metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3⋊C8.29D4, (C2×C6)⋊3Q16, C4⋊C4.70D6, C4.176(S3×D4), (C2×Q8).56D6, C6.39(C2×Q16), C22⋊Q8.6S3, C12.157(C2×D4), C6.Q16⋊41C2, (C2×C12).268D4, C3⋊3(C8.18D4), (C22×C6).97D4, C6.102(C4○D8), Q8⋊2Dic3⋊17C2, C6.SD16⋊39C2, C6.99(C4⋊D4), C22⋊2(C3⋊Q16), (C22×C4).359D6, (C6×Q8).50C22, C12.192(C4○D4), C4.65(D4⋊2S3), (C2×C12).370C23, C23.49(C3⋊D4), C12.48D4.12C2, C2.21(Q8.13D6), C4⋊Dic3.148C22, C2.20(C23.14D6), (C22×C12).174C22, (C2×Dic6).105C22, (C22×C3⋊C8).9C2, (C2×C3⋊Q16)⋊10C2, (C2×C6).501(C2×D4), C2.10(C2×C3⋊Q16), (C3×C22⋊Q8).5C2, (C2×C3⋊C8).251C22, (C2×C4).108(C3⋊D4), (C3×C4⋊C4).117C22, (C2×C4).470(C22×S3), C22.176(C2×C3⋊D4), SmallGroup(192,610)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊C8.29D4
G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b-1, bd=db, dcd=b4c-1 >
Subgroups: 272 in 114 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C2×Q8, C2×Q8, C3⋊C8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×C6, Q8⋊C4, C2.D8, C22⋊Q8, C22⋊Q8, C22×C8, C2×Q16, C2×C3⋊C8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C3⋊Q16, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×Q8, C8.18D4, C6.Q16, C6.SD16, Q8⋊2Dic3, C22×C3⋊C8, C12.48D4, C2×C3⋊Q16, C3×C22⋊Q8, C3⋊C8.29D4
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C2×Q16, C4○D8, C3⋊Q16, S3×D4, D4⋊2S3, C2×C3⋊D4, C8.18D4, C23.14D6, C2×C3⋊Q16, Q8.13D6, C3⋊C8.29D4
►On 96 points
(1 14 94)(2 95 15)(3 16 96)(4 89 9)(5 10 90)(6 91 11)(7 12 92)(8 93 13)(17 64 33)(18 34 57)(19 58 35)(20 36 59)(21 60 37)(22 38 61)(23 62 39)(24 40 63)(25 82 66)(26 67 83)(27 84 68)(28 69 85)(29 86 70)(30 71 87)(31 88 72)(32 65 81)(41 51 79)(42 80 52)(43 53 73)(44 74 54)(45 55 75)(46 76 56)(47 49 77)(48 78 50)
(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 27 78 24)(2 26 79 23)(3 25 80 22)(4 32 73 21)(5 31 74 20)(6 30 75 19)(7 29 76 18)(8 28 77 17)(9 65 53 60)(10 72 54 59)(11 71 55 58)(12 70 56 57)(13 69 49 64)(14 68 50 63)(15 67 51 62)(16 66 52 61)(33 93 85 47)(34 92 86 46)(35 91 87 45)(36 90 88 44)(37 89 81 43)(38 96 82 42)(39 95 83 41)(40 94 84 48)
(17 32)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(64 65)
G:=sub<Sym(96)| (1,14,94)(2,95,15)(3,16,96)(4,89,9)(5,10,90)(6,91,11)(7,12,92)(8,93,13)(17,64,33)(18,34,57)(19,58,35)(20,36,59)(21,60,37)(22,38,61)(23,62,39)(24,40,63)(25,82,66)(26,67,83)(27,84,68)(28,69,85)(29,86,70)(30,71,87)(31,88,72)(32,65,81)(41,51,79)(42,80,52)(43,53,73)(44,74,54)(45,55,75)(46,76,56)(47,49,77)(48,78,50), (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), (1,27,78,24)(2,26,79,23)(3,25,80,22)(4,32,73,21)(5,31,74,20)(6,30,75,19)(7,29,76,18)(8,28,77,17)(9,65,53,60)(10,72,54,59)(11,71,55,58)(12,70,56,57)(13,69,49,64)(14,68,50,63)(15,67,51,62)(16,66,52,61)(33,93,85,47)(34,92,86,46)(35,91,87,45)(36,90,88,44)(37,89,81,43)(38,96,82,42)(39,95,83,41)(40,94,84,48), (17,32)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,65)>;
G:=Group( (1,14,94)(2,95,15)(3,16,96)(4,89,9)(5,10,90)(6,91,11)(7,12,92)(8,93,13)(17,64,33)(18,34,57)(19,58,35)(20,36,59)(21,60,37)(22,38,61)(23,62,39)(24,40,63)(25,82,66)(26,67,83)(27,84,68)(28,69,85)(29,86,70)(30,71,87)(31,88,72)(32,65,81)(41,51,79)(42,80,52)(43,53,73)(44,74,54)(45,55,75)(46,76,56)(47,49,77)(48,78,50), (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), (1,27,78,24)(2,26,79,23)(3,25,80,22)(4,32,73,21)(5,31,74,20)(6,30,75,19)(7,29,76,18)(8,28,77,17)(9,65,53,60)(10,72,54,59)(11,71,55,58)(12,70,56,57)(13,69,49,64)(14,68,50,63)(15,67,51,62)(16,66,52,61)(33,93,85,47)(34,92,86,46)(35,91,87,45)(36,90,88,44)(37,89,81,43)(38,96,82,42)(39,95,83,41)(40,94,84,48), (17,32)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,65) );
G=PermutationGroup([[(1,14,94),(2,95,15),(3,16,96),(4,89,9),(5,10,90),(6,91,11),(7,12,92),(8,93,13),(17,64,33),(18,34,57),(19,58,35),(20,36,59),(21,60,37),(22,38,61),(23,62,39),(24,40,63),(25,82,66),(26,67,83),(27,84,68),(28,69,85),(29,86,70),(30,71,87),(31,88,72),(32,65,81),(41,51,79),(42,80,52),(43,53,73),(44,74,54),(45,55,75),(46,76,56),(47,49,77),(48,78,50)], [(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)], [(1,27,78,24),(2,26,79,23),(3,25,80,22),(4,32,73,21),(5,31,74,20),(6,30,75,19),(7,29,76,18),(8,28,77,17),(9,65,53,60),(10,72,54,59),(11,71,55,58),(12,70,56,57),(13,69,49,64),(14,68,50,63),(15,67,51,62),(16,66,52,61),(33,93,85,47),(34,92,86,46),(35,91,87,45),(36,90,88,44),(37,89,81,43),(38,96,82,42),(39,95,83,41),(40,94,84,48)], [(17,32),(18,25),(19,26),(20,27),(21,28),(22,29),(23,30),(24,31),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72),(64,65)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4○D4 | Q16 | C3⋊D4 | C3⋊D4 | C4○D8 | S3×D4 | D4⋊2S3 | C3⋊Q16 | Q8.13D6 |
| kernel | C3⋊C8.29D4 | C6.Q16 | C6.SD16 | Q8⋊2Dic3 | C22×C3⋊C8 | C12.48D4 | C2×C3⋊Q16 | C3×C22⋊Q8 | C22⋊Q8 | C3⋊C8 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×Q8 | C12 | C2×C6 | C2×C4 | C23 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C3⋊C8.29D4 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 63 | 0 | 0 | 0 | 0 | 0 |
| 5 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 7 | 0 | 0 |
| 0 | 0 | 14 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 |
| 64 | 7 | 0 | 0 | 0 | 0 |
| 72 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 66 | 0 | 0 |
| 0 | 0 | 59 | 66 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 13 | 72 | 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 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[63,5,0,0,0,0,0,51,0,0,0,0,0,0,66,14,0,0,0,0,7,7,0,0,0,0,0,0,27,0,0,0,0,0,0,46],[64,72,0,0,0,0,7,9,0,0,0,0,0,0,7,59,0,0,0,0,66,66,0,0,0,0,0,0,0,27,0,0,0,0,27,0],[1,13,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;
C3⋊C8.29D4 in GAP, Magma, Sage, TeX
C_3\rtimes C_8._{29}D_4 % in TeX
G:=Group("C3:C8.29D4"); // GroupNames label
G:=SmallGroup(192,610);
// by ID
G=gap.SmallGroup(192,610);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,219,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d=b^4*c^-1>;
// generators/relations