direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12.10D4, (C6×Q8).10C4, C12.209(C2×D4), (C2×C12).194D4, (C2×Q8).191D6, (C2×Q8).9Dic3, (C22×Q8).9S3, C6⋊2(C4.10D4), (C22×C12).10C4, (C22×C4).171D6, (C22×C4).8Dic3, C12.35(C22⋊C4), (C2×C12).476C23, (C6×Q8).202C22, C23.38(C2×Dic3), C4.12(C6.D4), C4.Dic3.46C22, C22.7(C22×Dic3), (C22×C12).202C22, C22.36(C6.D4), (Q8×C2×C6).3C2, C3⋊3(C2×C4.10D4), C4.93(C2×C3⋊D4), C6.79(C2×C22⋊C4), (C2×C12).121(C2×C4), (C2×C4).26(C2×Dic3), (C2×C4).199(C3⋊D4), (C2×C6).198(C22×C4), (C22×C6).137(C2×C4), (C2×C4).130(C22×S3), C2.15(C2×C6.D4), (C2×C4.Dic3).28C2, (C2×C6).116(C22⋊C4), SmallGroup(192,785)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C12.10D4
G = < a,b,c,d | a2=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b5, dcd-1=b9c3 >
Subgroups: 264 in 146 conjugacy classes, 71 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C3⋊C8, C2×C12, C2×C12, C2×C12, C3×Q8, C22×C6, C4.10D4, C2×M4(2), C22×Q8, C2×C3⋊C8, C4.Dic3, C4.Dic3, C22×C12, C22×C12, C6×Q8, C6×Q8, C2×C4.10D4, C12.10D4, C2×C4.Dic3, Q8×C2×C6, C2×C12.10D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C2×Dic3, C3⋊D4, C22×S3, C4.10D4, C2×C22⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C2×C4.10D4, C12.10D4, C2×C6.D4, C2×C12.10D4
►On 96 points
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 85)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 49)(48 50)(61 84)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)
(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 76 34 62 7 82 28 68)(2 75 35 61 8 81 29 67)(3 74 36 72 9 80 30 66)(4 73 25 71 10 79 31 65)(5 84 26 70 11 78 32 64)(6 83 27 69 12 77 33 63)(13 40 95 57 19 46 89 51)(14 39 96 56 20 45 90 50)(15 38 85 55 21 44 91 49)(16 37 86 54 22 43 92 60)(17 48 87 53 23 42 93 59)(18 47 88 52 24 41 94 58)
(1 22 10 19 7 16 4 13)(2 15 11 24 8 21 5 18)(3 20 12 17 9 14 6 23)(25 89 34 86 31 95 28 92)(26 94 35 91 32 88 29 85)(27 87 36 96 33 93 30 90)(37 68 46 65 43 62 40 71)(38 61 47 70 44 67 41 64)(39 66 48 63 45 72 42 69)(49 81 58 78 55 75 52 84)(50 74 59 83 56 80 53 77)(51 79 60 76 57 73 54 82)
G:=sub<Sym(96)| (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,85)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,49)(48,50)(61,84)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83), (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), (1,76,34,62,7,82,28,68)(2,75,35,61,8,81,29,67)(3,74,36,72,9,80,30,66)(4,73,25,71,10,79,31,65)(5,84,26,70,11,78,32,64)(6,83,27,69,12,77,33,63)(13,40,95,57,19,46,89,51)(14,39,96,56,20,45,90,50)(15,38,85,55,21,44,91,49)(16,37,86,54,22,43,92,60)(17,48,87,53,23,42,93,59)(18,47,88,52,24,41,94,58), (1,22,10,19,7,16,4,13)(2,15,11,24,8,21,5,18)(3,20,12,17,9,14,6,23)(25,89,34,86,31,95,28,92)(26,94,35,91,32,88,29,85)(27,87,36,96,33,93,30,90)(37,68,46,65,43,62,40,71)(38,61,47,70,44,67,41,64)(39,66,48,63,45,72,42,69)(49,81,58,78,55,75,52,84)(50,74,59,83,56,80,53,77)(51,79,60,76,57,73,54,82)>;
G:=Group( (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,85)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,49)(48,50)(61,84)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83), (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), (1,76,34,62,7,82,28,68)(2,75,35,61,8,81,29,67)(3,74,36,72,9,80,30,66)(4,73,25,71,10,79,31,65)(5,84,26,70,11,78,32,64)(6,83,27,69,12,77,33,63)(13,40,95,57,19,46,89,51)(14,39,96,56,20,45,90,50)(15,38,85,55,21,44,91,49)(16,37,86,54,22,43,92,60)(17,48,87,53,23,42,93,59)(18,47,88,52,24,41,94,58), (1,22,10,19,7,16,4,13)(2,15,11,24,8,21,5,18)(3,20,12,17,9,14,6,23)(25,89,34,86,31,95,28,92)(26,94,35,91,32,88,29,85)(27,87,36,96,33,93,30,90)(37,68,46,65,43,62,40,71)(38,61,47,70,44,67,41,64)(39,66,48,63,45,72,42,69)(49,81,58,78,55,75,52,84)(50,74,59,83,56,80,53,77)(51,79,60,76,57,73,54,82) );
G=PermutationGroup([[(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,86),(14,87),(15,88),(16,89),(17,90),(18,91),(19,92),(20,93),(21,94),(22,95),(23,96),(24,85),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(43,57),(44,58),(45,59),(46,60),(47,49),(48,50),(61,84),(62,73),(63,74),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83)], [(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)], [(1,76,34,62,7,82,28,68),(2,75,35,61,8,81,29,67),(3,74,36,72,9,80,30,66),(4,73,25,71,10,79,31,65),(5,84,26,70,11,78,32,64),(6,83,27,69,12,77,33,63),(13,40,95,57,19,46,89,51),(14,39,96,56,20,45,90,50),(15,38,85,55,21,44,91,49),(16,37,86,54,22,43,92,60),(17,48,87,53,23,42,93,59),(18,47,88,52,24,41,94,58)], [(1,22,10,19,7,16,4,13),(2,15,11,24,8,21,5,18),(3,20,12,17,9,14,6,23),(25,89,34,86,31,95,28,92),(26,94,35,91,32,88,29,85),(27,87,36,96,33,93,30,90),(37,68,46,65,43,62,40,71),(38,61,47,70,44,67,41,64),(39,66,48,63,45,72,42,69),(49,81,58,78,55,75,52,84),(50,74,59,83,56,80,53,77),(51,79,60,76,57,73,54,82)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 12 | ··· | 12 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | Dic3 | D6 | Dic3 | D6 | C3⋊D4 | C4.10D4 | C12.10D4 |
| kernel | C2×C12.10D4 | C12.10D4 | C2×C4.Dic3 | Q8×C2×C6 | C22×C12 | C6×Q8 | C22×Q8 | C2×C12 | C22×C4 | C22×C4 | C2×Q8 | C2×Q8 | C2×C4 | C6 | C2 |
| # reps | 1 | 4 | 2 | 1 | 4 | 4 | 1 | 4 | 2 | 1 | 2 | 2 | 8 | 2 | 4 |
Matrix representation of C2×C12.10D4 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 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 | 0 |
| 70 | 45 | 0 | 0 | 0 | 0 |
| 42 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 70 | 45 | 0 | 0 | 0 | 0 |
| 42 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 45 |
| 0 | 0 | 0 | 0 | 45 | 23 |
| 0 | 0 | 45 | 23 | 0 | 0 |
| 0 | 0 | 23 | 28 | 0 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[70,42,0,0,0,0,45,3,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[70,42,0,0,0,0,45,3,0,0,0,0,0,0,0,0,45,23,0,0,0,0,23,28,0,0,50,45,0,0,0,0,45,23,0,0] >;
C2×C12.10D4 in GAP, Magma, Sage, TeX
C_2\times C_{12}._{10}D_4 % in TeX
G:=Group("C2xC12.10D4"); // GroupNames label
G:=SmallGroup(192,785);
// by ID
G=gap.SmallGroup(192,785);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,184,297,136,1684,6278]);
// 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=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^9*c^3>;
// generators/relations