direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12.23D4, (C2×Q8)⋊32D6, D6⋊C4⋊74C22, (C2×C12).215D4, C12.259(C2×D4), C6⋊4(C4.4D4), (C22×Q8)⋊13S3, (C6×Q8)⋊36C22, (C2×C6).306C24, C6.154(C22×D4), (C22×C4).401D6, (C2×C12).646C23, (C4×Dic3)⋊69C22, (C22×D12).19C2, (C2×D12).278C22, (S3×C23).78C22, C23.352(C22×S3), C22.317(S3×C23), (C22×C6).424C23, (C22×S3).133C23, (C22×C12).439C22, C22.40(Q8⋊3S3), (C2×Dic3).289C23, (C22×Dic3).234C22, (Q8×C2×C6)⋊5C2, C3⋊5(C2×C4.4D4), (C2×D6⋊C4)⋊43C2, (C2×C4×Dic3)⋊13C2, C4.28(C2×C3⋊D4), C6.128(C2×C4○D4), (C2×C6).589(C2×D4), C2.35(C2×Q8⋊3S3), C2.27(C22×C3⋊D4), (C2×C6).201(C4○D4), (C2×C4).157(C3⋊D4), (C2×C4).243(C22×S3), C22.117(C2×C3⋊D4), SmallGroup(192,1373)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C12.23D4
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=b6c-1 >
Subgroups: 904 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C24, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C22×C6, C2×C42, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, C4×Dic3, D6⋊C4, C2×D12, C2×D12, C22×Dic3, C22×C12, C22×C12, C6×Q8, C6×Q8, S3×C23, C2×C4.4D4, C2×C4×Dic3, C2×D6⋊C4, C12.23D4, C22×D12, Q8×C2×C6, C2×C12.23D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C4.4D4, C22×D4, C2×C4○D4, Q8⋊3S3, C2×C3⋊D4, S3×C23, C2×C4.4D4, C12.23D4, C2×Q8⋊3S3, C22×C3⋊D4, C2×C12.23D4
►On 96 points
(1 34)(2 35)(3 36)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 59)(14 60)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 85)(47 86)(48 87)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 73)
(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 40 81 16)(2 45 82 21)(3 38 83 14)(4 43 84 19)(5 48 73 24)(6 41 74 17)(7 46 75 22)(8 39 76 15)(9 44 77 20)(10 37 78 13)(11 42 79 18)(12 47 80 23)(25 94 71 53)(26 87 72 58)(27 92 61 51)(28 85 62 56)(29 90 63 49)(30 95 64 54)(31 88 65 59)(32 93 66 52)(33 86 67 57)(34 91 68 50)(35 96 69 55)(36 89 70 60)
(1 28)(2 27)(3 26)(4 25)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 94)(14 93)(15 92)(16 91)(17 90)(18 89)(19 88)(20 87)(21 86)(22 85)(23 96)(24 95)(37 53)(38 52)(39 51)(40 50)(41 49)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 84)(72 83)
G:=sub<Sym(96)| (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,59)(14,60)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,85)(47,86)(48,87)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,73), (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), (1,40,81,16)(2,45,82,21)(3,38,83,14)(4,43,84,19)(5,48,73,24)(6,41,74,17)(7,46,75,22)(8,39,76,15)(9,44,77,20)(10,37,78,13)(11,42,79,18)(12,47,80,23)(25,94,71,53)(26,87,72,58)(27,92,61,51)(28,85,62,56)(29,90,63,49)(30,95,64,54)(31,88,65,59)(32,93,66,52)(33,86,67,57)(34,91,68,50)(35,96,69,55)(36,89,70,60), (1,28)(2,27)(3,26)(4,25)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,94)(14,93)(15,92)(16,91)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,96)(24,95)(37,53)(38,52)(39,51)(40,50)(41,49)(42,60)(43,59)(44,58)(45,57)(46,56)(47,55)(48,54)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,84)(72,83)>;
G:=Group( (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,59)(14,60)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,85)(47,86)(48,87)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,73), (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), (1,40,81,16)(2,45,82,21)(3,38,83,14)(4,43,84,19)(5,48,73,24)(6,41,74,17)(7,46,75,22)(8,39,76,15)(9,44,77,20)(10,37,78,13)(11,42,79,18)(12,47,80,23)(25,94,71,53)(26,87,72,58)(27,92,61,51)(28,85,62,56)(29,90,63,49)(30,95,64,54)(31,88,65,59)(32,93,66,52)(33,86,67,57)(34,91,68,50)(35,96,69,55)(36,89,70,60), (1,28)(2,27)(3,26)(4,25)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,94)(14,93)(15,92)(16,91)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,96)(24,95)(37,53)(38,52)(39,51)(40,50)(41,49)(42,60)(43,59)(44,58)(45,57)(46,56)(47,55)(48,54)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,84)(72,83) );
G=PermutationGroup([[(1,34),(2,35),(3,36),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,59),(14,60),(15,49),(16,50),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,57),(24,58),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,94),(44,95),(45,96),(46,85),(47,86),(48,87),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,81),(69,82),(70,83),(71,84),(72,73)], [(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)], [(1,40,81,16),(2,45,82,21),(3,38,83,14),(4,43,84,19),(5,48,73,24),(6,41,74,17),(7,46,75,22),(8,39,76,15),(9,44,77,20),(10,37,78,13),(11,42,79,18),(12,47,80,23),(25,94,71,53),(26,87,72,58),(27,92,61,51),(28,85,62,56),(29,90,63,49),(30,95,64,54),(31,88,65,59),(32,93,66,52),(33,86,67,57),(34,91,68,50),(35,96,69,55),(36,89,70,60)], [(1,28),(2,27),(3,26),(4,25),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,94),(14,93),(15,92),(16,91),(17,90),(18,89),(19,88),(20,87),(21,86),(22,85),(23,96),(24,95),(37,53),(38,52),(39,51),(40,50),(41,49),(42,60),(43,59),(44,58),(45,57),(46,56),(47,55),(48,54),(61,82),(62,81),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,84),(72,83)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | Q8⋊3S3 |
| kernel | C2×C12.23D4 | C2×C4×Dic3 | C2×D6⋊C4 | C12.23D4 | C22×D12 | Q8×C2×C6 | C22×Q8 | C2×C12 | C22×C4 | C2×Q8 | C2×C6 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 8 | 1 | 1 | 1 | 4 | 3 | 4 | 8 | 8 | 4 |
Matrix representation of C2×C12.23D4 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 3 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,11,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[8,3,0,0,0,0,5,5,0,0,0,0,0,0,0,8,0,0,0,0,5,0,0,0,0,0,0,0,9,11,0,0,0,0,2,4],[1,0,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;
C2×C12.23D4 in GAP, Magma, Sage, TeX
C_2\times C_{12}._{23}D_4 % in TeX
G:=Group("C2xC12.23D4"); // GroupNames label
G:=SmallGroup(192,1373);
// by ID
G=gap.SmallGroup(192,1373);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=b^6*c^-1>;
// generators/relations