metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.158D6, C6.322- (1+4), C6.1362+ (1+4), C4⋊C4.115D6, C12⋊D4⋊35C2, D6⋊Q8⋊38C2, C42.C2⋊14S3, C42⋊7S3⋊32C2, D6.D4⋊37C2, C2.61(D4○D12), (C2×C6).244C24, D6⋊C4.74C22, (C2×C12).191C23, (C4×C12).225C22, (C2×D12).167C22, Dic3⋊C4.55C22, C22.265(S3×C23), (C2×Dic6).42C22, (C22×S3).109C23, C2.33(Q8.15D6), C3⋊5(C22.56C24), (C2×Dic3).126C23, (S3×C2×C4).134C22, (C3×C42.C2)⋊17C2, (C3×C4⋊C4).199C22, (C2×C4).208(C22×S3), SmallGroup(192,1259)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 656 in 220 conjugacy classes, 91 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×11], C22, C22 [×12], S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×8], D4 [×6], Q8 [×2], C23 [×4], Dic3 [×4], C12 [×7], D6 [×12], C2×C6, C42, C22⋊C4 [×12], C4⋊C4 [×6], C4⋊C4 [×4], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], Dic6 [×2], C4×S3 [×4], D12 [×6], C2×Dic3 [×4], C2×C12, C2×C12 [×6], C22×S3 [×4], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C42.C2, Dic3⋊C4 [×4], D6⋊C4 [×12], C4×C12, C3×C4⋊C4 [×6], C2×Dic6 [×2], S3×C2×C4 [×4], C2×D12 [×6], C22.56C24, C42⋊7S3 [×2], D6.D4 [×4], C12⋊D4 [×4], D6⋊Q8 [×4], C3×C42.C2, C42.158D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 2+ (1+4) [×2], 2- (1+4), S3×C23, C22.56C24, Q8.15D6, D4○D12 [×2], C42.158D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c5 >
►On 96 points
Generators in S96
(1 54 7 60)(2 71 8 65)(3 56 9 50)(4 61 10 67)(5 58 11 52)(6 63 12 69)(13 66 19 72)(14 51 20 57)(15 68 21 62)(16 53 22 59)(17 70 23 64)(18 55 24 49)(25 84 31 78)(26 44 32 38)(27 74 33 80)(28 46 34 40)(29 76 35 82)(30 48 36 42)(37 85 43 91)(39 87 45 93)(41 89 47 95)(73 92 79 86)(75 94 81 88)(77 96 83 90)
(1 30 23 96)(2 85 24 31)(3 32 13 86)(4 87 14 33)(5 34 15 88)(6 89 16 35)(7 36 17 90)(8 91 18 25)(9 26 19 92)(10 93 20 27)(11 28 21 94)(12 95 22 29)(37 55 84 65)(38 66 73 56)(39 57 74 67)(40 68 75 58)(41 59 76 69)(42 70 77 60)(43 49 78 71)(44 72 79 50)(45 51 80 61)(46 62 81 52)(47 53 82 63)(48 64 83 54)
(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 92 7 86)(2 85 8 91)(3 90 9 96)(4 95 10 89)(5 88 11 94)(6 93 12 87)(13 36 19 30)(14 29 20 35)(15 34 21 28)(16 27 22 33)(17 32 23 26)(18 25 24 31)(37 65 43 71)(38 70 44 64)(39 63 45 69)(40 68 46 62)(41 61 47 67)(42 66 48 72)(49 84 55 78)(50 77 56 83)(51 82 57 76)(52 75 58 81)(53 80 59 74)(54 73 60 79)
G:=sub<Sym(96)| (1,54,7,60)(2,71,8,65)(3,56,9,50)(4,61,10,67)(5,58,11,52)(6,63,12,69)(13,66,19,72)(14,51,20,57)(15,68,21,62)(16,53,22,59)(17,70,23,64)(18,55,24,49)(25,84,31,78)(26,44,32,38)(27,74,33,80)(28,46,34,40)(29,76,35,82)(30,48,36,42)(37,85,43,91)(39,87,45,93)(41,89,47,95)(73,92,79,86)(75,94,81,88)(77,96,83,90), (1,30,23,96)(2,85,24,31)(3,32,13,86)(4,87,14,33)(5,34,15,88)(6,89,16,35)(7,36,17,90)(8,91,18,25)(9,26,19,92)(10,93,20,27)(11,28,21,94)(12,95,22,29)(37,55,84,65)(38,66,73,56)(39,57,74,67)(40,68,75,58)(41,59,76,69)(42,70,77,60)(43,49,78,71)(44,72,79,50)(45,51,80,61)(46,62,81,52)(47,53,82,63)(48,64,83,54), (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), (1,92,7,86)(2,85,8,91)(3,90,9,96)(4,95,10,89)(5,88,11,94)(6,93,12,87)(13,36,19,30)(14,29,20,35)(15,34,21,28)(16,27,22,33)(17,32,23,26)(18,25,24,31)(37,65,43,71)(38,70,44,64)(39,63,45,69)(40,68,46,62)(41,61,47,67)(42,66,48,72)(49,84,55,78)(50,77,56,83)(51,82,57,76)(52,75,58,81)(53,80,59,74)(54,73,60,79)>;
G:=Group( (1,54,7,60)(2,71,8,65)(3,56,9,50)(4,61,10,67)(5,58,11,52)(6,63,12,69)(13,66,19,72)(14,51,20,57)(15,68,21,62)(16,53,22,59)(17,70,23,64)(18,55,24,49)(25,84,31,78)(26,44,32,38)(27,74,33,80)(28,46,34,40)(29,76,35,82)(30,48,36,42)(37,85,43,91)(39,87,45,93)(41,89,47,95)(73,92,79,86)(75,94,81,88)(77,96,83,90), (1,30,23,96)(2,85,24,31)(3,32,13,86)(4,87,14,33)(5,34,15,88)(6,89,16,35)(7,36,17,90)(8,91,18,25)(9,26,19,92)(10,93,20,27)(11,28,21,94)(12,95,22,29)(37,55,84,65)(38,66,73,56)(39,57,74,67)(40,68,75,58)(41,59,76,69)(42,70,77,60)(43,49,78,71)(44,72,79,50)(45,51,80,61)(46,62,81,52)(47,53,82,63)(48,64,83,54), (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), (1,92,7,86)(2,85,8,91)(3,90,9,96)(4,95,10,89)(5,88,11,94)(6,93,12,87)(13,36,19,30)(14,29,20,35)(15,34,21,28)(16,27,22,33)(17,32,23,26)(18,25,24,31)(37,65,43,71)(38,70,44,64)(39,63,45,69)(40,68,46,62)(41,61,47,67)(42,66,48,72)(49,84,55,78)(50,77,56,83)(51,82,57,76)(52,75,58,81)(53,80,59,74)(54,73,60,79) );
G=PermutationGroup([(1,54,7,60),(2,71,8,65),(3,56,9,50),(4,61,10,67),(5,58,11,52),(6,63,12,69),(13,66,19,72),(14,51,20,57),(15,68,21,62),(16,53,22,59),(17,70,23,64),(18,55,24,49),(25,84,31,78),(26,44,32,38),(27,74,33,80),(28,46,34,40),(29,76,35,82),(30,48,36,42),(37,85,43,91),(39,87,45,93),(41,89,47,95),(73,92,79,86),(75,94,81,88),(77,96,83,90)], [(1,30,23,96),(2,85,24,31),(3,32,13,86),(4,87,14,33),(5,34,15,88),(6,89,16,35),(7,36,17,90),(8,91,18,25),(9,26,19,92),(10,93,20,27),(11,28,21,94),(12,95,22,29),(37,55,84,65),(38,66,73,56),(39,57,74,67),(40,68,75,58),(41,59,76,69),(42,70,77,60),(43,49,78,71),(44,72,79,50),(45,51,80,61),(46,62,81,52),(47,53,82,63),(48,64,83,54)], [(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)], [(1,92,7,86),(2,85,8,91),(3,90,9,96),(4,95,10,89),(5,88,11,94),(6,93,12,87),(13,36,19,30),(14,29,20,35),(15,34,21,28),(16,27,22,33),(17,32,23,26),(18,25,24,31),(37,65,43,71),(38,70,44,64),(39,63,45,69),(40,68,46,62),(41,61,47,67),(42,66,48,72),(49,84,55,78),(50,77,56,83),(51,82,57,76),(52,75,58,81),(53,80,59,74),(54,73,60,79)])
Matrix representation ►G ⊆ GL8(𝔽13)
| 3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 3 |
| 0 | 0 | 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 7 | 6 | 5 | 8 | 0 | 0 | 0 | 0 |
| 7 | 1 | 5 | 10 | 0 | 0 | 0 | 0 |
| 5 | 8 | 6 | 7 | 0 | 0 | 0 | 0 |
| 5 | 10 | 6 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 2 | 2 |
| 0 | 0 | 0 | 0 | 1 | 2 | 11 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 12 | 1 |
| 0 | 0 | 0 | 0 | 11 | 0 | 12 | 11 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(13))| [3,7,0,0,0,0,0,0,6,10,0,0,0,0,0,0,0,0,3,7,0,0,0,0,0,0,6,10,0,0,0,0,0,0,0,0,0,0,3,7,0,0,0,0,0,0,6,10,0,0,0,0,10,6,0,0,0,0,0,0,7,3,0,0],[0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[7,7,5,5,0,0,0,0,6,1,8,10,0,0,0,0,5,5,6,6,0,0,0,0,8,10,7,12,0,0,0,0,0,0,0,0,1,1,2,11,0,0,0,0,12,2,2,0,0,0,0,0,2,11,12,12,0,0,0,0,2,0,1,11],[0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0] >;
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4G | 4H | 4I | 4J | 4K | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | 2+ (1+4) | 2- (1+4) | Q8.15D6 | D4○D12 |
| kernel | C42.158D6 | C42⋊7S3 | D6.D4 | C12⋊D4 | D6⋊Q8 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 6 | 2 | 1 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{158}D_6 % in TeX
G:=Group("C4^2.158D6"); // GroupNames label
G:=SmallGroup(192,1259);
// by ID
G=gap.SmallGroup(192,1259);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,555,100,675,570,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