metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.56D6, (C2×D4).36D6, C4⋊D4.4S3, C6.95(C4○D8), C12.Q8⋊35C2, (C2×C12).261D4, (C22×C6).81D4, D4⋊Dic3⋊14C2, C12.55D4⋊9C2, C6.89(C8⋊C22), C4.Dic6⋊34C2, (C6×D4).52C22, (C22×C4).134D6, C12.182(C4○D4), C4.92(D4⋊2S3), (C2×C12).354C23, C23.29(C3⋊D4), C3⋊7(C23.19D4), C2.14(Q8.13D6), C2.10(D12⋊6C22), C23.26D6⋊15C2, C4⋊Dic3.336C22, (C22×C12).158C22, C6.79(C22.D4), C2.13(C23.23D6), (C2×C6).485(C2×D4), (C3×C4⋊D4).3C2, (C2×C3⋊C8).107C22, (C2×C4).170(C3⋊D4), (C3×C4⋊C4).103C22, (C2×C4).454(C22×S3), C22.160(C2×C3⋊D4), SmallGroup(192,594)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C22 — C22×C4 — C4⋊D4 |
Generators and relations for C12.Q8⋊C2
G = < a,b,c,d | a12=b4=d2=1, c2=a9b2, bab-1=dad=a7, cac-1=a5, cbc-1=a9b-1, dbd=a6b-1, dcd=a9c >
Subgroups: 272 in 106 conjugacy classes, 39 normal (all characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×4], C23, C23, Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×6], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], C22×C4, C2×D4, C2×D4, C3⋊C8 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C22×C6, C22×C6, C22⋊C8, D4⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C2×C3⋊C8 [×2], C4×Dic3, C4⋊Dic3 [×2], C6.D4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×D4, C23.19D4, C12.Q8, C4.Dic6, C12.55D4, D4⋊Dic3 [×2], C23.26D6, C3×C4⋊D4, C12.Q8⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C22.D4, C4○D8, C8⋊C22, D4⋊2S3 [×2], C2×C3⋊D4, C23.19D4, D12⋊6C22, C23.23D6, Q8.13D6, C12.Q8⋊C2
(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 33 42)(2 18 34 37)(3 13 35 44)(4 20 36 39)(5 15 25 46)(6 22 26 41)(7 17 27 48)(8 24 28 43)(9 19 29 38)(10 14 30 45)(11 21 31 40)(12 16 32 47)(49 72 86 84)(50 67 87 79)(51 62 88 74)(52 69 89 81)(53 64 90 76)(54 71 91 83)(55 66 92 78)(56 61 93 73)(57 68 94 80)(58 63 95 75)(59 70 96 82)(60 65 85 77)
(1 75 30 72 7 81 36 66)(2 80 31 65 8 74 25 71)(3 73 32 70 9 79 26 64)(4 78 33 63 10 84 27 69)(5 83 34 68 11 77 28 62)(6 76 35 61 12 82 29 67)(13 90 41 50 19 96 47 56)(14 95 42 55 20 89 48 49)(15 88 43 60 21 94 37 54)(16 93 44 53 22 87 38 59)(17 86 45 58 23 92 39 52)(18 91 46 51 24 85 40 57)
(2 8)(4 10)(6 12)(13 38)(14 45)(15 40)(16 47)(17 42)(18 37)(19 44)(20 39)(21 46)(22 41)(23 48)(24 43)(26 32)(28 34)(30 36)(49 89)(50 96)(51 91)(52 86)(53 93)(54 88)(55 95)(56 90)(57 85)(58 92)(59 87)(60 94)(61 70)(62 65)(63 72)(64 67)(66 69)(68 71)(73 82)(74 77)(75 84)(76 79)(78 81)(80 83)
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), (1,23,33,42)(2,18,34,37)(3,13,35,44)(4,20,36,39)(5,15,25,46)(6,22,26,41)(7,17,27,48)(8,24,28,43)(9,19,29,38)(10,14,30,45)(11,21,31,40)(12,16,32,47)(49,72,86,84)(50,67,87,79)(51,62,88,74)(52,69,89,81)(53,64,90,76)(54,71,91,83)(55,66,92,78)(56,61,93,73)(57,68,94,80)(58,63,95,75)(59,70,96,82)(60,65,85,77), (1,75,30,72,7,81,36,66)(2,80,31,65,8,74,25,71)(3,73,32,70,9,79,26,64)(4,78,33,63,10,84,27,69)(5,83,34,68,11,77,28,62)(6,76,35,61,12,82,29,67)(13,90,41,50,19,96,47,56)(14,95,42,55,20,89,48,49)(15,88,43,60,21,94,37,54)(16,93,44,53,22,87,38,59)(17,86,45,58,23,92,39,52)(18,91,46,51,24,85,40,57), (2,8)(4,10)(6,12)(13,38)(14,45)(15,40)(16,47)(17,42)(18,37)(19,44)(20,39)(21,46)(22,41)(23,48)(24,43)(26,32)(28,34)(30,36)(49,89)(50,96)(51,91)(52,86)(53,93)(54,88)(55,95)(56,90)(57,85)(58,92)(59,87)(60,94)(61,70)(62,65)(63,72)(64,67)(66,69)(68,71)(73,82)(74,77)(75,84)(76,79)(78,81)(80,83)>;
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), (1,23,33,42)(2,18,34,37)(3,13,35,44)(4,20,36,39)(5,15,25,46)(6,22,26,41)(7,17,27,48)(8,24,28,43)(9,19,29,38)(10,14,30,45)(11,21,31,40)(12,16,32,47)(49,72,86,84)(50,67,87,79)(51,62,88,74)(52,69,89,81)(53,64,90,76)(54,71,91,83)(55,66,92,78)(56,61,93,73)(57,68,94,80)(58,63,95,75)(59,70,96,82)(60,65,85,77), (1,75,30,72,7,81,36,66)(2,80,31,65,8,74,25,71)(3,73,32,70,9,79,26,64)(4,78,33,63,10,84,27,69)(5,83,34,68,11,77,28,62)(6,76,35,61,12,82,29,67)(13,90,41,50,19,96,47,56)(14,95,42,55,20,89,48,49)(15,88,43,60,21,94,37,54)(16,93,44,53,22,87,38,59)(17,86,45,58,23,92,39,52)(18,91,46,51,24,85,40,57), (2,8)(4,10)(6,12)(13,38)(14,45)(15,40)(16,47)(17,42)(18,37)(19,44)(20,39)(21,46)(22,41)(23,48)(24,43)(26,32)(28,34)(30,36)(49,89)(50,96)(51,91)(52,86)(53,93)(54,88)(55,95)(56,90)(57,85)(58,92)(59,87)(60,94)(61,70)(62,65)(63,72)(64,67)(66,69)(68,71)(73,82)(74,77)(75,84)(76,79)(78,81)(80,83) );
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)], [(1,23,33,42),(2,18,34,37),(3,13,35,44),(4,20,36,39),(5,15,25,46),(6,22,26,41),(7,17,27,48),(8,24,28,43),(9,19,29,38),(10,14,30,45),(11,21,31,40),(12,16,32,47),(49,72,86,84),(50,67,87,79),(51,62,88,74),(52,69,89,81),(53,64,90,76),(54,71,91,83),(55,66,92,78),(56,61,93,73),(57,68,94,80),(58,63,95,75),(59,70,96,82),(60,65,85,77)], [(1,75,30,72,7,81,36,66),(2,80,31,65,8,74,25,71),(3,73,32,70,9,79,26,64),(4,78,33,63,10,84,27,69),(5,83,34,68,11,77,28,62),(6,76,35,61,12,82,29,67),(13,90,41,50,19,96,47,56),(14,95,42,55,20,89,48,49),(15,88,43,60,21,94,37,54),(16,93,44,53,22,87,38,59),(17,86,45,58,23,92,39,52),(18,91,46,51,24,85,40,57)], [(2,8),(4,10),(6,12),(13,38),(14,45),(15,40),(16,47),(17,42),(18,37),(19,44),(20,39),(21,46),(22,41),(23,48),(24,43),(26,32),(28,34),(30,36),(49,89),(50,96),(51,91),(52,86),(53,93),(54,88),(55,95),(56,90),(57,85),(58,92),(59,87),(60,94),(61,70),(62,65),(63,72),(64,67),(66,69),(68,71),(73,82),(74,77),(75,84),(76,79),(78,81),(80,83)])
33 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F |
order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 |
size | 1 | 1 | 1 | 1 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 8 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 8 | 8 |
33 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | C3⋊D4 | C4○D8 | C8⋊C22 | D4⋊2S3 | D12⋊6C22 | Q8.13D6 |
kernel | C12.Q8⋊C2 | C12.Q8 | C4.Dic6 | C12.55D4 | D4⋊Dic3 | C23.26D6 | C3×C4⋊D4 | C4⋊D4 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C2×C4 | C23 | C6 | C6 | C4 | C2 | C2 |
# reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of C12.Q8⋊C2 ►in GL6(𝔽73)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 72 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 72 |
0 | 0 | 0 | 0 | 1 | 72 |
56 | 3 | 0 | 0 | 0 | 0 |
25 | 17 | 0 | 0 | 0 | 0 |
0 | 0 | 46 | 0 | 0 | 0 |
0 | 0 | 0 | 27 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
46 | 0 | 0 | 0 | 0 | 0 |
59 | 27 | 0 | 0 | 0 | 0 |
0 | 0 | 57 | 16 | 0 | 0 |
0 | 0 | 57 | 57 | 0 | 0 |
0 | 0 | 0 | 0 | 28 | 3 |
0 | 0 | 0 | 0 | 31 | 45 |
1 | 0 | 0 | 0 | 0 | 0 |
60 | 72 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 72 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[56,25,0,0,0,0,3,17,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[46,59,0,0,0,0,0,27,0,0,0,0,0,0,57,57,0,0,0,0,16,57,0,0,0,0,0,0,28,31,0,0,0,0,3,45],[1,60,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C12.Q8⋊C2 in GAP, Magma, Sage, TeX
C_{12}.Q_8\rtimes C_2
% in TeX
G:=Group("C12.Q8:C2");
// GroupNames label
G:=SmallGroup(192,594);
// by ID
G=gap.SmallGroup(192,594);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,254,219,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^4=d^2=1,c^2=a^9*b^2,b*a*b^-1=d*a*d=a^7,c*a*c^-1=a^5,c*b*c^-1=a^9*b^-1,d*b*d=a^6*b^-1,d*c*d=a^9*c>;
// generators/relations