direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C8⋊8D4, C24⋊34D4, C8⋊8(C3×D4), C4.Q8⋊6C6, (C2×C6)⋊7SD16, C4.57(C6×D4), C22⋊Q8⋊3C6, D4⋊C4⋊1C6, (C22×C8)⋊13C6, Q8⋊C4⋊1C6, C4⋊D4.3C6, C2.9(C6×SD16), (C22×C24)⋊23C2, (C6×SD16)⋊30C2, (C2×SD16)⋊13C6, (C2×C12).363D4, C12.464(C2×D4), C6.89(C2×SD16), C22⋊2(C3×SD16), C22.89(C6×D4), C23.30(C3×D4), C6.123(C4○D8), (C22×C6).129D4, C12.262(C4○D4), C6.148(C4⋊D4), (C2×C12).924C23, (C2×C24).363C22, (C6×D4).189C22, (C6×Q8).163C22, (C22×C12).591C22, C4⋊C4.5(C2×C6), C4.7(C3×C4○D4), (C2×C8).93(C2×C6), (C3×C4.Q8)⋊21C2, C2.10(C3×C4○D8), (C2×C4).53(C3×D4), (C2×Q8).8(C2×C6), (C3×D4⋊C4)⋊1C2, (C3×Q8⋊C4)⋊1C2, (C2×D4).12(C2×C6), (C2×C6).645(C2×D4), C2.17(C3×C4⋊D4), (C3×C22⋊Q8)⋊30C2, (C3×C4⋊D4).13C2, (C2×C4).99(C22×C6), (C3×C4⋊C4).227C22, (C22×C4).127(C2×C6), SmallGroup(192,898)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊8D4
G = < a,b,c,d | a3=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b3, dcd=c-1 >
Subgroups: 234 in 124 conjugacy classes, 58 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C3×SD16, C22×C12, C6×D4, C6×D4, C6×Q8, C8⋊8D4, C3×D4⋊C4, C3×Q8⋊C4, C3×C4.Q8, C3×C4⋊D4, C3×C22⋊Q8, C22×C24, C6×SD16, C3×C8⋊8D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, SD16, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C2×SD16, C4○D8, C3×SD16, C6×D4, C3×C4○D4, C8⋊8D4, C3×C4⋊D4, C6×SD16, C3×C4○D8, C3×C8⋊8D4
►On 96 points
(1 31 71)(2 32 72)(3 25 65)(4 26 66)(5 27 67)(6 28 68)(7 29 69)(8 30 70)(9 21 60)(10 22 61)(11 23 62)(12 24 63)(13 17 64)(14 18 57)(15 19 58)(16 20 59)(33 73 81)(34 74 82)(35 75 83)(36 76 84)(37 77 85)(38 78 86)(39 79 87)(40 80 88)(41 55 89)(42 56 90)(43 49 91)(44 50 92)(45 51 93)(46 52 94)(47 53 95)(48 54 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 23 79 47)(2 18 80 42)(3 21 73 45)(4 24 74 48)(5 19 75 43)(6 22 76 46)(7 17 77 41)(8 20 78 44)(9 33 93 65)(10 36 94 68)(11 39 95 71)(12 34 96 66)(13 37 89 69)(14 40 90 72)(15 35 91 67)(16 38 92 70)(25 60 81 51)(26 63 82 54)(27 58 83 49)(28 61 84 52)(29 64 85 55)(30 59 86 50)(31 62 87 53)(32 57 88 56)
(2 4)(3 7)(6 8)(9 89)(10 92)(11 95)(12 90)(13 93)(14 96)(15 91)(16 94)(17 45)(18 48)(19 43)(20 46)(21 41)(22 44)(23 47)(24 42)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(49 58)(50 61)(51 64)(52 59)(53 62)(54 57)(55 60)(56 63)(65 69)(66 72)(68 70)(73 77)(74 80)(76 78)(81 85)(82 88)(84 86)
G:=sub<Sym(96)| (1,31,71)(2,32,72)(3,25,65)(4,26,66)(5,27,67)(6,28,68)(7,29,69)(8,30,70)(9,21,60)(10,22,61)(11,23,62)(12,24,63)(13,17,64)(14,18,57)(15,19,58)(16,20,59)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,55,89)(42,56,90)(43,49,91)(44,50,92)(45,51,93)(46,52,94)(47,53,95)(48,54,96), (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), (1,23,79,47)(2,18,80,42)(3,21,73,45)(4,24,74,48)(5,19,75,43)(6,22,76,46)(7,17,77,41)(8,20,78,44)(9,33,93,65)(10,36,94,68)(11,39,95,71)(12,34,96,66)(13,37,89,69)(14,40,90,72)(15,35,91,67)(16,38,92,70)(25,60,81,51)(26,63,82,54)(27,58,83,49)(28,61,84,52)(29,64,85,55)(30,59,86,50)(31,62,87,53)(32,57,88,56), (2,4)(3,7)(6,8)(9,89)(10,92)(11,95)(12,90)(13,93)(14,96)(15,91)(16,94)(17,45)(18,48)(19,43)(20,46)(21,41)(22,44)(23,47)(24,42)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(49,58)(50,61)(51,64)(52,59)(53,62)(54,57)(55,60)(56,63)(65,69)(66,72)(68,70)(73,77)(74,80)(76,78)(81,85)(82,88)(84,86)>;
