direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C6×C8.C22, C24.46C23, C12.83C24, Q16⋊3(C2×C6), C4.67(C6×D4), (C2×Q16)⋊11C6, (C6×Q16)⋊25C2, SD16⋊2(C2×C6), (C2×SD16)⋊5C6, C8.1(C22×C6), C4.6(C23×C6), (C6×SD16)⋊16C2, (C2×C12).526D4, C12.330(C2×D4), (C6×M4(2))⋊9C2, (C2×M4(2))⋊4C6, M4(2)⋊4(C2×C6), (C22×Q8)⋊17C6, (C6×Q8)⋊55C22, D4.3(C22×C6), C22.24(C6×D4), C23.56(C3×D4), Q8.7(C22×C6), (C3×Q16)⋊17C22, (C3×D4).36C23, (C22×C6).173D4, C6.204(C22×D4), (C3×Q8).37C23, (C2×C24).208C22, (C2×C12).976C23, (C3×SD16)⋊18C22, (C6×D4).329C22, (C3×M4(2))⋊25C22, (C22×C12).466C22, (Q8×C2×C6)⋊21C2, C2.28(D4×C2×C6), (C2×C8).32(C2×C6), (C2×Q8)⋊17(C2×C6), (C6×C4○D4).26C2, (C2×C4○D4).18C6, C4○D4.20(C2×C6), (C2×D4).75(C2×C6), (C2×C6).420(C2×D4), (C2×C4).137(C3×D4), (C2×C4).46(C22×C6), (C22×C4).82(C2×C6), (C3×C4○D4).58C22, SmallGroup(192,1463)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C8.C22
G = < a,b,c,d | a6=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >
Subgroups: 370 in 258 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C2×C24, C3×M4(2), C3×SD16, C3×Q16, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C6×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, C2×C8.C22, C6×M4(2), C6×SD16, C6×Q16, C3×C8.C22, Q8×C2×C6, C6×C4○D4, C6×C8.C22
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C8.C22, C22×D4, C6×D4, C23×C6, C2×C8.C22, C3×C8.C22, D4×C2×C6, C6×C8.C22
►On 96 points
(1 73 85 48 19 50)(2 74 86 41 20 51)(3 75 87 42 21 52)(4 76 88 43 22 53)(5 77 81 44 23 54)(6 78 82 45 24 55)(7 79 83 46 17 56)(8 80 84 47 18 49)(9 27 90 39 64 67)(10 28 91 40 57 68)(11 29 92 33 58 69)(12 30 93 34 59 70)(13 31 94 35 60 71)(14 32 95 36 61 72)(15 25 96 37 62 65)(16 26 89 38 63 66)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 21)(18 24)(20 22)(25 27)(26 30)(29 31)(33 35)(34 38)(37 39)(41 43)(42 46)(45 47)(49 55)(51 53)(52 56)(58 60)(59 63)(62 64)(65 67)(66 70)(69 71)(74 76)(75 79)(78 80)(82 84)(83 87)(86 88)(89 93)(90 96)(92 94)
(1 70)(2 67)(3 72)(4 69)(5 66)(6 71)(7 68)(8 65)(9 74)(10 79)(11 76)(12 73)(13 78)(14 75)(15 80)(16 77)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(25 84)(26 81)(27 86)(28 83)(29 88)(30 85)(31 82)(32 87)(41 90)(42 95)(43 92)(44 89)(45 94)(46 91)(47 96)(48 93)(49 62)(50 59)(51 64)(52 61)(53 58)(54 63)(55 60)(56 57)
G:=sub<Sym(96)| (1,73,85,48,19,50)(2,74,86,41,20,51)(3,75,87,42,21,52)(4,76,88,43,22,53)(5,77,81,44,23,54)(6,78,82,45,24,55)(7,79,83,46,17,56)(8,80,84,47,18,49)(9,27,90,39,64,67)(10,28,91,40,57,68)(11,29,92,33,58,69)(12,30,93,34,59,70)(13,31,94,35,60,71)(14,32,95,36,61,72)(15,25,96,37,62,65)(16,26,89,38,63,66), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,21)(18,24)(20,22)(25,27)(26,30)(29,31)(33,35)(34,38)(37,39)(41,43)(42,46)(45,47)(49,55)(51,53)(52,56)(58,60)(59,63)(62,64)(65,67)(66,70)(69,71)(74,76)(75,79)(78,80)(82,84)(83,87)(86,88)(89,93)(90,96)(92,94), (1,70)(2,67)(3,72)(4,69)(5,66)(6,71)(7,68)(8,65)(9,74)(10,79)(11,76)(12,73)(13,78)(14,75)(15,80)(16,77)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(25,84)(26,81)(27,86)(28,83)(29,88)(30,85)(31,82)(32,87)(41,90)(42,95)(43,92)(44,89)(45,94)(46,91)(47,96)(48,93)(49,62)(50,59)(51,64)(52,61)(53,58)(54,63)(55,60)(56,57)>;
G:=Group( (1,73,85,48,19,50)(2,74,86,41,20,51)(3,75,87,42,21,52)(4,76,88,43,22,53)(5,77,81,44,23,54)(6,78,82,45,24,55)(7,79,83,46,17,56)(8,80,84,47,18,49)(9,27,90,39,64,67)(10,28,91,40,57,68)(11,29,92,33,58,69)(12,30,93,34,59,70)(13,31,94,35,60,71)(14,32,95,36,61,72)(15,25,96,37,62,65)(16,26,89,38,63,66), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,21)(18,24)(20,22)(25,27)(26,30)(29,31)(33,35)(34,38)(37,39)(41,43)(42,46)(45,47)(49,55)(51,53)(52,56)(58,60)(59,63)(62,64)(65,67)(66,70)(69,71)(74,76)(75,79)(78,80)(82,84)(83,87)(86,88)(89,93)(90,96)(92,94), (1,70)(2,67)(3,72)(4,69)(5,66)(6,71)(7,68)(8,65)(9,74)(10,79)(11,76)(12,73)(13,78)(14,75)(15,80)(16,77)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(25,84)(26,81)(27,86)(28,83)(29,88)(30,85)(31,82)(32,87)(41,90)(42,95)(43,92)(44,89)(45,94)(46,91)(47,96)(48,93)(49,62)(50,59)(51,64)(52,61)(53,58)(54,63)(55,60)(56,57) );
G=PermutationGroup([[(1,73,85,48,19,50),(2,74,86,41,20,51),(3,75,87,42,21,52),(4,76,88,43,22,53),(5,77,81,44,23,54),(6,78,82,45,24,55),(7,79,83,46,17,56),(8,80,84,47,18,49),(9,27,90,39,64,67),(10,28,91,40,57,68),(11,29,92,33,58,69),(12,30,93,34,59,70),(13,31,94,35,60,71),(14,32,95,36,61,72),(15,25,96,37,62,65),(16,26,89,38,63,66)], [(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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,21),(18,24),(20,22),(25,27),(26,30),(29,31),(33,35),(34,38),(37,39),(41,43),(42,46),(45,47),(49,55),(51,53),(52,56),(58,60),(59,63),(62,64),(65,67),(66,70),(69,71),(74,76),(75,79),(78,80),(82,84),(83,87),(86,88),(89,93),(90,96),(92,94)], [(1,70),(2,67),(3,72),(4,69),(5,66),(6,71),(7,68),(8,65),(9,74),(10,79),(11,76),(12,73),(13,78),(14,75),(15,80),(16,77),(17,40),(18,37),(19,34),(20,39),(21,36),(22,33),(23,38),(24,35),(25,84),(26,81),(27,86),(28,83),(29,88),(30,85),(31,82),(32,87),(41,90),(42,95),(43,92),(44,89),(45,94),(46,91),(47,96),(48,93),(49,62),(50,59),(51,64),(52,61),(53,58),(54,63),(55,60),(56,57)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12T | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C3×D4 | C3×D4 | C8.C22 | C3×C8.C22 |
| kernel | C6×C8.C22 | C6×M4(2) | C6×SD16 | C6×Q16 | C3×C8.C22 | Q8×C2×C6 | C6×C4○D4 | C2×C8.C22 | C2×M4(2) | C2×SD16 | C2×Q16 | C8.C22 | C22×Q8 | C2×C4○D4 | C2×C12 | C22×C6 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 3 | 1 | 6 | 2 | 2 | 4 |
Matrix representation of C6×C8.C22 ►in GL6(𝔽73)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 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 | 1 |
| 30 | 12 | 0 | 0 | 0 | 0 |
| 4 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 13 | 53 | 20 |
| 0 | 0 | 60 | 7 | 0 | 20 |
| 0 | 0 | 7 | 13 | 0 | 6 |
| 0 | 0 | 7 | 6 | 60 | 6 |
| 1 | 58 | 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 | 72 | 0 |
| 0 | 0 | 0 | 1 | 72 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 72 | 1 | 72 | 2 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,4,0,0,0,0,12,43,0,0,0,0,0,0,60,60,7,7,0,0,13,7,13,6,0,0,53,0,0,60,0,0,20,20,6,6],[1,0,0,0,0,0,58,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,1,0,0,0,0,72,72,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,2,0,72] >;
C6×C8.C22 in GAP, Magma, Sage, TeX
C_6\times C_8.C_2^2
% in TeX
G:=Group("C6xC8.C2^2"); // GroupNames label
G:=SmallGroup(192,1463);
// by ID
G=gap.SmallGroup(192,1463);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,680,2102,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations