)(18 22)(25 31)(26 36)(27 35)(28 34)(29 33)(30 32)(37 41)(38 40)(43 45)(46 48)
G:=sub<Sym(48)| (1,31,4,26)(2,35,5,30)(3,33,6,28)(7,34,10,29)(8,32,11,27)(9,36,12,25)(13,41,16,43)(14,47,17,39)(15,37,18,45)(19,46,22,38)(20,42,23,44)(21,48,24,40), (1,22,10,13)(2,20,11,17)(3,24,12,15)(4,19,7,16)(5,23,8,14)(6,21,9,18)(25,37,33,40)(26,46,34,43)(27,39,35,42)(28,48,36,45)(29,41,31,38)(30,44,32,47), (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)(7,12)(8,11)(9,10)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22)(25,31)(26,36)(27,35)(28,34)(29,33)(30,32)(37,41)(38,40)(43,45)(46,48)>;
G:=Group( (1,31,4,26)(2,35,5,30)(3,33,6,28)(7,34,10,29)(8,32,11,27)(9,36,12,25)(13,41,16,43)(14,47,17,39)(15,37,18,45)(19,46,22,38)(20,42,23,44)(21,48,24,40), (1,22,10,13)(2,20,11,17)(3,24,12,15)(4,19,7,16)(5,23,8,14)(6,21,9,18)(25,37,33,40)(26,46,34,43)(27,39,35,42)(28,48,36,45)(29,41,31,38)(30,44,32,47), (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)(7,12)(8,11)(9,10)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22)(25,31)(26,36)(27,35)(28,34)(29,33)(30,32)(37,41)(38,40)(43,45)(46,48) );
G=PermutationGroup([(1,31,4,26),(2,35,5,30),(3,33,6,28),(7,34,10,29),(8,32,11,27),(9,36,12,25),(13,41,16,43),(14,47,17,39),(15,37,18,45),(19,46,22,38),(20,42,23,44),(21,48,24,40)], [(1,22,10,13),(2,20,11,17),(3,24,12,15),(4,19,7,16),(5,23,8,14),(6,21,9,18),(25,37,33,40),(26,46,34,43),(27,39,35,42),(28,48,36,45),(29,41,31,38),(30,44,32,47)], [(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),(7,12),(8,11),(9,10),(13,21),(14,20),(15,19),(16,24),(17,23),(18,22),(25,31),(26,36),(27,35),(28,34),(29,33),(30,32),(37,41),(38,40),(43,45),(46,48)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 7 | 1 |
| 0 | 0 | 0 | 12 | 12 | 6 |
| 0 | 0 | 10 | 7 | 1 | 0 |
| 0 | 0 | 6 | 3 | 0 | 1 |
| 9 | 3 | 0 | 0 | 0 | 0 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 | 11 | 0 |
| 0 | 0 | 7 | 10 | 0 | 11 |
| 0 | 0 | 12 | 0 | 10 | 7 |
| 0 | 0 | 0 | 12 | 6 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 7 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 10 | 3 | 12 | 12 |
| 0 | 0 | 10 | 7 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 7 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 3 | 12 | 12 |
| 0 | 0 | 6 | 3 | 0 | 1 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,10,6,0,0,0,12,7,3,0,0,7,12,1,0,0,0,1,6,0,1],[9,3,0,0,0,0,3,4,0,0,0,0,0,0,3,7,12,0,0,0,6,10,0,12,0,0,11,0,10,6,0,0,0,11,7,3],[1,7,0,0,0,0,0,12,0,0,0,0,0,0,1,12,10,10,0,0,1,0,3,7,0,0,0,0,12,1,0,0,0,0,12,0],[1,7,0,0,0,0,0,12,0,0,0,0,0,0,12,0,10,6,0,0,12,1,3,3,0,0,0,0,12,0,0,0,0,0,12,1] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | 2+ (1+4) | S3×C4○D4 | D4○D12 |
| kernel | C42⋊25D6 | C4×D12 | C42⋊7S3 | S3×C22⋊C4 | D6⋊D4 | C23.9D6 | Dic3⋊D4 | C23.11D6 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | D6⋊Q8 | C4⋊C4⋊S3 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | D6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 4 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{25}D_6 % in TeX
G:=Group("C4^2:25D6"); // GroupNames label
G:=SmallGroup(192,1263);
// by ID
G=gap.SmallGroup(192,1263);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,675,570,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations