direct product, metabelian, supersoluble, monomial, rational, 2-hyperelementary
Aliases: S3×C4⋊Q8, C42.239D6, C4⋊2(S3×Q8), (C4×S3)⋊4Q8, C12⋊2(C2×Q8), C12⋊Q8⋊43C2, C4.36(S3×D4), C4⋊C4.216D6, Dic3⋊2(C2×Q8), (C4×S3).25D4, D6.62(C2×D4), C12.68(C2×D4), D6.16(C2×Q8), C12⋊2Q8⋊35C2, (S3×C42).9C2, (C2×Q8).166D6, C6.97(C22×D4), C6.46(C22×Q8), (C2×C6).267C24, Dic3.10(C2×D4), Dic3⋊Q8⋊25C2, (C2×C12).100C23, (C4×C12).208C22, (C6×Q8).134C22, Dic3⋊C4.59C22, C4⋊Dic3.250C22, C22.288(S3×C23), (C22×S3).260C23, (C4×Dic3).257C22, (C2×Dic3).139C23, (C2×Dic6).187C22, C3⋊3(C2×C4⋊Q8), (C3×C4⋊Q8)⋊9C2, C2.70(C2×S3×D4), (C2×S3×Q8).7C2, C2.29(C2×S3×Q8), (S3×C4⋊C4).12C2, (S3×C2×C4).141C22, (C2×C4).92(C22×S3), (C3×C4⋊C4).210C22, SmallGroup(192,1282)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C4⋊Q8
G = < a,b,c,d,e | a3=b2=c4=d4=1, e2=d2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 656 in 290 conjugacy classes, 131 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, C2×C6, C42, C42, 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, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, S3×Q8, C6×Q8, C2×C4⋊Q8, C12⋊2Q8, S3×C42, C12⋊Q8, S3×C4⋊C4, Dic3⋊Q8, C3×C4⋊Q8, C2×S3×Q8, S3×C4⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C24, C22×S3, C4⋊Q8, C22×D4, C22×Q8, S3×D4, S3×Q8, S3×C23, C2×C4⋊Q8, C2×S3×D4, C2×S3×Q8, S3×C4⋊Q8
►On 96 points
Generators in S96
(1 7 53)(2 8 54)(3 5 55)(4 6 56)(9 30 80)(10 31 77)(11 32 78)(12 29 79)(13 33 58)(14 34 59)(15 35 60)(16 36 57)(17 67 87)(18 68 88)(19 65 85)(20 66 86)(21 42 92)(22 43 89)(23 44 90)(24 41 91)(25 45 70)(26 46 71)(27 47 72)(28 48 69)(37 64 82)(38 61 83)(39 62 84)(40 63 81)(49 76 94)(50 73 95)(51 74 96)(52 75 93)
(1 3)(2 4)(5 53)(6 54)(7 55)(8 56)(9 11)(10 12)(13 60)(14 57)(15 58)(16 59)(17 65)(18 66)(19 67)(20 68)(21 23)(22 24)(25 72)(26 69)(27 70)(28 71)(29 77)(30 78)(31 79)(32 80)(33 35)(34 36)(37 84)(38 81)(39 82)(40 83)(41 89)(42 90)(43 91)(44 92)(45 47)(46 48)(49 96)(50 93)(51 94)(52 95)(61 63)(62 64)(73 75)(74 76)(85 87)(86 88)
(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 22 10 85)(2 23 11 86)(3 24 12 87)(4 21 9 88)(5 41 29 17)(6 42 30 18)(7 43 31 19)(8 44 32 20)(13 49 37 25)(14 50 38 26)(15 51 39 27)(16 52 40 28)(33 76 64 45)(34 73 61 46)(35 74 62 47)(36 75 63 48)(53 89 77 65)(54 90 78 66)(55 91 79 67)(56 92 80 68)(57 93 81 69)(58 94 82 70)(59 95 83 71)(60 96 84 72)
(1 62 10 35)(2 61 11 34)(3 64 12 33)(4 63 9 36)(5 82 29 58)(6 81 30 57)(7 84 31 60)(8 83 32 59)(13 55 37 79)(14 54 38 78)(15 53 39 77)(16 56 40 80)(17 70 41 94)(18 69 42 93)(19 72 43 96)(20 71 44 95)(21 75 88 48)(22 74 85 47)(23 73 86 46)(24 76 87 45)(25 91 49 67)(26 90 50 66)(27 89 51 65)(28 92 52 68)
G:=sub<Sym(96)| (1,7,53)(2,8,54)(3,5,55)(4,6,56)(9,30,80)(10,31,77)(11,32,78)(12,29,79)(13,33,58)(14,34,59)(15,35,60)(16,36,57)(17,67,87)(18,68,88)(19,65,85)(20,66,86)(21,42,92)(22,43,89)(23,44,90)(24,41,91)(25,45,70)(26,46,71)(27,47,72)(28,48,69)(37,64,82)(38,61,83)(39,62,84)(40,63,81)(49,76,94)(50,73,95)(51,74,96)(52,75,93), (1,3)(2,4)(5,53)(6,54)(7,55)(8,56)(9,11)(10,12)(13,60)(14,57)(15,58)(16,59)(17,65)(18,66)(19,67)(20,68)(21,23)(22,24)(25,72)(26,69)(27,70)(28,71)(29,77)(30,78)(31,79)(32,80)(33,35)(34,36)(37,84)(38,81)(39,82)(40,83)(41,89)(42,90)(43,91)(44,92)(45,47)(46,48)(49,96)(50,93)(51,94)(52,95)(61,63)(62,64)(73,75)(74,76)(85,87)(86,88), (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), (1,22,10,85)(2,23,11,86)(3,24,12,87)(4,21,9,88)(5,41,29,17)(6,42,30,18)(7,43,31,19)(8,44,32,20)(13,49,37,25)(14,50,38,26)(15,51,39,27)(16,52,40,28)(33,76,64,45)(34,73,61,46)(35,74,62,47)(36,75,63,48)(53,89,77,65)(54,90,78,66)(55,91,79,67)(56,92,80,68)(57,93,81,69)(58,94,82,70)(59,95,83,71)(60,96,84,72), (1,62,10,35)(2,61,11,34)(3,64,12,33)(4,63,9,36)(5,82,29,58)(6,81,30,57)(7,84,31,60)(8,83,32,59)(13,55,37,79)(14,54,38,78)(15,53,39,77)(16,56,40,80)(17,70,41,94)(18,69,42,93)(19,72,43,96)(20,71,44,95)(21,75,88,48)(22,74,85,47)(23,73,86,46)(24,76,87,45)(25,91,49,67)(26,90,50,66)(27,89,51,65)(28,92,52,68)>;
G:=Group( (1,7,53)(2,8,54)(3,5,55)(4,6,56)(9,30,80)(10,31,77)(11,32,78)(12,29,79)(13,33,58)(14,34,59)(15,35,60)(16,36,57)(17,67,87)(18,68,88)(19,65,85)(20,66,86)(21,42,92)(22,43,89)(23,44,90)(24,41,91)(25,45,70)(26,46,71)(27,47,72)(28,48,69)(37,64,82)(38,61,83)(39,62,84)(40,63,81)(49,76,94)(50,73,95)(51,74,96)(52,75,93), (1,3)(2,4)(5,53)(6,54)(7,55)(8,56)(9,11)(10,12)(13,60)(14,57)(15,58)(16,59)(17,65)(18,66)(19,67)(20,68)(21,23)(22,24)(25,72)(26,69)(27,70)(28,71)(29,77)(30,78)(31,79)(32,80)(33,35)(34,36)(37,84)(38,81)(39,82)(40,83)(41,89)(42,90)(43,91)(44,92)(45,47)(46,48)(49,96)(50,93)(51,94)(52,95)(61,63)(62,64)(73,75)(74,76)(85,87)(86,88), (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), (1,22,10,85)(2,23,11,86)(3,24,12,87)(4,21,9,88)(5,41,29,17)(6,42,30,18)(7,43,31,19)(8,44,32,20)(13,49,37,25)(14,50,38,26)(15,51,39,27)(16,52,40,28)(33,76,64,45)(34,73,61,46)(35,74,62,47)(36,75,63,48)(53,89,77,65)(54,90,78,66)(55,91,79,67)(56,92,80,68)(57,93,81,69)(58,94,82,70)(59,95,83,71)(60,96,84,72), (1,62,10,35)(2,61,11,34)(3,64,12,33)(4,63,9,36)(5,82,29,58)(6,81,30,57)(7,84,31,60)(8,83,32,59)(13,55,37,79)(14,54,38,78)(15,53,39,77)(16,56,40,80)(17,70,41,94)(18,69,42,93)(19,72,43,96)(20,71,44,95)(21,75,88,48)(22,74,85,47)(23,73,86,46)(24,76,87,45)(25,91,49,67)(26,90,50,66)(27,89,51,65)(28,92,52,68) );
