metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊20D6, C6.1242+ (1+4), (C4×S3)⋊4D4, (C2×Q8)⋊21D6, C4.32(S3×D4), C22⋊C4⋊20D6, D6.45(C2×D4), C4.4D4⋊8S3, C12.61(C2×D4), Dic3⋊D4⋊39C2, D6⋊D4⋊23C2, C4⋊D12⋊14C2, C12⋊3D4⋊24C2, (C4×C12)⋊22C22, D6⋊C4⋊23C22, (C2×D4).171D6, (C2×D12)⋊9C22, (C6×Q8)⋊12C22, C6.88(C22×D4), C42⋊2S3⋊19C2, C2.48(D4○D12), (C2×C6).218C24, Dic3.50(C2×D4), C12.23D4⋊21C2, (C2×C12).186C23, Dic3⋊C4⋊55C22, C3⋊4(C22.29C24), (C4×Dic3)⋊35C22, (C6×D4).153C22, C23.50(C22×S3), (C22×C6).48C23, (S3×C23).63C22, C22.239(S3×C23), (C22×S3).213C23, (C2×Dic3).113C23, (C2×S3×D4)⋊16C2, C2.61(C2×S3×D4), (S3×C2×C4)⋊25C22, (C2×Q8⋊3S3)⋊10C2, (C3×C4.4D4)⋊10C2, (C2×C3⋊D4)⋊22C22, (C3×C22⋊C4)⋊28C22, (C2×C4).193(C22×S3), SmallGroup(192,1233)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1104 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×8], C22, C22 [×30], S3 [×6], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], C2×C4 [×11], D4 [×22], Q8 [×2], C23 [×2], C23 [×13], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×4], D6 [×2], D6 [×22], C2×C6, C2×C6 [×6], C42, C42, C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×18], C2×Q8, C4○D4 [×4], C24 [×2], C4×S3 [×4], C4×S3 [×4], D12 [×12], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×8], C2×C12, C2×C12 [×4], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×S3 [×4], C22×S3 [×8], C22×C6 [×2], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4, C4.4D4, C4⋊1D4 [×2], C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×2], D6⋊C4 [×6], C4×C12, C3×C22⋊C4 [×4], S3×C2×C4, S3×C2×C4 [×2], C2×D12 [×2], C2×D12 [×6], S3×D4 [×4], Q8⋊3S3 [×4], C2×C3⋊D4 [×6], C6×D4, C6×Q8, S3×C23 [×2], C22.29C24, C42⋊2S3, C4⋊D12, D6⋊D4 [×4], Dic3⋊D4 [×4], C12⋊3D4, C12.23D4, C3×C4.4D4, C2×S3×D4, C2×Q8⋊3S3, C42⋊20D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, 2+ (1+4) [×2], S3×D4 [×2], S3×C23, C22.29C24, C2×S3×D4, D4○D12 [×2], C42⋊20D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=a2b-1, dbd=b-1, dcd=c-1 >
►On 48 points
(1 28 9 25)(2 26 7 29)(3 30 8 27)(4 24 12 21)(5 22 10 19)(6 20 11 23)(13 42 35 46)(14 47 36 37)(15 38 31 48)(16 43 32 39)(17 40 33 44)(18 45 34 41)
(1 46 5 39)(2 44 6 37)(3 48 4 41)(7 40 11 47)(8 38 12 45)(9 42 10 43)(13 22 16 28)(14 26 17 20)(15 24 18 30)(19 32 25 35)(21 34 27 31)(23 36 29 33)
(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 11)(5 10)(6 12)(13 16)(14 15)(17 18)(20 24)(21 23)(26 30)(27 29)(31 36)(32 35)(33 34)(37 38)(39 42)(40 41)(43 46)(44 45)(47 48)
G:=sub<Sym(48)| (1,28,9,25)(2,26,7,29)(3,30,8,27)(4,24,12,21)(5,22,10,19)(6,20,11,23)(13,42,35,46)(14,47,36,37)(15,38,31,48)(16,43,32,39)(17,40,33,44)(18,45,34,41), (1,46,5,39)(2,44,6,37)(3,48,4,41)(7,40,11,47)(8,38,12,45)(9,42,10,43)(13,22,16,28)(14,26,17,20)(15,24,18,30)(19,32,25,35)(21,34,27,31)(23,36,29,33), (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)(4,11)(5,10)(6,12)(13,16)(14,15)(17,18)(20,24)(21,23)(26,30)(27,29)(31,36)(32,35)(33,34)(37,38)(39,42)(40,41)(43,46)(44,45)(47,48)>;
G:=Group( (1,28,9,25)(2,26,7,29)(3,30,8,27)(4,24,12,21)(5,22,10,19)(6,20,11,23)(13,42,35,46)(14,47,36,37)(15,38,31,48)(16,43,32,39)(17,40,33,44)(18,45,34,41), (1,46,5,39)(2,44,6,37)(3,48,4,41)(7,40,11,47)(8,38,12,45)(9,42,10,43)(13,22,16,28)(14,26,17,20)(15,24,18,30)(19,32,25,35)(21,34,27,31)(23,36,29,33), (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)(4,11)(5,10)(6,12)(13,16)(14,15)(17,18)(20,24)(21,23)(26,30)(27,29)(31,36)(32,35)(33,34)(37,38)(39,42)(40,41)(43,46)(44,45)(47,48) );
G=PermutationGroup([(1,28,9,25),(2,26,7,29),(3,30,8,27),(4,24,12,21),(5,22,10,19),(6,20,11,23),(13,42,35,46),(14,47,36,37),(15,38,31,48),(16,43,32,39),(17,40,33,44),(18,45,34,41)], [(1,46,5,39),(2,44,6,37),(3,48,4,41),(7,40,11,47),(8,38,12,45),(9,42,10,43),(13,22,16,28),(14,26,17,20),(15,24,18,30),(19,32,25,35),(21,34,27,31),(23,36,29,33)], [(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),(4,11),(5,10),(6,12),(13,16),(14,15),(17,18),(20,24),(21,23),(26,30),(27,29),(31,36),(32,35),(33,34),(37,38),(39,42),(40,41),(43,46),(44,45),(47,48)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,3,6,0,0,0,0,7,10,0,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;
36 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 | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | 2+ (1+4) | S3×D4 | D4○D12 |
| kernel | C42⋊20D6 | C42⋊2S3 | C4⋊D12 | D6⋊D4 | Dic3⋊D4 | C12⋊3D4 | C12.23D4 | C3×C4.4D4 | C2×S3×D4 | C2×Q8⋊3S3 | C4.4D4 | C4×S3 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 1 | 1 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{20}D_6 % in TeX
G:=Group("C4^2:20D6"); // GroupNames label
G:=SmallGroup(192,1233);
// by ID
G=gap.SmallGroup(192,1233);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,675,570,297,192,6278]);
// 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,c*b*c^-1=a^2*b^-1,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations