direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C4⋊SD16, C12⋊15SD16, C4⋊C8⋊8C6, Q8⋊2(C3×D4), (C4×Q8)⋊7C6, (C3×Q8)⋊12D4, C4⋊3(C3×SD16), C4.32(C6×D4), (Q8×C12)⋊23C2, C4⋊1D4.3C6, D4⋊C4⋊10C6, C2.7(C6×SD16), (C6×SD16)⋊28C2, (C2×SD16)⋊11C6, (C2×C12).322D4, C12.393(C2×D4), C42.15(C2×C6), C6.87(C2×SD16), C22.84(C6×D4), C12.342(C4○D4), C6.143(C4⋊D4), C6.135(C8⋊C22), (C4×C12).257C22, (C2×C24).299C22, (C2×C12).919C23, (C6×D4).186C22, (C6×Q8).262C22, (C3×C4⋊C8)⋊27C2, C4⋊C4.52(C2×C6), (C2×C8).36(C2×C6), C4.41(C3×C4○D4), (C2×D4).10(C2×C6), (C2×C6).640(C2×D4), (C2×C4).128(C3×D4), C2.12(C3×C4⋊D4), C2.10(C3×C8⋊C22), (C2×Q8).59(C2×C6), (C3×D4⋊C4)⋊34C2, (C3×C4⋊1D4).10C2, (C2×C4).94(C22×C6), (C3×C4⋊C4).373C22, SmallGroup(192,893)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4⋊SD16
G = < a,b,c,d | a3=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c3 >
Subgroups: 266 in 128 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C12, C12, C2×C6, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×C6, D4⋊C4, C4⋊C8, C4×Q8, C4⋊1D4, C2×SD16, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×SD16, C6×D4, C6×D4, C6×Q8, C4⋊SD16, C3×D4⋊C4, C3×C4⋊C8, Q8×C12, C3×C4⋊1D4, C6×SD16, C3×C4⋊SD16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, SD16, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C2×SD16, C8⋊C22, C3×SD16, C6×D4, C3×C4○D4, C4⋊SD16, C3×C4⋊D4, C6×SD16, C3×C8⋊C22, C3×C4⋊SD16
►On 96 points
(1 34 12)(2 35 13)(3 36 14)(4 37 15)(5 38 16)(6 39 9)(7 40 10)(8 33 11)(17 53 42)(18 54 43)(19 55 44)(20 56 45)(21 49 46)(22 50 47)(23 51 48)(24 52 41)(25 93 72)(26 94 65)(27 95 66)(28 96 67)(29 89 68)(30 90 69)(31 91 70)(32 92 71)(57 78 86)(58 79 87)(59 80 88)(60 73 81)(61 74 82)(62 75 83)(63 76 84)(64 77 85)
(1 74 22 66)(2 67 23 75)(3 76 24 68)(4 69 17 77)(5 78 18 70)(6 71 19 79)(7 80 20 72)(8 65 21 73)(9 92 44 58)(10 59 45 93)(11 94 46 60)(12 61 47 95)(13 96 48 62)(14 63 41 89)(15 90 42 64)(16 57 43 91)(25 40 88 56)(26 49 81 33)(27 34 82 50)(28 51 83 35)(29 36 84 52)(30 53 85 37)(31 38 86 54)(32 55 87 39)
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 23)(19 21)(20 24)(25 84)(26 87)(27 82)(28 85)(29 88)(30 83)(31 86)(32 81)(33 39)(35 37)(36 40)(41 45)(42 48)(44 46)(49 55)(51 53)(52 56)(57 91)(58 94)(59 89)(60 92)(61 95)(62 90)(63 93)(64 96)(65 79)(66 74)(67 77)(68 80)(69 75)(70 78)(71 73)(72 76)
G:=sub<Sym(96)| (1,34,12)(2,35,13)(3,36,14)(4,37,15)(5,38,16)(6,39,9)(7,40,10)(8,33,11)(17,53,42)(18,54,43)(19,55,44)(20,56,45)(21,49,46)(22,50,47)(23,51,48)(24,52,41)(25,93,72)(26,94,65)(27,95,66)(28,96,67)(29,89,68)(30,90,69)(31,91,70)(32,92,71)(57,78,86)(58,79,87)(59,80,88)(60,73,81)(61,74,82)(62,75,83)(63,76,84)(64,77,85), (1,74,22,66)(2,67,23,75)(3,76,24,68)(4,69,17,77)(5,78,18,70)(6,71,19,79)(7,80,20,72)(8,65,21,73)(9,92,44,58)(10,59,45,93)(11,94,46,60)(12,61,47,95)(13,96,48,62)(14,63,41,89)(15,90,42,64)(16,57,43,91)(25,40,88,56)(26,49,81,33)(27,34,82,50)(28,51,83,35)(29,36,84,52)(30,53,85,37)(31,38,86,54)(32,55,87,39), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)(25,84)(26,87)(27,82)(28,85)(29,88)(30,83)(31,86)(32,81)(33,39)(35,37)(36,40)(41,45)(42,48)(44,46)(49,55)(51,53)(52,56)(57,91)(58,94)(59,89)(60,92)(61,95)(62,90)(63,93)(64,96)(65,79)(66,74)(67,77)(68,80)(69,75)(70,78)(71,73)(72,76)>;
