metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.38C24, D12.33C23, 2+ (1+4)⋊9S3, Dic6.33C23, Q8○D12⋊9C2, C4○D4.32D6, (C3×D4).37D4, C3⋊6(D4○SD16), C3⋊C8.17C23, (C3×Q8).37D4, D4⋊S3⋊21C22, (C2×D4).118D6, Q8.13D6⋊9C2, C12.270(C2×D4), C4.38(S3×C23), Q8.14D6⋊11C2, D12⋊6C22⋊12C2, D4.19(C3⋊D4), C4○D12⋊11C22, D4.Dic3⋊11C2, D4.S3⋊21C22, Q8.26(C3⋊D4), (C3×D4).26C23, D4.26(C22×S3), C3⋊Q16⋊18C22, C6.172(C22×D4), (C3×Q8).26C23, Q8.36(C22×S3), (C2×C12).119C23, Q8⋊2S3⋊22C22, (C2×Dic6)⋊43C22, (C6×D4).169C22, C4.Dic3⋊17C22, (C3×2+ (1+4))⋊3C2, (C2×C3⋊C8)⋊25C22, (C2×C6).86(C2×D4), C4.76(C2×C3⋊D4), (C2×D4.S3)⋊32C2, C22.7(C2×C3⋊D4), C2.45(C22×C3⋊D4), (C2×C4).103(C22×S3), (C3×C4○D4).29C22, SmallGroup(192,1395)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 600 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×7], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], S3, C6, C6 [×6], C8 [×4], C2×C4 [×3], C2×C4 [×9], D4 [×6], D4 [×10], Q8 [×2], Q8 [×6], C23 [×3], Dic3 [×3], C12, C12 [×3], C12, D6, C2×C6 [×3], C2×C6 [×6], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×3], C2×D4 [×3], C2×Q8 [×4], C4○D4, C4○D4 [×3], C4○D4 [×7], C3⋊C8, C3⋊C8 [×3], Dic6 [×3], Dic6 [×3], C4×S3 [×3], D12, C2×Dic3 [×3], C3⋊D4 [×3], C2×C12 [×3], C2×C12 [×3], C3×D4 [×6], C3×D4 [×6], C3×Q8 [×2], C22×C6 [×3], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ (1+4), 2- (1+4), C2×C3⋊C8 [×3], C4.Dic3 [×3], D4⋊S3 [×3], D4.S3 [×9], Q8⋊2S3, C3⋊Q16 [×3], C2×Dic6 [×3], C4○D12 [×3], D4⋊2S3 [×3], S3×Q8, C6×D4 [×3], C6×D4 [×3], C3×C4○D4, C3×C4○D4 [×3], C3×C4○D4, D4○SD16, D12⋊6C22 [×3], C2×D4.S3 [×3], D4.Dic3, Q8.13D6 [×3], Q8.14D6 [×3], Q8○D12, C3×2+ (1+4), D12.33C23
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C3⋊D4 [×6], S3×C23, D4○SD16, C22×C3⋊D4, D12.33C23
Generators and relations
G = < a,b,c,d,e | a12=b2=1, c2=d2=e2=a6, bab=a-1, ac=ca, ad=da, eae-1=a7, bc=cb, bd=db, ebe-1=a3b, dcd-1=a6c, ce=ec, de=ed >
►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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 48)(22 47)(23 46)(24 45)
(1 16 7 22)(2 17 8 23)(3 18 9 24)(4 19 10 13)(5 20 11 14)(6 21 12 15)(25 45 31 39)(26 46 32 40)(27 47 33 41)(28 48 34 42)(29 37 35 43)(30 38 36 44)
(1 4 7 10)(2 5 8 11)(3 6 9 12)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)
(1 22 7 16)(2 17 8 23)(3 24 9 18)(4 19 10 13)(5 14 11 20)(6 21 12 15)(25 48 31 42)(26 43 32 37)(27 38 33 44)(28 45 34 39)(29 40 35 46)(30 47 36 41)
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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,36)(11,35)(12,34)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,48)(22,47)(23,46)(24,45), (1,16,7,22)(2,17,8,23)(3,18,9,24)(4,19,10,13)(5,20,11,14)(6,21,12,15)(25,45,31,39)(26,46,32,40)(27,47,33,41)(28,48,34,42)(29,37,35,43)(30,38,36,44), (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48), (1,22,7,16)(2,17,8,23)(3,24,9,18)(4,19,10,13)(5,14,11,20)(6,21,12,15)(25,48,31,42)(26,43,32,37)(27,38,33,44)(28,45,34,39)(29,40,35,46)(30,47,36,41)>;