G=PermutationGroup([[(1,7,53),(2,8,54),(3,5,55),(4,6,56),(9,30,80),(10,31,77),(11,32,78),(12,29,79),(13,33,58),(14,34,59),(15,35,60),(16,36,57),(17,67,87),(18,68,88),(19,65,85),(20,66,86),(21,42,92),(22,43,89),(23,44,90),(24,41,91),(25,45,70),(26,46,71),(27,47,72),(28,48,69),(37,64,82),(38,61,83),(39,62,84),(40,63,81),(49,76,94),(50,73,95),(51,74,96),(52,75,93)], [(1,3),(2,4),(5,53),(6,54),(7,55),(8,56),(9,11),(10,12),(13,60),(14,57),(15,58),(16,59),(17,65),(18,66),(19,67),(20,68),(21,23),(22,24),(25,72),(26,69),(27,70),(28,71),(29,77),(30,78),(31,79),(32,80),(33,35),(34,36),(37,84),(38,81),(39,82),(40,83),(41,89),(42,90),(43,91),(44,92),(45,47),(46,48),(49,96),(50,93),(51,94),(52,95),(61,63),(62,64),(73,75),(74,76),(85,87),(86,88)], [(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)], [(1,22,10,85),(2,23,11,86),(3,24,12,87),(4,21,9,88),(5,41,29,17),(6,42,30,18),(7,43,31,19),(8,44,32,20),(13,49,37,25),(14,50,38,26),(15,51,39,27),(16,52,40,28),(33,76,64,45),(34,73,61,46),(35,74,62,47),(36,75,63,48),(53,89,77,65),(54,90,78,66),(55,91,79,67),(56,92,80,68),(57,93,81,69),(58,94,82,70),(59,95,83,71),(60,96,84,72)], [(1,62,10,35),(2,61,11,34),(3,64,12,33),(4,63,9,36),(5,82,29,58),(6,81,30,57),(7,84,31,60),(8,83,32,59),(13,55,37,79),(14,54,38,78),(15,53,39,77),(16,56,40,80),(17,70,41,94),(18,69,42,93),(19,72,43,96),(20,71,44,95),(21,75,88,48),(22,74,85,47),(23,73,86,46),(24,76,87,45),(25,91,49,67),(26,90,50,66),(27,89,51,65),(28,92,52,68)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 4Q | 4R | 4S | 4T | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 3 | 3 | 3 | 3 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | D6 | S3×D4 | S3×Q8 |
| kernel | S3×C4⋊Q8 | C12⋊2Q8 | S3×C42 | C12⋊Q8 | S3×C4⋊C4 | Dic3⋊Q8 | C3×C4⋊Q8 | C2×S3×Q8 | C4⋊Q8 | C4×S3 | C4×S3 | C42 | C4⋊C4 | C2×Q8 | C4 | C4 |
| # reps | 1 | 1 | 1 | 4 | 4 | 2 | 1 | 2 | 1 | 4 | 8 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of S3×C4⋊Q8 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 10 | 4 | 0 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 10 | 4 | 0 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 9 |
| 0 | 0 | 0 | 0 | 9 | 3 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 3 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 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))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,4,0,0,0,0,4,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,4,0,0,0,0,4,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,10,9,0,0,0,0,9,3],[4,3,0,0,0,0,3,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
S3×C4⋊Q8 in GAP, Magma, Sage, TeX
S_3\times C_4\rtimes Q_8
% in TeX
G:=Group("S3xC4:Q8"); // GroupNames label
G:=SmallGroup(192,1282);
// by ID
G=gap.SmallGroup(192,1282);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,570,185,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=d^4=1,e^2=d^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations