metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.115D6, C6.222+ 1+4, C6.622- 1+4, (C4×D4)⋊23S3, C12⋊Q8⋊16C2, (D4×C12)⋊25C2, C4⋊C4.286D6, C4.D12⋊16C2, (C4×Dic6)⋊35C2, (C2×D4).222D6, C42⋊2S3⋊14C2, D6⋊3D4.10C2, C4.45(C4○D12), (C2×C6).105C24, C2.19(Q8○D12), C22⋊C4.117D6, (C22×C4).229D6, C23.8D6⋊9C2, C12.112(C4○D4), C12.48D4⋊12C2, C2.23(D4⋊6D6), C23.12D6⋊10C2, (C2×C12).163C23, (C4×C12).159C22, D6⋊C4.123C22, (C6×D4).264C22, C23.28D6⋊3C2, C4⋊Dic3.40C22, C23.11D6⋊10C2, (C22×S3).39C23, (C22×C6).175C23, C22.130(S3×C23), C23.112(C22×S3), (C22×C12).82C22, (C2×Dic3).46C23, (C2×Dic6).27C22, (C4×Dic3).77C22, Dic3⋊C4.135C22, C3⋊2(C22.36C24), C6.D4.15C22, C6.47(C2×C4○D4), C2.54(C2×C4○D12), (S3×C2×C4).67C22, (C3×C4⋊C4).333C22, (C2×C4).287(C22×S3), (C2×C3⋊D4).18C22, (C3×C22⋊C4).128C22, SmallGroup(192,1120)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.115D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=b2c-1 >
Subgroups: 520 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C22×C12, C6×D4, C22.36C24, C4×Dic6, C42⋊2S3, C23.8D6, C23.11D6, C12⋊Q8, C4.D12, C12.48D4, C23.28D6, C23.12D6, D6⋊3D4, D4×C12, C42.115D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, C4○D12, S3×C23, C22.36C24, C2×C4○D12, D4⋊6D6, Q8○D12, C42.115D6
►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 73 28 58)(8 59 29 74)(9 75 30 60)(10 55 25 76)(11 77 26 56)(12 57 27 78)(13 40 81 54)(14 49 82 41)(15 42 83 50)(16 51 84 37)(17 38 79 52)(18 53 80 39)(31 88 43 91)(32 92 44 89)(33 90 45 93)(34 94 46 85)(35 86 47 95)(36 96 48 87)
(1 42 8 31)(2 37 9 32)(3 38 10 33)(4 39 11 34)(5 40 12 35)(6 41 7 36)(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 6 8 7)(2 12 9 5)(3 4 10 11)(13 84 95 89)(14 88 96 83)(15 82 91 87)(16 86 92 81)(17 80 93 85)(18 90 94 79)(19 24 29 28)(20 27 30 23)(21 22 25 26)(31 48 42 49)(32 54 37 47)(33 46 38 53)(34 52 39 45)(35 44 40 51)(36 50 41 43)(55 56 66 61)(57 60 62 65)(58 64 63 59)(67 72 74 73)(68 78 75 71)(69 70 76 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)(7,73,28,58)(8,59,29,74)(9,75,30,60)(10,55,25,76)(11,77,26,56)(12,57,27,78)(13,40,81,54)(14,49,82,41)(15,42,83,50)(16,51,84,37)(17,38,79,52)(18,53,80,39)(31,88,43,91)(32,92,44,89)(33,90,45,93)(34,94,46,85)(35,86,47,95)(36,96,48,87), (1,42,8,31)(2,37,9,32)(3,38,10,33)(4,39,11,34)(5,40,12,35)(6,41,7,36)(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), (1,6,8,7)(2,12,9,5)(3,4,10,11)(13,84,95,89)(14,88,96,83)(15,82,91,87)(16,86,92,81)(17,80,93,85)(18,90,94,79)(19,24,29,28)(20,27,30,23)(21,22,25,26)(31,48,42,49)(32,54,37,47)(33,46,38,53)(34,52,39,45)(35,44,40,51)(36,50,41,43)(55,56,66,61)(57,60,62,65)(58,64,63,59)(67,72,74,73)(68,78,75,71)(69,70,76,77)>;
