metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2)⋊28D6, C12.72C24, C24.54C23, (S3×D4).C4, (S3×Q8).C4, (C2×C8)⋊23D6, C8○D4⋊12S3, D4⋊2S3.C4, Q8⋊3S3.C4, C8○D12⋊17C2, C4○D4.59D6, D4.13(C4×S3), C3⋊C8.33C23, Q8.19(C4×S3), (S3×C8)⋊12C22, D12.C4⋊13C2, (C2×C24)⋊32C22, D12.21(C2×C4), C3⋊2(Q8○M4(2)), D4.Dic3⋊8C2, C4.71(S3×C23), C6.35(C23×C4), C8.56(C22×S3), C8⋊S3⋊22C22, (S3×M4(2))⋊11C2, (C4×S3).37C23, C12.39(C22×C4), Dic6.22(C2×C4), D6.16(C22×C4), (C2×C12).514C23, C4○D12.52C22, C4.Dic3⋊27C22, (C3×M4(2))⋊33C22, Dic3.16(C22×C4), C4.39(S3×C2×C4), C22.5(S3×C2×C4), (C3×C8○D4)⋊14C2, (C2×C3⋊C8)⋊13C22, (S3×C4○D4).3C2, C3⋊D4.2(C2×C4), (C2×C8⋊S3)⋊28C2, C2.36(S3×C22×C4), (C4×S3).11(C2×C4), (C3×D4).17(C2×C4), (C2×C6).5(C22×C4), (C3×Q8).18(C2×C4), (S3×C2×C4).154C22, (C22×S3).28(C2×C4), (C2×C4).607(C22×S3), (C2×Dic3).38(C2×C4), (C3×C4○D4).44C22, SmallGroup(192,1309)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 512 in 258 conjugacy classes, 147 normal (24 characteristic)
C1, C2, C2 [×7], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], S3 [×4], C6, C6 [×3], C8, C8 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], Dic3, Dic3 [×3], C12, C12 [×3], D6, D6 [×3], D6 [×3], C2×C6 [×3], C2×C8 [×3], C2×C8 [×9], M4(2) [×3], M4(2) [×13], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], C3⋊C8, C3⋊C8 [×3], C24, C24 [×3], Dic6 [×3], C4×S3, C4×S3 [×9], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×3], C2×M4(2) [×6], C8○D4, C8○D4 [×7], C2×C4○D4, S3×C8 [×6], C8⋊S3, C8⋊S3 [×9], C2×C3⋊C8 [×3], C4.Dic3 [×3], C2×C24 [×3], C3×M4(2) [×3], S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D4⋊2S3 [×3], S3×Q8, Q8⋊3S3, C3×C4○D4, Q8○M4(2), C2×C8⋊S3 [×3], C8○D12 [×3], S3×M4(2) [×3], D12.C4 [×3], D4.Dic3, C3×C8○D4, S3×C4○D4, M4(2)⋊28D6
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, S3×C2×C4 [×6], S3×C23, Q8○M4(2), S3×C22×C4, M4(2)⋊28D6
Generators and relations
G = < a,b,c,d | a8=b2=c6=d2=1, bab=cac-1=a5, ad=da, cbc-1=dbd=a4b, dcd=c-1 >
►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 19)(2 24)(3 21)(4 18)(5 23)(6 20)(7 17)(8 22)(9 28)(10 25)(11 30)(12 27)(13 32)(14 29)(15 26)(16 31)(33 45)(34 42)(35 47)(36 44)(37 41)(38 46)(39 43)(40 48)
(1 46 31 21 36 10)(2 43 32 18 37 15)(3 48 25 23 38 12)(4 45 26 20 39 9)(5 42 27 17 40 14)(6 47 28 22 33 11)(7 44 29 19 34 16)(8 41 30 24 35 13)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)
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), (1,19)(2,24)(3,21)(4,18)(5,23)(6,20)(7,17)(8,22)(9,28)(10,25)(11,30)(12,27)(13,32)(14,29)(15,26)(16,31)(33,45)(34,42)(35,47)(36,44)(37,41)(38,46)(39,43)(40,48), (1,46,31,21,36,10)(2,43,32,18,37,15)(3,48,25,23,38,12)(4,45,26,20,39,9)(5,42,27,17,40,14)(6,47,28,22,33,11)(7,44,29,19,34,16)(8,41,30,24,35,13), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42)>;
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), (1,19)(2,24)(3,21)(4,18)(5,23)(6,20)(7,17)(8,22)(9,28)(10,25)(11,30)(12,27)(13,32)(14,29)(15,26)(16,31)(33,45)(34,42)(35,47)(36,44)(37,41)(38,46)(39,43)(40,48), (1,46,31,21,36,10)(2,43,32,18,37,15)(3,48,25,23,38,12)(4,45,26,20,39,9)(5,42,27,17,40,14)(6,47,28,22,33,11)(7,44,29,19,34,16)(8,41,30,24,35,13), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42) );
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)], [(1,19),(2,24),(3,21),(4,18),(5,23),(6,20),(7,17),(8,22),(9,28),(10,25),(11,30),(12,27),(13,32),(14,29),(15,26),(16,31),(33,45),(34,42),(35,47),(36,44),(37,41),(38,46),(39,43),(40,48)], [(1,46,31,21,36,10),(2,43,32,18,37,15),(3,48,25,23,38,12),(4,45,26,20,39,9),(5,42,27,17,40,14),(6,47,28,22,33,11),(7,44,29,19,34,16),(8,41,30,24,35,13)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,41),(40,42)])
Matrix representation ►G ⊆ GL4(𝔽5) generated by
| 0 | 0 | 0 | 1 |
| 0 | 3 | 3 | 0 |
| 0 | 1 | 2 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 1 | 0 | 0 | 2 |
| 0 | 3 | 3 | 0 |
| 0 | 1 | 2 | 0 |
| 1 | 0 | 0 | 4 |
| 3 | 0 | 0 | 4 |
| 0 | 0 | 3 | 0 |
| 0 | 4 | 3 | 0 |
| 4 | 0 | 0 | 1 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(5))| [0,0,0,2,0,3,1,0,0,3,2,0,1,0,0,0],[0,4,1,0,4,0,0,3,0,0,0,3,0,0,2,0],[0,1,3,0,1,0,0,0,2,0,0,3,0,4,4,0],[0,4,0,0,4,0,0,0,3,0,0,3,0,1,2,0] >;
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 8A | ··· | 8H | 8I | ··· | 8P | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D | 24E | ··· | 24J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D6 | D6 | D6 | C4×S3 | C4×S3 | Q8○M4(2) | M4(2)⋊28D6 |
| kernel | M4(2)⋊28D6 | C2×C8⋊S3 | C8○D12 | S3×M4(2) | D12.C4 | D4.Dic3 | C3×C8○D4 | S3×C4○D4 | S3×D4 | D4⋊2S3 | S3×Q8 | Q8⋊3S3 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 1 | 3 | 3 | 1 | 6 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
M_{4(2)}\rtimes_{28}D_6 % in TeX
G:=Group("M4(2):28D6"); // GroupNames label
G:=SmallGroup(192,1309);
// by ID
G=gap.SmallGroup(192,1309);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,1123,80,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^5,a*d=d*a,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations