metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.232D6, (C4×S3)⋊6Q8, C12⋊Q8⋊48C2, (C4×Q8)⋊15S3, D6.1(C2×Q8), C4.59(S3×Q8), C4⋊C4.297D6, (Q8×C12)⋊13C2, (C4×Dic6)⋊39C2, (S3×C42).6C2, D6⋊Q8.5C2, (C2×Q8).201D6, C12.117(C2×Q8), Dic3.2(C2×Q8), C4.47(C4○D12), C6.30(C22×Q8), Dic3.Q8⋊46C2, C42⋊2S3.4C2, (C2×C6).122C24, D6⋊3Q8.14C2, C4.D12.14C2, Dic3⋊Q8⋊33C2, C12.117(C4○D4), (C4×C12).174C22, (C2×C12).499C23, D6⋊C4.102C22, Dic3.4(C4○D4), (C6×Q8).222C22, C4⋊Dic3.307C22, C22.143(S3×C23), Dic3⋊C4.155C22, (C22×S3).179C23, C3⋊3(C23.37C23), (C2×Dic6).291C22, (C4×Dic3).253C22, (C2×Dic3).216C23, C2.13(C2×S3×Q8), C6.54(C2×C4○D4), C2.30(S3×C4○D4), C2.61(C2×C4○D12), (S3×C2×C4).295C22, (C3×C4⋊C4).350C22, (C2×C4).584(C22×S3), SmallGroup(192,1137)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.232D6
G = < a,b,c,d | a4=b4=1, c6=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b2c5 >
Subgroups: 472 in 222 conjugacy classes, 109 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C2×C42, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, C6×Q8, C23.37C23, C4×Dic6, C4×Dic6, S3×C42, C42⋊2S3, C12⋊Q8, Dic3.Q8, D6⋊Q8, C4.D12, Dic3⋊Q8, D6⋊3Q8, Q8×C12, C42.232D6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C24, C22×S3, C22×Q8, C2×C4○D4, C4○D12, S3×Q8, S3×C23, C23.37C23, C2×C4○D12, C2×S3×Q8, S3×C4○D4, C42.232D6
►On 96 points
(1 59 7 53)(2 54 8 60)(3 49 9 55)(4 56 10 50)(5 51 11 57)(6 58 12 52)(13 76 19 82)(14 83 20 77)(15 78 21 84)(16 73 22 79)(17 80 23 74)(18 75 24 81)(25 38 31 44)(26 45 32 39)(27 40 33 46)(28 47 34 41)(29 42 35 48)(30 37 36 43)(61 90 67 96)(62 85 68 91)(63 92 69 86)(64 87 70 93)(65 94 71 88)(66 89 72 95)
(1 95 26 13)(2 96 27 14)(3 85 28 15)(4 86 29 16)(5 87 30 17)(6 88 31 18)(7 89 32 19)(8 90 33 20)(9 91 34 21)(10 92 35 22)(11 93 36 23)(12 94 25 24)(37 80 51 70)(38 81 52 71)(39 82 53 72)(40 83 54 61)(41 84 55 62)(42 73 56 63)(43 74 57 64)(44 75 58 65)(45 76 59 66)(46 77 60 67)(47 78 49 68)(48 79 50 69)
(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 32 25)(2 36 33 5)(3 4 34 35)(7 12 26 31)(8 30 27 11)(9 10 28 29)(13 18 89 94)(14 93 90 17)(15 16 91 92)(19 24 95 88)(20 87 96 23)(21 22 85 86)(37 46 57 54)(38 53 58 45)(39 44 59 52)(40 51 60 43)(41 42 49 50)(47 48 55 56)(61 80 77 64)(62 63 78 79)(65 76 81 72)(66 71 82 75)(67 74 83 70)(68 69 84 73)
G:=sub<Sym(96)| (1,59,7,53)(2,54,8,60)(3,49,9,55)(4,56,10,50)(5,51,11,57)(6,58,12,52)(13,76,19,82)(14,83,20,77)(15,78,21,84)(16,73,22,79)(17,80,23,74)(18,75,24,81)(25,38,31,44)(26,45,32,39)(27,40,33,46)(28,47,34,41)(29,42,35,48)(30,37,36,43)(61,90,67,96)(62,85,68,91)(63,92,69,86)(64,87,70,93)(65,94,71,88)(66,89,72,95), (1,95,26,13)(2,96,27,14)(3,85,28,15)(4,86,29,16)(5,87,30,17)(6,88,31,18)(7,89,32,19)(8,90,33,20)(9,91,34,21)(10,92,35,22)(11,93,36,23)(12,94,25,24)(37,80,51,70)(38,81,52,71)(39,82,53,72)(40,83,54,61)(41,84,55,62)(42,73,56,63)(43,74,57,64)(44,75,58,65)(45,76,59,66)(46,77,60,67)(47,78,49,68)(48,79,50,69), (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), (1,6,32,25)(2,36,33,5)(3,4,34,35)(7,12,26,31)(8,30,27,11)(9,10,28,29)(13,18,89,94)(14,93,90,17)(15,16,91,92)(19,24,95,88)(20,87,96,23)(21,22,85,86)(37,46,57,54)(38,53,58,45)(39,44,59,52)(40,51,60,43)(41,42,49,50)(47,48,55,56)(61,80,77,64)(62,63,78,79)(65,76,81,72)(66,71,82,75)(67,74,83,70)(68,69,84,73)>;