G:=Group( (1,31,71)(2,32,72)(3,25,65)(4,26,66)(5,27,67)(6,28,68)(7,29,69)(8,30,70)(9,21,60)(10,22,61)(11,23,62)(12,24,63)(13,17,64)(14,18,57)(15,19,58)(16,20,59)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,55,89)(42,56,90)(43,49,91)(44,50,92)(45,51,93)(46,52,94)(47,53,95)(48,54,96), (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), (1,23,79,47)(2,18,80,42)(3,21,73,45)(4,24,74,48)(5,19,75,43)(6,22,76,46)(7,17,77,41)(8,20,78,44)(9,33,93,65)(10,36,94,68)(11,39,95,71)(12,34,96,66)(13,37,89,69)(14,40,90,72)(15,35,91,67)(16,38,92,70)(25,60,81,51)(26,63,82,54)(27,58,83,49)(28,61,84,52)(29,64,85,55)(30,59,86,50)(31,62,87,53)(32,57,88,56), (2,4)(3,7)(6,8)(9,89)(10,92)(11,95)(12,90)(13,93)(14,96)(15,91)(16,94)(17,45)(18,48)(19,43)(20,46)(21,41)(22,44)(23,47)(24,42)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(49,58)(50,61)(51,64)(52,59)(53,62)(54,57)(55,60)(56,63)(65,69)(66,72)(68,70)(73,77)(74,80)(76,78)(81,85)(82,88)(84,86) );
G=PermutationGroup([[(1,31,71),(2,32,72),(3,25,65),(4,26,66),(5,27,67),(6,28,68),(7,29,69),(8,30,70),(9,21,60),(10,22,61),(11,23,62),(12,24,63),(13,17,64),(14,18,57),(15,19,58),(16,20,59),(33,73,81),(34,74,82),(35,75,83),(36,76,84),(37,77,85),(38,78,86),(39,79,87),(40,80,88),(41,55,89),(42,56,90),(43,49,91),(44,50,92),(45,51,93),(46,52,94),(47,53,95),(48,54,96)], [(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)], [(1,23,79,47),(2,18,80,42),(3,21,73,45),(4,24,74,48),(5,19,75,43),(6,22,76,46),(7,17,77,41),(8,20,78,44),(9,33,93,65),(10,36,94,68),(11,39,95,71),(12,34,96,66),(13,37,89,69),(14,40,90,72),(15,35,91,67),(16,38,92,70),(25,60,81,51),(26,63,82,54),(27,58,83,49),(28,61,84,52),(29,64,85,55),(30,59,86,50),(31,62,87,53),(32,57,88,56)], [(2,4),(3,7),(6,8),(9,89),(10,92),(11,95),(12,90),(13,93),(14,96),(15,91),(16,94),(17,45),(18,48),(19,43),(20,46),(21,41),(22,44),(23,47),(24,42),(25,29),(26,32),(28,30),(33,37),(34,40),(36,38),(49,58),(50,61),(51,64),(52,59),(53,62),(54,57),(55,60),(56,63),(65,69),(66,72),(68,70),(73,77),(74,80),(76,78),(81,85),(82,88),(84,86)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 8A | ··· | 8H | 12A | ··· | 12H | 12I | ··· | 12N | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D4 | C4○D4 | SD16 | C3×D4 | C3×D4 | C3×D4 | C4○D8 | C3×C4○D4 | C3×SD16 | C3×C4○D8 |
| kernel | C3×C8⋊8D4 | C3×D4⋊C4 | C3×Q8⋊C4 | C3×C4.Q8 | C3×C4⋊D4 | C3×C22⋊Q8 | C22×C24 | C6×SD16 | C8⋊8D4 | D4⋊C4 | Q8⋊C4 | C4.Q8 | C4⋊D4 | C22⋊Q8 | C22×C8 | C2×SD16 | C24 | C2×C12 | C22×C6 | C12 | C2×C6 | C8 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C3×C8⋊8D4 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 6 | 67 | 0 | 0 |
| 6 | 6 | 0 | 0 |
| 0 | 0 | 6 | 67 |
| 0 | 0 | 6 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 46 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[6,6,0,0,67,6,0,0,0,0,6,6,0,0,67,6],[1,0,0,0,0,72,0,0,0,0,0,46,0,0,46,0],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72] >;
C3×C8⋊8D4 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes_8D_4
% in TeX
G:=Group("C3xC8:8D4"); // GroupNames label
G:=SmallGroup(192,898);
// by ID
G=gap.SmallGroup(192,898);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,176,1094,4204,172]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^3,d*c*d=c^-1>;
// generators/relations