metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.228D6, (C4×D4)⋊10S3, (C4×S3)⋊10D4, C12⋊Q8⋊47C2, D6.1(C2×D4), (C4×D12)⋊26C2, (D4×C12)⋊12C2, C4⋊2(C4○D12), C12⋊1(C4○D4), (S3×C42)⋊4C2, C4⋊C4.282D6, C4.220(S3×D4), C12⋊D4⋊43C2, Dic3⋊D4⋊48C2, D6⋊3D4⋊45C2, C12⋊3D4⋊32C2, (C2×D4).212D6, C12.379(C2×D4), (C2×C6).92C24, Dic3.3(C2×D4), C6.48(C22×D4), Dic3⋊1(C4○D4), C22⋊C4.109D6, (C22×C4).223D6, (C4×C12).151C22, (C2×C12).492C23, D6⋊C4.122C22, (C6×D4).255C22, C23.11D6⋊50C2, (C2×D12).259C22, (C22×S3).30C23, C4⋊Dic3.363C22, C22.117(S3×C23), C23.103(C22×S3), (C22×C6).162C23, C3⋊2(C22.26C24), Dic3⋊C4.110C22, (C22×C12).106C22, (C2×Dic6).238C22, (C2×Dic3).203C23, (C4×Dic3).251C22, C6.D4.105C22, C2.20(C2×S3×D4), (C4×C3⋊D4)⋊4C2, (C2×C4○D12)⋊6C2, C2.21(S3×C4○D4), C6.40(C2×C4○D4), C2.44(C2×C4○D12), (S3×C2×C4).292C22, (C3×C4⋊C4).325C22, (C2×C4).578(C22×S3), (C2×C3⋊D4).112C22, (C3×C22⋊C4).121C22, SmallGroup(192,1107)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 808 in 310 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×4], C4 [×10], C22, C22 [×16], S3 [×4], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×21], D4 [×20], Q8 [×4], C23 [×2], C23 [×3], Dic3 [×4], Dic3 [×3], C12 [×4], C12 [×3], D6 [×2], D6 [×8], C2×C6, C2×C6 [×6], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×5], C2×D4, C2×D4 [×9], C2×Q8 [×2], C4○D4 [×8], Dic6 [×4], C4×S3 [×4], C4×S3 [×8], D12 [×6], C2×Dic3 [×3], C2×Dic3 [×2], C3⋊D4 [×12], C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×2], C22×S3, C22×S3 [×2], C22×C6 [×2], C2×C42, C4×D4, C4×D4 [×3], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C2×C4○D4 [×2], C4×Dic3 [×3], Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×4], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6 [×2], S3×C2×C4 [×3], S3×C2×C4 [×2], C2×D12, C2×D12 [×2], C4○D12 [×8], C2×C3⋊D4 [×6], C22×C12 [×2], C6×D4, C22.26C24, S3×C42, C4×D12, Dic3⋊D4 [×2], C23.11D6 [×2], C12⋊Q8, C12⋊D4, C4×C3⋊D4 [×2], D6⋊3D4, C12⋊3D4, D4×C12, C2×C4○D12 [×2], C42.228D6
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, C22.26C24, C2×C4○D12, C2×S3×D4, S3×C4○D4, C42.228D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b2c-1 >
►On 96 points
(1 64 19 67)(2 68 20 65)(3 66 21 69)(4 70 22 61)(5 62 23 71)(6 72 24 63)(7 75 30 60)(8 55 25 76)(9 77 26 56)(10 57 27 78)(11 73 28 58)(12 59 29 74)(13 38 81 54)(14 49 82 39)(15 40 83 50)(16 51 84 41)(17 42 79 52)(18 53 80 37)(31 90 45 93)(32 94 46 85)(33 86 47 95)(34 96 48 87)(35 88 43 91)(36 92 44 89)
(1 40 12 35)(2 41 7 36)(3 42 8 31)(4 37 9 32)(5 38 10 33)(6 39 11 34)(13 78 95 71)(14 73 96 72)(15 74 91 67)(16 75 92 68)(17 76 93 69)(18 77 94 70)(19 50 29 43)(20 51 30 44)(21 52 25 45)(22 53 26 46)(23 54 27 47)(24 49 28 48)(55 90 66 79)(56 85 61 80)(57 86 62 81)(58 87 63 82)(59 88 64 83)(60 89 65 84)
(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)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 90 12 79)(2 84 7 89)(3 88 8 83)(4 82 9 87)(5 86 10 81)(6 80 11 85)(13 23 95 27)(14 26 96 22)(15 21 91 25)(16 30 92 20)(17 19 93 29)(18 28 94 24)(31 59 42 64)(32 63 37 58)(33 57 38 62)(34 61 39 56)(35 55 40 66)(36 65 41 60)(43 76 50 69)(44 68 51 75)(45 74 52 67)(46 72 53 73)(47 78 54 71)(48 70 49 77)
G:=sub<Sym(96)| (1,64,19,67)(2,68,20,65)(3,66,21,69)(4,70,22,61)(5,62,23,71)(6,72,24,63)(7,75,30,60)(8,55,25,76)(9,77,26,56)(10,57,27,78)(11,73,28,58)(12,59,29,74)(13,38,81,54)(14,49,82,39)(15,40,83,50)(16,51,84,41)(17,42,79,52)(18,53,80,37)(31,90,45,93)(32,94,46,85)(33,86,47,95)(34,96,48,87)(35,88,43,91)(36,92,44,89), (1,40,12,35)(2,41,7,36)(3,42,8,31)(4,37,9,32)(5,38,10,33)(6,39,11,34)(13,78,95,71)(14,73,96,72)(15,74,91,67)(16,75,92,68)(17,76,93,69)(18,77,94,70)(19,50,29,43)(20,51,30,44)(21,52,25,45)(22,53,26,46)(23,54,27,47)(24,49,28,48)(55,90,66,79)(56,85,61,80)(57,86,62,81)(58,87,63,82)(59,88,64,83)(60,89,65,84), (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,90,12,79)(2,84,7,89)(3,88,8,83)(4,82,9,87)(5,86,10,81)(6,80,11,85)(13,23,95,27)(14,26,96,22)(15,21,91,25)(16,30,92,20)(17,19,93,29)(18,28,94,24)(31,59,42,64)(32,63,37,58)(33,57,38,62)(34,61,39,56)(35,55,40,66)(36,65,41,60)(43,76,50,69)(44,68,51,75)(45,74,52,67)(46,72,53,73)(47,78,54,71)(48,70,49,77)>;
G:=Group( (1,64,19,67)(2,68,20,65)(3,66,21,69)(4,70,22,61)(5,62,23,71)(6,72,24,63)(7,75,30,60)(8,55,25,76)(9,77,26,56)(10,57,27,78)(11,73,28,58)(12,59,29,74)(13,38,81,54)(14,49,82,39)(15,40,83,50)(16,51,84,41)(17,42,79,52)(18,53,80,37)(31,90,45,93)(32,94,46,85)(33,86,47,95)(34,96,48,87)(35,88,43,91)(36,92,44,89), (1,40,12,35)(2,41,7,36)(3,42,8,31)(4,37,9,32)(5,38,10,33)(6,39,11,34)(13,78,95,71)(14,73,96,72)(15,74,91,67)(16,75,92,68)(17,76,93,69)(18,77,94,70)(19,50,29,43)(20,51,30,44)(21,52,25,45)(22,53,26,46)(23,54,27,47)(24,49,28,48)(55,90,66,79)(56,85,61,80)(57,86,62,81)(58,87,63,82)(59,88,64,83)(60,89,65,84), (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,90,12,79)(2,84,7,89)(3,88,8,83)(4,82,9,87)(5,86,10,81)(6,80,11,85)(13,23,95,27)(14,26,96,22)(15,21,91,25)(16,30,92,20)(17,19,93,29)(18,28,94,24)(31,59,42,64)(32,63,37,58)(33,57,38,62)(34,61,39,56)(35,55,40,66)(36,65,41,60)(43,76,50,69)(44,68,51,75)(45,74,52,67)(46,72,53,73)(47,78,54,71)(48,70,49,77) );
G=PermutationGroup([(1,64,19,67),(2,68,20,65),(3,66,21,69),(4,70,22,61),(5,62,23,71),(6,72,24,63),(7,75,30,60),(8,55,25,76),(9,77,26,56),(10,57,27,78),(11,73,28,58),(12,59,29,74),(13,38,81,54),(14,49,82,39),(15,40,83,50),(16,51,84,41),(17,42,79,52),(18,53,80,37),(31,90,45,93),(32,94,46,85),(33,86,47,95),(34,96,48,87),(35,88,43,91),(36,92,44,89)], [(1,40,12,35),(2,41,7,36),(3,42,8,31),(4,37,9,32),(5,38,10,33),(6,39,11,34),(13,78,95,71),(14,73,96,72),(15,74,91,67),(16,75,92,68),(17,76,93,69),(18,77,94,70),(19,50,29,43),(20,51,30,44),(21,52,25,45),(22,53,26,46),(23,54,27,47),(24,49,28,48),(55,90,66,79),(56,85,61,80),(57,86,62,81),(58,87,63,82),(59,88,64,83),(60,89,65,84)], [(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),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,90,12,79),(2,84,7,89),(3,88,8,83),(4,82,9,87),(5,86,10,81),(6,80,11,85),(13,23,95,27),(14,26,96,22),(15,21,91,25),(16,30,92,20),(17,19,93,29),(18,28,94,24),(31,59,42,64),(32,63,37,58),(33,57,38,62),(34,61,39,56),(35,55,40,66),(36,65,41,60),(43,76,50,69),(44,68,51,75),(45,74,52,67),(46,72,53,73),(47,78,54,71),(48,70,49,77)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 9 | 2 | 0 | 0 |
| 11 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 8 | 0 | 0 | 0 |
| 8 | 5 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[8,0,0,0,0,8,0,0,0,0,12,0,0,0,0,12],[9,11,0,0,2,11,0,0,0,0,0,1,0,0,1,0],[8,8,0,0,0,5,0,0,0,0,0,12,0,0,12,0] >;
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 4Q | 4R | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 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 | 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 | S3 | D4 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | S3×D4 | S3×C4○D4 |
| kernel | C42.228D6 | S3×C42 | C4×D12 | Dic3⋊D4 | C23.11D6 | C12⋊Q8 | C12⋊D4 | C4×C3⋊D4 | D6⋊3D4 | C12⋊3D4 | D4×C12 | C2×C4○D12 | C4×D4 | C4×S3 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | Dic3 | C12 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 4 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{228}D_6 % in TeX
G:=Group("C4^2.228D6"); // GroupNames label
G:=SmallGroup(192,1107);
// by ID
G=gap.SmallGroup(192,1107);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,675,570,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^-1>;
// generators/relations