metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.236D6, (C4×S3)⋊3Q8, C12⋊Q8⋊34C2, D6.3(C2×Q8), C4.38(S3×Q8), C4⋊C4.204D6, C12.49(C2×Q8), (S3×C42).8C2, D6⋊Q8.1C2, C42.C2⋊15S3, C6.41(C22×Q8), (C2×C12).86C23, (C2×C6).232C24, D6⋊C4.38C22, C12.6Q8⋊22C2, Dic3.16(C2×Q8), Dic6⋊C4⋊34C2, (C4×C12).192C22, Dic3.12(C4○D4), Dic3⋊C4.50C22, C4⋊Dic3.239C22, C22.253(S3×C23), (C22×S3).219C23, C3⋊5(C23.37C23), (C4×Dic3).139C22, (C2×Dic3).311C23, (C2×Dic6).178C22, C2.24(C2×S3×Q8), C2.84(S3×C4○D4), C6.195(C2×C4○D4), C4⋊C4⋊7S3.11C2, (C3×C42.C2)⋊5C2, (S3×C2×C4).249C22, (C2×C4).77(C22×S3), (C3×C4⋊C4).187C22, SmallGroup(192,1247)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.236D6
G = < a,b,c,d | a4=b4=1, c6=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=a2b-1, dcd-1=a2c5 >
Subgroups: 480 in 222 conjugacy classes, 107 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C23.37C23, C12.6Q8, S3×C42, Dic6⋊C4, C12⋊Q8, C4⋊C4⋊7S3, D6⋊Q8, C3×C42.C2, C42.236D6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C24, C22×S3, C22×Q8, C2×C4○D4, S3×Q8, S3×C23, C23.37C23, C2×S3×Q8, S3×C4○D4, C42.236D6
►On 96 points
(1 54 35 76)(2 77 36 55)(3 56 25 78)(4 79 26 57)(5 58 27 80)(6 81 28 59)(7 60 29 82)(8 83 30 49)(9 50 31 84)(10 73 32 51)(11 52 33 74)(12 75 34 53)(13 85 71 47)(14 48 72 86)(15 87 61 37)(16 38 62 88)(17 89 63 39)(18 40 64 90)(19 91 65 41)(20 42 66 92)(21 93 67 43)(22 44 68 94)(23 95 69 45)(24 46 70 96)
(1 23 7 17)(2 64 8 70)(3 13 9 19)(4 66 10 72)(5 15 11 21)(6 68 12 62)(14 26 20 32)(16 28 22 34)(18 30 24 36)(25 71 31 65)(27 61 33 67)(29 63 35 69)(37 74 43 80)(38 59 44 53)(39 76 45 82)(40 49 46 55)(41 78 47 84)(42 51 48 57)(50 91 56 85)(52 93 58 87)(54 95 60 89)(73 86 79 92)(75 88 81 94)(77 90 83 96)
(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 29 34)(2 33 30 5)(3 4 31 32)(7 12 35 28)(8 27 36 11)(9 10 25 26)(13 14 65 66)(15 24 67 64)(16 63 68 23)(17 22 69 62)(18 61 70 21)(19 20 71 72)(37 46 93 90)(38 89 94 45)(39 44 95 88)(40 87 96 43)(41 42 85 86)(47 48 91 92)(49 80 77 52)(50 51 78 79)(53 76 81 60)(54 59 82 75)(55 74 83 58)(56 57 84 73)
G:=sub<Sym(96)| (1,54,35,76)(2,77,36,55)(3,56,25,78)(4,79,26,57)(5,58,27,80)(6,81,28,59)(7,60,29,82)(8,83,30,49)(9,50,31,84)(10,73,32,51)(11,52,33,74)(12,75,34,53)(13,85,71,47)(14,48,72,86)(15,87,61,37)(16,38,62,88)(17,89,63,39)(18,40,64,90)(19,91,65,41)(20,42,66,92)(21,93,67,43)(22,44,68,94)(23,95,69,45)(24,46,70,96), (1,23,7,17)(2,64,8,70)(3,13,9,19)(4,66,10,72)(5,15,11,21)(6,68,12,62)(14,26,20,32)(16,28,22,34)(18,30,24,36)(25,71,31,65)(27,61,33,67)(29,63,35,69)(37,74,43,80)(38,59,44,53)(39,76,45,82)(40,49,46,55)(41,78,47,84)(42,51,48,57)(50,91,56,85)(52,93,58,87)(54,95,60,89)(73,86,79,92)(75,88,81,94)(77,90,83,96), (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), (1,6,29,34)(2,33,30,5)(3,4,31,32)(7,12,35,28)(8,27,36,11)(9,10,25,26)(13,14,65,66)(15,24,67,64)(16,63,68,23)(17,22,69,62)(18,61,70,21)(19,20,71,72)(37,46,93,90)(38,89,94,45)(39,44,95,88)(40,87,96,43)(41,42,85,86)(47,48,91,92)(49,80,77,52)(50,51,78,79)(53,76,81,60)(54,59,82,75)(55,74,83,58)(56,57,84,73)>;
G:=Group( (1,54,35,76)(2,77,36,55)(3,56,25,78)(4,79,26,57)(5,58,27,80)(6,81,28,59)(7,60,29,82)(8,83,30,49)(9,50,31,84)(10,73,32,51)(11,52,33,74)(12,75,34,53)(13,85,71,47)(14,48,72,86)(15,87,61,37)(16,38,62,88)(17,89,63,39)(18,40,64,90)(19,91,65,41)(20,42,66,92)(21,93,67,43)(22,44,68,94)(23,95,69,45)(24,46,70,96), (1,23,7,17)(2,64,8,70)(3,13,9,19)(4,66,10,72)(5,15,11,21)(6,68,12,62)(14,26,20,32)(16,28,22,34)(18,30,24,36)(25,71,31,65)(27,61,33,67)(29,63,35,69)(37,74,43,80)(38,59,44,53)(39,76,45,82)(40,49,46,55)(41,78,47,84)(42,51,48,57)(50,91,56,85)(52,93,58,87)(54,95,60,89)(73,86,79,92)(75,88,81,94)(77,90,83,96), (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), (1,6,29,34)(2,33,30,5)(3,4,31,32)(7,12,35,28)(8,27,36,11)(9,10,25,26)(13,14,65,66)(15,24,67,64)(16,63,68,23)(17,22,69,62)(18,61,70,21)(19,20,71,72)(37,46,93,90)(38,89,94,45)(39,44,95,88)(40,87,96,43)(41,42,85,86)(47,48,91,92)(49,80,77,52)(50,51,78,79)(53,76,81,60)(54,59,82,75)(55,74,83,58)(56,57,84,73) );
G=PermutationGroup([[(1,54,35,76),(2,77,36,55),(3,56,25,78),(4,79,26,57),(5,58,27,80),(6,81,28,59),(7,60,29,82),(8,83,30,49),(9,50,31,84),(10,73,32,51),(11,52,33,74),(12,75,34,53),(13,85,71,47),(14,48,72,86),(15,87,61,37),(16,38,62,88),(17,89,63,39),(18,40,64,90),(19,91,65,41),(20,42,66,92),(21,93,67,43),(22,44,68,94),(23,95,69,45),(24,46,70,96)], [(1,23,7,17),(2,64,8,70),(3,13,9,19),(4,66,10,72),(5,15,11,21),(6,68,12,62),(14,26,20,32),(16,28,22,34),(18,30,24,36),(25,71,31,65),(27,61,33,67),(29,63,35,69),(37,74,43,80),(38,59,44,53),(39,76,45,82),(40,49,46,55),(41,78,47,84),(42,51,48,57),(50,91,56,85),(52,93,58,87),(54,95,60,89),(73,86,79,92),(75,88,81,94),(77,90,83,96)], [(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)], [(1,6,29,34),(2,33,30,5),(3,4,31,32),(7,12,35,28),(8,27,36,11),(9,10,25,26),(13,14,65,66),(15,24,67,64),(16,63,68,23),(17,22,69,62),(18,61,70,21),(19,20,71,72),(37,46,93,90),(38,89,94,45),(39,44,95,88),(40,87,96,43),(41,42,85,86),(47,48,91,92),(49,80,77,52),(50,51,78,79),(53,76,81,60),(54,59,82,75),(55,74,83,58),(56,57,84,73)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 4U | 4V | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| 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 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | C4○D4 | S3×Q8 | S3×C4○D4 |
| kernel | C42.236D6 | C12.6Q8 | S3×C42 | Dic6⋊C4 | C12⋊Q8 | C4⋊C4⋊7S3 | D6⋊Q8 | C3×C42.C2 | C42.C2 | C4×S3 | C42 | C4⋊C4 | Dic3 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 1 | 1 | 4 | 1 | 6 | 8 | 2 | 4 |
Matrix representation of C42.236D6 ►in GL6(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 9 |
| 0 | 0 | 0 | 0 | 9 | 10 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,9,0,0,0,0,9,10],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
C42.236D6 in GAP, Magma, Sage, TeX
C_4^2._{236}D_6 % in TeX
G:=Group("C4^2.236D6"); // GroupNames label
G:=SmallGroup(192,1247);
// by ID
G=gap.SmallGroup(192,1247);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,1123,570,409,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=a^2*c^5>;
// generators/relations