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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,36)(11,35)(12,34)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,48)(22,47)(23,46)(24,45), (1,16,7,22)(2,17,8,23)(3,18,9,24)(4,19,10,13)(5,20,11,14)(6,21,12,15)(25,45,31,39)(26,46,32,40)(27,47,33,41)(28,48,34,42)(29,37,35,43)(30,38,36,44), (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48), (1,22,7,16)(2,17,8,23)(3,24,9,18)(4,19,10,13)(5,14,11,20)(6,21,12,15)(25,48,31,42)(26,43,32,37)(27,38,33,44)(28,45,34,39)(29,40,35,46)(30,47,36,41) );
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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,36),(11,35),(12,34),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,48),(22,47),(23,46),(24,45)], [(1,16,7,22),(2,17,8,23),(3,18,9,24),(4,19,10,13),(5,20,11,14),(6,21,12,15),(25,45,31,39),(26,46,32,40),(27,47,33,41),(28,48,34,42),(29,37,35,43),(30,38,36,44)], [(1,4,7,10),(2,5,8,11),(3,6,9,12),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,40,43,46),(38,41,44,47),(39,42,45,48)], [(1,22,7,16),(2,17,8,23),(3,24,9,18),(4,19,10,13),(5,14,11,20),(6,21,12,15),(25,48,31,42),(26,43,32,37),(27,38,33,44),(28,45,34,39),(29,40,35,46),(30,47,36,41)])
Matrix representation ►G ⊆ GL6(𝔽73)
| 64 | 0 | 0 | 0 | 0 | 0 |
| 71 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 72 | 2 |
| 0 | 0 | 0 | 1 | 72 | 1 |
| 55 | 7 | 0 | 0 | 0 | 0 |
| 6 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 67 | 0 | 61 |
| 0 | 0 | 67 | 6 | 61 | 12 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 12 | 67 | 6 | 61 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 1 | 72 | 1 | 71 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 72 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 72 | 2 |
| 0 | 0 | 72 | 0 | 72 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 72 | 1 | 71 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 1 |
G:=sub<GL(6,GF(73))| [64,71,0,0,0,0,0,8,0,0,0,0,0,0,0,72,72,0,0,0,1,0,1,1,0,0,0,0,72,72,0,0,0,0,2,1],[55,6,0,0,0,0,7,18,0,0,0,0,0,0,6,67,6,12,0,0,67,6,6,67,0,0,0,61,0,6,0,0,61,12,0,61],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0,1,0,0,72,1,0,72,0,0,0,71,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,72,72,0,0,72,0,1,0,0,0,0,0,72,72,0,0,0,0,2,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,72,72,0,0,0,72,0,1,0,0,1,1,0,0,0,0,0,71,0,1] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | ··· | 6J | 8A | 8B | 8C | 8D | 8E | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 12 | 2 | 4 | ··· | 4 | 6 | 6 | 12 | 12 | 12 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C3⋊D4 | C3⋊D4 | D4○SD16 | D12.33C23 |
| kernel | D12.33C23 | D12⋊6C22 | C2×D4.S3 | D4.Dic3 | Q8.13D6 | Q8.14D6 | Q8○D12 | C3×2+ (1+4) | 2+ (1+4) | C3×D4 | C3×Q8 | C2×D4 | C4○D4 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 1 | 1 | 3 | 1 | 3 | 4 | 6 | 2 | 2 | 1 |
In GAP, Magma, Sage, TeX
D_{12}._{33}C_2^3 % in TeX
G:=Group("D12.33C2^3"); // GroupNames label
G:=SmallGroup(192,1395);
// by ID
G=gap.SmallGroup(192,1395);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,675,136,1684,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=b^2=1,c^2=d^2=e^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^7,b*c=c*b,b*d=d*b,e*b*e^-1=a^3*b,d*c*d^-1=a^6*c,c*e=e*c,d*e=e*d>;
// generators/relations