G:=Group( (1,59,7,53)(2,54,8,60)(3,49,9,55)(4,56,10,50)(5,51,11,57)(6,58,12,52)(13,76,19,82)(14,83,20,77)(15,78,21,84)(16,73,22,79)(17,80,23,74)(18,75,24,81)(25,38,31,44)(26,45,32,39)(27,40,33,46)(28,47,34,41)(29,42,35,48)(30,37,36,43)(61,90,67,96)(62,85,68,91)(63,92,69,86)(64,87,70,93)(65,94,71,88)(66,89,72,95), (1,95,26,13)(2,96,27,14)(3,85,28,15)(4,86,29,16)(5,87,30,17)(6,88,31,18)(7,89,32,19)(8,90,33,20)(9,91,34,21)(10,92,35,22)(11,93,36,23)(12,94,25,24)(37,80,51,70)(38,81,52,71)(39,82,53,72)(40,83,54,61)(41,84,55,62)(42,73,56,63)(43,74,57,64)(44,75,58,65)(45,76,59,66)(46,77,60,67)(47,78,49,68)(48,79,50,69), (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), (1,6,32,25)(2,36,33,5)(3,4,34,35)(7,12,26,31)(8,30,27,11)(9,10,28,29)(13,18,89,94)(14,93,90,17)(15,16,91,92)(19,24,95,88)(20,87,96,23)(21,22,85,86)(37,46,57,54)(38,53,58,45)(39,44,59,52)(40,51,60,43)(41,42,49,50)(47,48,55,56)(61,80,77,64)(62,63,78,79)(65,76,81,72)(66,71,82,75)(67,74,83,70)(68,69,84,73) );
G=PermutationGroup([[(1,59,7,53),(2,54,8,60),(3,49,9,55),(4,56,10,50),(5,51,11,57),(6,58,12,52),(13,76,19,82),(14,83,20,77),(15,78,21,84),(16,73,22,79),(17,80,23,74),(18,75,24,81),(25,38,31,44),(26,45,32,39),(27,40,33,46),(28,47,34,41),(29,42,35,48),(30,37,36,43),(61,90,67,96),(62,85,68,91),(63,92,69,86),(64,87,70,93),(65,94,71,88),(66,89,72,95)], [(1,95,26,13),(2,96,27,14),(3,85,28,15),(4,86,29,16),(5,87,30,17),(6,88,31,18),(7,89,32,19),(8,90,33,20),(9,91,34,21),(10,92,35,22),(11,93,36,23),(12,94,25,24),(37,80,51,70),(38,81,52,71),(39,82,53,72),(40,83,54,61),(41,84,55,62),(42,73,56,63),(43,74,57,64),(44,75,58,65),(45,76,59,66),(46,77,60,67),(47,78,49,68),(48,79,50,69)], [(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)], [(1,6,32,25),(2,36,33,5),(3,4,34,35),(7,12,26,31),(8,30,27,11),(9,10,28,29),(13,18,89,94),(14,93,90,17),(15,16,91,92),(19,24,95,88),(20,87,96,23),(21,22,85,86),(37,46,57,54),(38,53,58,45),(39,44,59,52),(40,51,60,43),(41,42,49,50),(47,48,55,56),(61,80,77,64),(62,63,78,79),(65,76,81,72),(66,71,82,75),(67,74,83,70),(68,69,84,73)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 4S | 4T | 4U | 4V | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | S3×Q8 | S3×C4○D4 |
| kernel | C42.232D6 | C4×Dic6 | S3×C42 | C42⋊2S3 | C12⋊Q8 | Dic3.Q8 | D6⋊Q8 | C4.D12 | Dic3⋊Q8 | D6⋊3Q8 | Q8×C12 | C4×Q8 | C4×S3 | C42 | C4⋊C4 | C2×Q8 | Dic3 | C12 | C4 | C4 | C2 |
| # reps | 1 | 3 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 1 | 4 | 4 | 8 | 2 | 2 |
Matrix representation of C42.232D6 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 3 |
| 0 | 0 | 3 | 4 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 2 | 0 | 0 |
| 11 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 2 | 11 | 0 | 0 |
| 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,3,0,0,3,4],[8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[11,11,0,0,2,9,0,0,0,0,0,12,0,0,1,0],[2,9,0,0,11,11,0,0,0,0,0,12,0,0,1,0] >;
C42.232D6 in GAP, Magma, Sage, TeX
C_4^2._{232}D_6 % in TeX
G:=Group("C4^2.232D6"); // GroupNames label
G:=SmallGroup(192,1137);
// by ID
G=gap.SmallGroup(192,1137);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,675,570,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2,d^2=a^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^5>;
// generators/relations