direct product, metabelian, supersoluble, monomial
Aliases: C4×C32⋊2Q8, C12⋊4Dic6, C62.87C23, (C3×C12)⋊7Q8, C32⋊7(C4×Q8), C3⋊1(C4×Dic6), (C2×C12).309D6, Dic3.6(C4×S3), C6.22(C2×Dic6), C6.36(C4○D12), (Dic3×C12).4C2, (C4×Dic3).10S3, (C6×C12).240C22, (C2×Dic3).102D6, C2.5(D6.D6), Dic3⋊Dic3.15C2, C62.C22.14C2, (C6×Dic3).116C22, C2.23(C4×S32), C6.22(S3×C2×C4), (C2×C4).142S32, C22.43(C2×S32), (C3×C6).38(C2×Q8), C2.1(C2×C32⋊2Q8), (C3×C6).54(C4○D4), C3⋊Dic3.35(C2×C4), (C4×C3⋊Dic3).20C2, (C3×C6).21(C22×C4), (C2×C32⋊2Q8).9C2, (C2×C6).106(C22×S3), (C3×Dic3).14(C2×C4), (C2×C3⋊Dic3).140C22, SmallGroup(288,565)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C32⋊2Q8
G = < a,b,c,d,e | a4=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 458 in 155 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C32, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C4×Q8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, C32⋊2Q8, C6×Dic3, C2×C3⋊Dic3, C6×C12, C4×Dic6, Dic3⋊Dic3, C62.C22, Dic3×C12, C4×C3⋊Dic3, C2×C32⋊2Q8, C4×C32⋊2Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, Dic6, C4×S3, C22×S3, C4×Q8, S32, C2×Dic6, S3×C2×C4, C4○D12, C32⋊2Q8, C2×S32, C4×Dic6, D6.D6, C4×S32, C2×C32⋊2Q8, C4×C32⋊2Q8
►On 96 points
Generators in S96
(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 25 9)(2 26 10)(3 27 11)(4 28 12)(5 19 77)(6 20 78)(7 17 79)(8 18 80)(13 96 82)(14 93 83)(15 94 84)(16 95 81)(21 43 37)(22 44 38)(23 41 39)(24 42 40)(29 64 66)(30 61 67)(31 62 68)(32 63 65)(33 49 47)(34 50 48)(35 51 45)(36 52 46)(53 90 88)(54 91 85)(55 92 86)(56 89 87)(57 71 73)(58 72 74)(59 69 75)(60 70 76)
(1 25 9)(2 26 10)(3 27 11)(4 28 12)(5 19 77)(6 20 78)(7 17 79)(8 18 80)(13 96 82)(14 93 83)(15 94 84)(16 95 81)(21 43 37)(22 44 38)(23 41 39)(24 42 40)(29 66 64)(30 67 61)(31 68 62)(32 65 63)(33 47 49)(34 48 50)(35 45 51)(36 46 52)(53 88 90)(54 85 91)(55 86 92)(56 87 89)(57 73 71)(58 74 72)(59 75 69)(60 76 70)
(1 63 39 47)(2 64 40 48)(3 61 37 45)(4 62 38 46)(5 72 96 88)(6 69 93 85)(7 70 94 86)(8 71 95 87)(9 65 41 33)(10 66 42 34)(11 67 43 35)(12 68 44 36)(13 53 77 74)(14 54 78 75)(15 55 79 76)(16 56 80 73)(17 60 84 92)(18 57 81 89)(19 58 82 90)(20 59 83 91)(21 51 27 30)(22 52 28 31)(23 49 25 32)(24 50 26 29)
(1 85 39 69)(2 86 40 70)(3 87 37 71)(4 88 38 72)(5 62 96 46)(6 63 93 47)(7 64 94 48)(8 61 95 45)(9 91 41 59)(10 92 42 60)(11 89 43 57)(12 90 44 58)(13 52 77 31)(14 49 78 32)(15 50 79 29)(16 51 80 30)(17 66 84 34)(18 67 81 35)(19 68 82 36)(20 65 83 33)(21 73 27 56)(22 74 28 53)(23 75 25 54)(24 76 26 55)
G:=sub<Sym(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), (1,25,9)(2,26,10)(3,27,11)(4,28,12)(5,19,77)(6,20,78)(7,17,79)(8,18,80)(13,96,82)(14,93,83)(15,94,84)(16,95,81)(21,43,37)(22,44,38)(23,41,39)(24,42,40)(29,64,66)(30,61,67)(31,62,68)(32,63,65)(33,49,47)(34,50,48)(35,51,45)(36,52,46)(53,90,88)(54,91,85)(55,92,86)(56,89,87)(57,71,73)(58,72,74)(59,69,75)(60,70,76), (1,25,9)(2,26,10)(3,27,11)(4,28,12)(5,19,77)(6,20,78)(7,17,79)(8,18,80)(13,96,82)(14,93,83)(15,94,84)(16,95,81)(21,43,37)(22,44,38)(23,41,39)(24,42,40)(29,66,64)(30,67,61)(31,68,62)(32,65,63)(33,47,49)(34,48,50)(35,45,51)(36,46,52)(53,88,90)(54,85,91)(55,86,92)(56,87,89)(57,73,71)(58,74,72)(59,75,69)(60,76,70), (1,63,39,47)(2,64,40,48)(3,61,37,45)(4,62,38,46)(5,72,96,88)(6,69,93,85)(7,70,94,86)(8,71,95,87)(9,65,41,33)(10,66,42,34)(11,67,43,35)(12,68,44,36)(13,53,77,74)(14,54,78,75)(15,55,79,76)(16,56,80,73)(17,60,84,92)(18,57,81,89)(19,58,82,90)(20,59,83,91)(21,51,27,30)(22,52,28,31)(23,49,25,32)(24,50,26,29), (1,85,39,69)(2,86,40,70)(3,87,37,71)(4,88,38,72)(5,62,96,46)(6,63,93,47)(7,64,94,48)(8,61,95,45)(9,91,41,59)(10,92,42,60)(11,89,43,57)(12,90,44,58)(13,52,77,31)(14,49,78,32)(15,50,79,29)(16,51,80,30)(17,66,84,34)(18,67,81,35)(19,68,82,36)(20,65,83,33)(21,73,27,56)(22,74,28,53)(23,75,25,54)(24,76,26,55)>;
G:=Group( (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), (1,25,9)(2,26,10)(3,27,11)(4,28,12)(5,19,77)(6,20,78)(7,17,79)(8,18,80)(13,96,82)(14,93,83)(15,94,84)(16,95,81)(21,43,37)(22,44,38)(23,41,39)(24,42,40)(29,64,66)(30,61,67)(31,62,68)(32,63,65)(33,49,47)(34,50,48)(35,51,45)(36,52,46)(53,90,88)(54,91,85)(55,92,86)(56,89,87)(57,71,73)(58,72,74)(59,69,75)(60,70,76), (1,25,9)(2,26,10)(3,27,11)(4,28,12)(5,19,77)(6,20,78)(7,17,79)(8,18,80)(13,96,82)(14,93,83)(15,94,84)(16,95,81)(21,43,37)(22,44,38)(23,41,39)(24,42,40)(29,66,64)(30,67,61)(31,68,62)(32,65,63)(33,47,49)(34,48,50)(35,45,51)(36,46,52)(53,88,90)(54,85,91)(55,86,92)(56,87,89)(57,73,71)(58,74,72)(59,75,69)(60,76,70), (1,63,39,47)(2,64,40,48)(3,61,37,45)(4,62,38,46)(5,72,96,88)(6,69,93,85)(7,70,94,86)(8,71,95,87)(9,65,41,33)(10,66,42,34)(11,67,43,35)(12,68,44,36)(13,53,77,74)(14,54,78,75)(15,55,79,76)(16,56,80,73)(17,60,84,92)(18,57,81,89)(19,58,82,90)(20,59,83,91)(21,51,27,30)(22,52,28,31)(23,49,25,32)(24,50,26,29), (1,85,39,69)(2,86,40,70)(3,87,37,71)(4,88,38,72)(5,62,96,46)(6,63,93,47)(7,64,94,48)(8,61,95,45)(9,91,41,59)(10,92,42,60)(11,89,43,57)(12,90,44,58)(13,52,77,31)(14,49,78,32)(15,50,79,29)(16,51,80,30)(17,66,84,34)(18,67,81,35)(19,68,82,36)(20,65,83,33)(21,73,27,56)(22,74,28,53)(23,75,25,54)(24,76,26,55) );
G=PermutationGroup([[(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)], [(1,25,9),(2,26,10),(3,27,11),(4,28,12),(5,19,77),(6,20,78),(7,17,79),(8,18,80),(13,96,82),(14,93,83),(15,94,84),(16,95,81),(21,43,37),(22,44,38),(23,41,39),(24,42,40),(29,64,66),(30,61,67),(31,62,68),(32,63,65),(33,49,47),(34,50,48),(35,51,45),(36,52,46),(53,90,88),(54,91,85),(55,92,86),(56,89,87),(57,71,73),(58,72,74),(59,69,75),(60,70,76)], [(1,25,9),(2,26,10),(3,27,11),(4,28,12),(5,19,77),(6,20,78),(7,17,79),(8,18,80),(13,96,82),(14,93,83),(15,94,84),(16,95,81),(21,43,37),(22,44,38),(23,41,39),(24,42,40),(29,66,64),(30,67,61),(31,68,62),(32,65,63),(33,47,49),(34,48,50),(35,45,51),(36,46,52),(53,88,90),(54,85,91),(55,86,92),(56,87,89),(57,73,71),(58,74,72),(59,75,69),(60,76,70)], [(1,63,39,47),(2,64,40,48),(3,61,37,45),(4,62,38,46),(5,72,96,88),(6,69,93,85),(7,70,94,86),(8,71,95,87),(9,65,41,33),(10,66,42,34),(11,67,43,35),(12,68,44,36),(13,53,77,74),(14,54,78,75),(15,55,79,76),(16,56,80,73),(17,60,84,92),(18,57,81,89),(19,58,82,90),(20,59,83,91),(21,51,27,30),(22,52,28,31),(23,49,25,32),(24,50,26,29)], [(1,85,39,69),(2,86,40,70),(3,87,37,71),(4,88,38,72),(5,62,96,46),(6,63,93,47),(7,64,94,48),(8,61,95,45),(9,91,41,59),(10,92,42,60),(11,89,43,57),(12,90,44,58),(13,52,77,31),(14,49,78,32),(15,50,79,29),(16,51,80,30),(17,66,84,34),(18,67,81,35),(19,68,82,36),(20,65,83,33),(21,73,27,56),(22,74,28,53),(23,75,25,54),(24,76,26,55)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 6A | ··· | 6F | 6G | 6H | 6I | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 12M | ··· | 12AB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 6 | ··· | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | Q8 | D6 | D6 | C4○D4 | C4×S3 | Dic6 | C4○D12 | S32 | C32⋊2Q8 | C2×S32 | D6.D6 | C4×S32 |
| kernel | C4×C32⋊2Q8 | Dic3⋊Dic3 | C62.C22 | Dic3×C12 | C4×C3⋊Dic3 | C2×C32⋊2Q8 | C32⋊2Q8 | C4×Dic3 | C3×C12 | C2×Dic3 | C2×C12 | C3×C6 | Dic3 | C12 | C6 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 2 | 2 | 4 | 2 | 2 | 8 | 8 | 8 | 1 | 2 | 1 | 2 | 2 |
Matrix representation of C4×C32⋊2Q8 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 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 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 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 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 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 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 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 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[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],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C4×C32⋊2Q8 in GAP, Magma, Sage, TeX
C_4\times C_3^2\rtimes_2Q_8
% in TeX
G:=Group("C4xC3^2:2Q8"); // GroupNames label
G:=SmallGroup(288,565);
// by ID
G=gap.SmallGroup(288,565);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,120,58,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations