direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12.55D4, C24.5Dic3, C23⋊3(C3⋊C8), (C22×C6)⋊4C8, C6⋊2(C22⋊C8), (C23×C4).7S3, (C23×C6).8C4, (C2×C12).499D4, C12.449(C2×D4), C6.29(C22×C8), (C22×C12).33C4, (C23×C12).18C2, (C22×C4).480D6, (C2×C6).27M4(2), C6.43(C2×M4(2)), C12.97(C22⋊C4), (C2×C12).870C23, C23.46(C2×Dic3), (C22×C4).15Dic3, C4.31(C6.D4), (C22×C12).562C22, C22.10(C4.Dic3), C22.24(C22×Dic3), C22.31(C6.D4), (C2×C6)⋊8(C2×C8), C3⋊3(C2×C22⋊C8), C22⋊3(C2×C3⋊C8), C2.9(C22×C3⋊C8), (C22×C3⋊C8)⋊21C2, (C2×C3⋊C8)⋊45C22, C4.140(C2×C3⋊D4), C6.66(C2×C22⋊C4), (C2×C12).279(C2×C4), C2.5(C2×C4.Dic3), C2.1(C2×C6.D4), (C2×C4).279(C3⋊D4), (C2×C4).812(C22×S3), (C2×C6).190(C22×C4), (C22×C6).133(C2×C4), (C2×C4).104(C2×Dic3), (C2×C6).107(C22⋊C4), SmallGroup(192,765)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C12.55D4
G = < a,b,c,d | a2=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b3c3 >
Subgroups: 344 in 202 conjugacy classes, 103 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C22×C4, C22×C4, C22×C4, C24, C3⋊C8, C2×C12, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C22⋊C8, C22×C8, C23×C4, C2×C3⋊C8, C2×C3⋊C8, C22×C12, C22×C12, C22×C12, C23×C6, C2×C22⋊C8, C12.55D4, C22×C3⋊C8, C23×C12, C2×C12.55D4
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C3⋊C8, C2×Dic3, C3⋊D4, C22×S3, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C2×C3⋊C8, C4.Dic3, C6.D4, C22×Dic3, C2×C3⋊D4, C2×C22⋊C8, C12.55D4, C22×C3⋊C8, C2×C4.Dic3, C2×C6.D4, C2×C12.55D4
►On 96 points
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 49)(34 50)(35 51)(36 52)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 61)(46 62)(47 63)(48 64)(73 94)(74 95)(75 96)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(82 91)(83 92)(84 93)
(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 65 51 81 7 71 57 75)(2 70 52 74 8 64 58 80)(3 63 53 79 9 69 59 73)(4 68 54 84 10 62 60 78)(5 61 55 77 11 67 49 83)(6 66 56 82 12 72 50 76)(13 46 32 87 19 40 26 93)(14 39 33 92 20 45 27 86)(15 44 34 85 21 38 28 91)(16 37 35 90 22 43 29 96)(17 42 36 95 23 48 30 89)(18 47 25 88 24 41 31 94)
(1 84 10 81 7 78 4 75)(2 77 11 74 8 83 5 80)(3 82 12 79 9 76 6 73)(13 90 22 87 19 96 16 93)(14 95 23 92 20 89 17 86)(15 88 24 85 21 94 18 91)(25 44 34 41 31 38 28 47)(26 37 35 46 32 43 29 40)(27 42 36 39 33 48 30 45)(49 64 58 61 55 70 52 67)(50 69 59 66 56 63 53 72)(51 62 60 71 57 68 54 65)
G:=sub<Sym(96)| (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,49)(34,50)(35,51)(36,52)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,61)(46,62)(47,63)(48,64)(73,94)(74,95)(75,96)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93), (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), (1,65,51,81,7,71,57,75)(2,70,52,74,8,64,58,80)(3,63,53,79,9,69,59,73)(4,68,54,84,10,62,60,78)(5,61,55,77,11,67,49,83)(6,66,56,82,12,72,50,76)(13,46,32,87,19,40,26,93)(14,39,33,92,20,45,27,86)(15,44,34,85,21,38,28,91)(16,37,35,90,22,43,29,96)(17,42,36,95,23,48,30,89)(18,47,25,88,24,41,31,94), (1,84,10,81,7,78,4,75)(2,77,11,74,8,83,5,80)(3,82,12,79,9,76,6,73)(13,90,22,87,19,96,16,93)(14,95,23,92,20,89,17,86)(15,88,24,85,21,94,18,91)(25,44,34,41,31,38,28,47)(26,37,35,46,32,43,29,40)(27,42,36,39,33,48,30,45)(49,64,58,61,55,70,52,67)(50,69,59,66,56,63,53,72)(51,62,60,71,57,68,54,65)>;