G:=Group( (1,34,12)(2,35,13)(3,36,14)(4,37,15)(5,38,16)(6,39,9)(7,40,10)(8,33,11)(17,53,42)(18,54,43)(19,55,44)(20,56,45)(21,49,46)(22,50,47)(23,51,48)(24,52,41)(25,93,72)(26,94,65)(27,95,66)(28,96,67)(29,89,68)(30,90,69)(31,91,70)(32,92,71)(57,78,86)(58,79,87)(59,80,88)(60,73,81)(61,74,82)(62,75,83)(63,76,84)(64,77,85), (1,74,22,66)(2,67,23,75)(3,76,24,68)(4,69,17,77)(5,78,18,70)(6,71,19,79)(7,80,20,72)(8,65,21,73)(9,92,44,58)(10,59,45,93)(11,94,46,60)(12,61,47,95)(13,96,48,62)(14,63,41,89)(15,90,42,64)(16,57,43,91)(25,40,88,56)(26,49,81,33)(27,34,82,50)(28,51,83,35)(29,36,84,52)(30,53,85,37)(31,38,86,54)(32,55,87,39), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)(25,84)(26,87)(27,82)(28,85)(29,88)(30,83)(31,86)(32,81)(33,39)(35,37)(36,40)(41,45)(42,48)(44,46)(49,55)(51,53)(52,56)(57,91)(58,94)(59,89)(60,92)(61,95)(62,90)(63,93)(64,96)(65,79)(66,74)(67,77)(68,80)(69,75)(70,78)(71,73)(72,76) );
G=PermutationGroup([[(1,34,12),(2,35,13),(3,36,14),(4,37,15),(5,38,16),(6,39,9),(7,40,10),(8,33,11),(17,53,42),(18,54,43),(19,55,44),(20,56,45),(21,49,46),(22,50,47),(23,51,48),(24,52,41),(25,93,72),(26,94,65),(27,95,66),(28,96,67),(29,89,68),(30,90,69),(31,91,70),(32,92,71),(57,78,86),(58,79,87),(59,80,88),(60,73,81),(61,74,82),(62,75,83),(63,76,84),(64,77,85)], [(1,74,22,66),(2,67,23,75),(3,76,24,68),(4,69,17,77),(5,78,18,70),(6,71,19,79),(7,80,20,72),(8,65,21,73),(9,92,44,58),(10,59,45,93),(11,94,46,60),(12,61,47,95),(13,96,48,62),(14,63,41,89),(15,90,42,64),(16,57,43,91),(25,40,88,56),(26,49,81,33),(27,34,82,50),(28,51,83,35),(29,36,84,52),(30,53,85,37),(31,38,86,54),(32,55,87,39)], [(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)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,23),(19,21),(20,24),(25,84),(26,87),(27,82),(28,85),(29,88),(30,83),(31,86),(32,81),(33,39),(35,37),(36,40),(41,45),(42,48),(44,46),(49,55),(51,53),(52,56),(57,91),(58,94),(59,89),(60,92),(61,95),(62,90),(63,93),(64,96),(65,79),(66,74),(67,77),(68,80),(69,75),(70,78),(71,73),(72,76)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12R | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
57 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 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | SD16 | C4○D4 | C3×D4 | C3×D4 | C3×SD16 | C3×C4○D4 | C8⋊C22 | C3×C8⋊C22 |
| kernel | C3×C4⋊SD16 | C3×D4⋊C4 | C3×C4⋊C8 | Q8×C12 | C3×C4⋊1D4 | C6×SD16 | C4⋊SD16 | D4⋊C4 | C4⋊C8 | C4×Q8 | C4⋊1D4 | C2×SD16 | C2×C12 | C3×Q8 | C12 | C12 | C2×C4 | Q8 | C4 | C4 | C6 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 1 | 2 |
Matrix representation of C3×C4⋊SD16 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 72 | 72 |
| 67 | 67 | 0 | 0 |
| 6 | 67 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,72,0,0,2,72],[67,6,0,0,67,67,0,0,0,0,1,72,0,0,0,72],[1,0,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;
C3×C4⋊SD16 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes {\rm SD}_{16} % in TeX
G:=Group("C3xC4:SD16"); // GroupNames label
G:=SmallGroup(192,893);
// by ID
G=gap.SmallGroup(192,893);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,176,1094,520,6053,1531,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^3>;
// generators/relations