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)(7,73,28,58)(8,59,29,74)(9,75,30,60)(10,55,25,76)(11,77,26,56)(12,57,27,78)(13,40,81,54)(14,49,82,41)(15,42,83,50)(16,51,84,37)(17,38,79,52)(18,53,80,39)(31,88,43,91)(32,92,44,89)(33,90,45,93)(34,94,46,85)(35,86,47,95)(36,96,48,87), (1,42,8,31)(2,37,9,32)(3,38,10,33)(4,39,11,34)(5,40,12,35)(6,41,7,36)(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), (1,6,8,7)(2,12,9,5)(3,4,10,11)(13,84,95,89)(14,88,96,83)(15,82,91,87)(16,86,92,81)(17,80,93,85)(18,90,94,79)(19,24,29,28)(20,27,30,23)(21,22,25,26)(31,48,42,49)(32,54,37,47)(33,46,38,53)(34,52,39,45)(35,44,40,51)(36,50,41,43)(55,56,66,61)(57,60,62,65)(58,64,63,59)(67,72,74,73)(68,78,75,71)(69,70,76,77) );
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),(7,73,28,58),(8,59,29,74),(9,75,30,60),(10,55,25,76),(11,77,26,56),(12,57,27,78),(13,40,81,54),(14,49,82,41),(15,42,83,50),(16,51,84,37),(17,38,79,52),(18,53,80,39),(31,88,43,91),(32,92,44,89),(33,90,45,93),(34,94,46,85),(35,86,47,95),(36,96,48,87)], [(1,42,8,31),(2,37,9,32),(3,38,10,33),(4,39,11,34),(5,40,12,35),(6,41,7,36),(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)], [(1,6,8,7),(2,12,9,5),(3,4,10,11),(13,84,95,89),(14,88,96,83),(15,82,91,87),(16,86,92,81),(17,80,93,85),(18,90,94,79),(19,24,29,28),(20,27,30,23),(21,22,25,26),(31,48,42,49),(32,54,37,47),(33,46,38,53),(34,52,39,45),(35,44,40,51),(36,50,41,43),(55,56,66,61),(57,60,62,65),(58,64,63,59),(67,72,74,73),(68,78,75,71),(69,70,76,77)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4F | 4G | 4H | 4I | ··· | 4O | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ 1+4 | 2- 1+4 | D4⋊6D6 | Q8○D12 |
| kernel | C42.115D6 | C4×Dic6 | C42⋊2S3 | C23.8D6 | C23.11D6 | C12⋊Q8 | C4.D12 | C12.48D4 | C23.28D6 | C23.12D6 | D6⋊3D4 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C42.115D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 11 | 0 |
| 0 | 0 | 0 | 9 | 0 | 11 |
| 0 | 0 | 2 | 0 | 4 | 0 |
| 0 | 0 | 0 | 2 | 0 | 4 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 9 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 9 | 2 | 0 | 0 | 0 | 0 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,0,2,0,0,0,0,9,0,2,0,0,11,0,4,0,0,0,0,11,0,4],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,0,0,0,0,11,9,0,0,0,0,4,2],[9,11,0,0,0,0,2,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,1,0,0,0,0,12,12,0,0],[2,4,0,0,0,0,2,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,1,0,0,0,0,0,12,12,0,0] >;
C42.115D6 in GAP, Magma, Sage, TeX
C_4^2._{115}D_6 % in TeX
G:=Group("C4^2.115D6"); // GroupNames label
G:=SmallGroup(192,1120);
// by ID
G=gap.SmallGroup(192,1120);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,100,675,570,6278]);
// 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,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^-1>;
// generators/relations