G:=Group( (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,49)(34,50)(35,51)(36,52)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,61)(46,62)(47,63)(48,64)(73,94)(74,95)(75,96)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93), (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), (1,65,51,81,7,71,57,75)(2,70,52,74,8,64,58,80)(3,63,53,79,9,69,59,73)(4,68,54,84,10,62,60,78)(5,61,55,77,11,67,49,83)(6,66,56,82,12,72,50,76)(13,46,32,87,19,40,26,93)(14,39,33,92,20,45,27,86)(15,44,34,85,21,38,28,91)(16,37,35,90,22,43,29,96)(17,42,36,95,23,48,30,89)(18,47,25,88,24,41,31,94), (1,84,10,81,7,78,4,75)(2,77,11,74,8,83,5,80)(3,82,12,79,9,76,6,73)(13,90,22,87,19,96,16,93)(14,95,23,92,20,89,17,86)(15,88,24,85,21,94,18,91)(25,44,34,41,31,38,28,47)(26,37,35,46,32,43,29,40)(27,42,36,39,33,48,30,45)(49,64,58,61,55,70,52,67)(50,69,59,66,56,63,53,72)(51,62,60,71,57,68,54,65) );
G=PermutationGroup([[(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,13),(11,14),(12,15),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,49),(34,50),(35,51),(36,52),(37,65),(38,66),(39,67),(40,68),(41,69),(42,70),(43,71),(44,72),(45,61),(46,62),(47,63),(48,64),(73,94),(74,95),(75,96),(76,85),(77,86),(78,87),(79,88),(80,89),(81,90),(82,91),(83,92),(84,93)], [(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)], [(1,65,51,81,7,71,57,75),(2,70,52,74,8,64,58,80),(3,63,53,79,9,69,59,73),(4,68,54,84,10,62,60,78),(5,61,55,77,11,67,49,83),(6,66,56,82,12,72,50,76),(13,46,32,87,19,40,26,93),(14,39,33,92,20,45,27,86),(15,44,34,85,21,38,28,91),(16,37,35,90,22,43,29,96),(17,42,36,95,23,48,30,89),(18,47,25,88,24,41,31,94)], [(1,84,10,81,7,78,4,75),(2,77,11,74,8,83,5,80),(3,82,12,79,9,76,6,73),(13,90,22,87,19,96,16,93),(14,95,23,92,20,89,17,86),(15,88,24,85,21,94,18,91),(25,44,34,41,31,38,28,47),(26,37,35,46,32,43,29,40),(27,42,36,39,33,48,30,45),(49,64,58,61,55,70,52,67),(50,69,59,66,56,63,53,72),(51,62,60,71,57,68,54,65)]])
72 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6O | 8A | ··· | 8P | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | Dic3 | D6 | Dic3 | M4(2) | C3⋊D4 | C3⋊C8 | C4.Dic3 |
| kernel | C2×C12.55D4 | C12.55D4 | C22×C3⋊C8 | C23×C12 | C22×C12 | C23×C6 | C22×C6 | C23×C4 | C2×C12 | C22×C4 | C22×C4 | C24 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 4 | 2 | 1 | 6 | 2 | 16 | 1 | 4 | 3 | 3 | 1 | 4 | 8 | 8 | 8 |
Matrix representation of C2×C12.55D4 ►in GL5(𝔽73)
| 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 |
| 0 | 27 | 46 | 0 | 0 |
| 0 | 0 | 0 | 24 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 27 | 0 | 0 | 0 | 0 |
| 0 | 45 | 27 | 0 | 0 |
| 0 | 72 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 46 | 0 |
| 46 | 0 | 0 | 0 | 0 |
| 0 | 45 | 27 | 0 | 0 |
| 0 | 72 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 27 | 0 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,0,27,0,0,0,46,46,0,0,0,0,0,24,0,0,0,0,0,3],[27,0,0,0,0,0,45,72,0,0,0,27,28,0,0,0,0,0,0,46,0,0,0,72,0],[46,0,0,0,0,0,45,72,0,0,0,27,28,0,0,0,0,0,0,27,0,0,0,72,0] >;
C2×C12.55D4 in GAP, Magma, Sage, TeX
C_2\times C_{12}._{55}D_4 % in TeX
G:=Group("C2xC12.55D4"); // GroupNames label
G:=SmallGroup(192,765);
// by ID
G=gap.SmallGroup(192,765);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^3*c^3>;
// generators/relations