direct product, metabelian, supersoluble, monomial, A-group
Aliases: C22×C19⋊C6, D38⋊3C6, C38⋊(C2×C6), C19⋊C3⋊C23, D19⋊(C2×C6), C19⋊(C22×C6), (C2×C38)⋊4C6, (C22×D19)⋊3C3, (C2×C19⋊C3)⋊C22, (C22×C19⋊C3)⋊2C2, SmallGroup(456,44)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C22×C19⋊C6 |
Generators and relations for C22×C19⋊C6
G = < a,b,c,d | a2=b2=c19=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c12 >
Subgroups: 550 in 64 conjugacy classes, 37 normal (8 characteristic)
C1, C2, C2, C3, C22, C22, C6, C23, C2×C6, C19, C22×C6, D19, C38, C19⋊C3, D38, C2×C38, C19⋊C6, C2×C19⋊C3, C22×D19, C2×C19⋊C6, C22×C19⋊C3, C22×C19⋊C6
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C22×C6, C19⋊C6, C2×C19⋊C6, C22×C19⋊C6
►On 76 points
Generators in S76
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)
(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)
(1 58)(2 66 8 76 12 70)(3 74 15 75 4 63)(5 71 10 73 7 68)(6 60 17 72 18 61)(9 65 19 69 13 59)(11 62 14 67 16 64)(20 39)(21 47 27 57 31 51)(22 55 34 56 23 44)(24 52 29 54 26 49)(25 41 36 53 37 42)(28 46 38 50 32 40)(30 43 33 48 35 45)
G:=sub<Sym(76)| (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76), (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), (1,58)(2,66,8,76,12,70)(3,74,15,75,4,63)(5,71,10,73,7,68)(6,60,17,72,18,61)(9,65,19,69,13,59)(11,62,14,67,16,64)(20,39)(21,47,27,57,31,51)(22,55,34,56,23,44)(24,52,29,54,26,49)(25,41,36,53,37,42)(28,46,38,50,32,40)(30,43,33,48,35,45)>;
G:=Group( (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76), (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), (1,58)(2,66,8,76,12,70)(3,74,15,75,4,63)(5,71,10,73,7,68)(6,60,17,72,18,61)(9,65,19,69,13,59)(11,62,14,67,16,64)(20,39)(21,47,27,57,31,51)(22,55,34,56,23,44)(24,52,29,54,26,49)(25,41,36,53,37,42)(28,46,38,50,32,40)(30,43,33,48,35,45) );
G=PermutationGroup([[(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(31,69),(32,70),(33,71),(34,72),(35,73),(36,74),(37,75),(38,76)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(39,58),(40,59),(41,60),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,68),(50,69),(51,70),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76)], [(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)], [(1,58),(2,66,8,76,12,70),(3,74,15,75,4,63),(5,71,10,73,7,68),(6,60,17,72,18,61),(9,65,19,69,13,59),(11,62,14,67,16,64),(20,39),(21,47,27,57,31,51),(22,55,34,56,23,44),(24,52,29,54,26,49),(25,41,36,53,37,42),(28,46,38,50,32,40),(30,43,33,48,35,45)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 6A | ··· | 6N | 19A | 19B | 19C | 38A | ··· | 38I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 19 | 19 | 19 | 38 | ··· | 38 |
| size | 1 | 1 | 1 | 1 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | ··· | 19 | 6 | 6 | 6 | 6 | ··· | 6 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 |
| type | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C19⋊C6 | C2×C19⋊C6 |
| kernel | C22×C19⋊C6 | C2×C19⋊C6 | C22×C19⋊C3 | C22×D19 | D38 | C2×C38 | C22 | C2 |
| # reps | 1 | 6 | 1 | 2 | 12 | 2 | 3 | 9 |
Matrix representation of C22×C19⋊C6 ►in GL7(𝔽229)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 228 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 228 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 228 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 228 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 228 |
| 228 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 91 | 9 | 220 | 138 | 109 | 228 |
| 0 | 92 | 9 | 220 | 138 | 109 | 228 |
| 0 | 91 | 10 | 220 | 138 | 109 | 228 |
| 0 | 91 | 9 | 221 | 138 | 109 | 228 |
| 0 | 91 | 9 | 220 | 139 | 109 | 228 |
| 0 | 91 | 9 | 220 | 138 | 110 | 228 |
| 134 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 102 | 167 | 81 | 8 | 109 |
| 0 | 0 | 0 | 0 | 0 | 228 | 0 |
| 0 | 81 | 56 | 73 | 37 | 201 | 120 |
| 0 | 200 | 148 | 102 | 28 | 90 | 228 |
| 0 | 0 | 0 | 228 | 0 | 0 | 0 |
| 0 | 90 | 28 | 102 | 148 | 200 | 119 |
G:=sub<GL(7,GF(229))| [1,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228],[228,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,91,92,91,91,91,91,0,9,9,10,9,9,9,0,220,220,220,221,220,220,0,138,138,138,138,139,138,0,109,109,109,109,109,110,0,228,228,228,228,228,228],[134,0,0,0,0,0,0,0,9,0,81,200,0,90,0,102,0,56,148,0,28,0,167,0,73,102,228,102,0,81,0,37,28,0,148,0,8,228,201,90,0,200,0,109,0,120,228,0,119] >;
C22×C19⋊C6 in GAP, Magma, Sage, TeX
C_2^2\times C_{19}\rtimes C_6 % in TeX
G:=Group("C2^2xC19:C6"); // GroupNames label
G:=SmallGroup(456,44);
// by ID
G=gap.SmallGroup(456,44);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,10804,544]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^19=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^12>;
// generators/relations