metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.237D6, C4⋊C4.206D6, (S3×C42)⋊11C2, C4.D12⋊36C2, C42.C2⋊16S3, Dic3⋊5D4⋊35C2, C42⋊7S3⋊24C2, D6.10(C4○D4), C12⋊D4.11C2, D6.D4⋊33C2, (C2×C6).235C24, D6⋊C4.60C22, Dic6⋊C4⋊35C2, C12.128(C4○D4), (C2×C12).506C23, (C4×C12).195C22, C4.19(Q8⋊3S3), Dic3.44(C4○D4), (C2×D12).163C22, Dic3⋊C4.51C22, C4⋊Dic3.241C22, C22.256(S3×C23), (C22×S3).101C23, (C4×Dic3).142C22, (C2×Dic3).312C23, (C2×Dic6).179C22, C3⋊10(C23.36C23), C4⋊C4⋊S3⋊33C2, C4⋊C4⋊7S3⋊35C2, C2.86(S3×C4○D4), C6.197(C2×C4○D4), (C3×C42.C2)⋊8C2, (S3×C2×C4).125C22, (C2×C4).79(C22×S3), C2.22(C2×Q8⋊3S3), (C3×C4⋊C4).190C22, SmallGroup(192,1250)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 576 in 234 conjugacy classes, 99 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×12], C22, C22 [×10], S3 [×4], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×15], D4 [×6], Q8 [×2], C23 [×3], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×6], D6 [×2], D6 [×8], C2×C6, C42, C42 [×5], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic6 [×2], C4×S3 [×10], D12 [×6], C2×Dic3 [×3], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×4], C22×S3, C22×S3 [×2], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C4×Dic3 [×3], C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×10], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×4], C2×Dic6, S3×C2×C4 [×3], S3×C2×C4 [×2], C2×D12, C2×D12 [×2], C23.36C23, S3×C42, C42⋊7S3, Dic6⋊C4, C4⋊C4⋊7S3 [×2], Dic3⋊5D4, Dic3⋊5D4 [×2], D6.D4 [×2], C12⋊D4, C4.D12, C4⋊C4⋊S3 [×2], C3×C42.C2, C42.237D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×6], C24, C22×S3 [×7], C2×C4○D4 [×3], Q8⋊3S3 [×2], S3×C23, C23.36C23, C2×Q8⋊3S3, S3×C4○D4 [×2], C42.237D6
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c6=a2, ab=ba, cac-1=dad=ab2, cbc-1=dbd=a2b, dcd=a2c5 >
►On 96 points
(1 69 7 63)(2 90 8 96)(3 71 9 65)(4 92 10 86)(5 61 11 67)(6 94 12 88)(13 54 19 60)(14 33 20 27)(15 56 21 50)(16 35 22 29)(17 58 23 52)(18 25 24 31)(26 76 32 82)(28 78 34 84)(30 80 36 74)(37 93 43 87)(38 62 44 68)(39 95 45 89)(40 64 46 70)(41 85 47 91)(42 66 48 72)(49 77 55 83)(51 79 57 73)(53 81 59 75)
(1 81 45 18)(2 76 46 13)(3 83 47 20)(4 78 48 15)(5 73 37 22)(6 80 38 17)(7 75 39 24)(8 82 40 19)(9 77 41 14)(10 84 42 21)(11 79 43 16)(12 74 44 23)(25 69 59 89)(26 64 60 96)(27 71 49 91)(28 66 50 86)(29 61 51 93)(30 68 52 88)(31 63 53 95)(32 70 54 90)(33 65 55 85)(34 72 56 92)(35 67 57 87)(36 62 58 94)
(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 9)(2 8)(3 7)(4 6)(10 12)(14 24)(15 23)(16 22)(17 21)(18 20)(25 49)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(38 48)(39 47)(40 46)(41 45)(42 44)(61 93)(62 92)(63 91)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 96)(71 95)(72 94)(73 79)(74 78)(75 77)(80 84)(81 83)
G:=sub<Sym(96)| (1,69,7,63)(2,90,8,96)(3,71,9,65)(4,92,10,86)(5,61,11,67)(6,94,12,88)(13,54,19,60)(14,33,20,27)(15,56,21,50)(16,35,22,29)(17,58,23,52)(18,25,24,31)(26,76,32,82)(28,78,34,84)(30,80,36,74)(37,93,43,87)(38,62,44,68)(39,95,45,89)(40,64,46,70)(41,85,47,91)(42,66,48,72)(49,77,55,83)(51,79,57,73)(53,81,59,75), (1,81,45,18)(2,76,46,13)(3,83,47,20)(4,78,48,15)(5,73,37,22)(6,80,38,17)(7,75,39,24)(8,82,40,19)(9,77,41,14)(10,84,42,21)(11,79,43,16)(12,74,44,23)(25,69,59,89)(26,64,60,96)(27,71,49,91)(28,66,50,86)(29,61,51,93)(30,68,52,88)(31,63,53,95)(32,70,54,90)(33,65,55,85)(34,72,56,92)(35,67,57,87)(36,62,58,94), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(14,24)(15,23)(16,22)(17,21)(18,20)(25,49)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(38,48)(39,47)(40,46)(41,45)(42,44)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,96)(71,95)(72,94)(73,79)(74,78)(75,77)(80,84)(81,83)>;
G:=Group( (1,69,7,63)(2,90,8,96)(3,71,9,65)(4,92,10,86)(5,61,11,67)(6,94,12,88)(13,54,19,60)(14,33,20,27)(15,56,21,50)(16,35,22,29)(17,58,23,52)(18,25,24,31)(26,76,32,82)(28,78,34,84)(30,80,36,74)(37,93,43,87)(38,62,44,68)(39,95,45,89)(40,64,46,70)(41,85,47,91)(42,66,48,72)(49,77,55,83)(51,79,57,73)(53,81,59,75), (1,81,45,18)(2,76,46,13)(3,83,47,20)(4,78,48,15)(5,73,37,22)(6,80,38,17)(7,75,39,24)(8,82,40,19)(9,77,41,14)(10,84,42,21)(11,79,43,16)(12,74,44,23)(25,69,59,89)(26,64,60,96)(27,71,49,91)(28,66,50,86)(29,61,51,93)(30,68,52,88)(31,63,53,95)(32,70,54,90)(33,65,55,85)(34,72,56,92)(35,67,57,87)(36,62,58,94), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(14,24)(15,23)(16,22)(17,21)(18,20)(25,49)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(38,48)(39,47)(40,46)(41,45)(42,44)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,96)(71,95)(72,94)(73,79)(74,78)(75,77)(80,84)(81,83) );
G=PermutationGroup([(1,69,7,63),(2,90,8,96),(3,71,9,65),(4,92,10,86),(5,61,11,67),(6,94,12,88),(13,54,19,60),(14,33,20,27),(15,56,21,50),(16,35,22,29),(17,58,23,52),(18,25,24,31),(26,76,32,82),(28,78,34,84),(30,80,36,74),(37,93,43,87),(38,62,44,68),(39,95,45,89),(40,64,46,70),(41,85,47,91),(42,66,48,72),(49,77,55,83),(51,79,57,73),(53,81,59,75)], [(1,81,45,18),(2,76,46,13),(3,83,47,20),(4,78,48,15),(5,73,37,22),(6,80,38,17),(7,75,39,24),(8,82,40,19),(9,77,41,14),(10,84,42,21),(11,79,43,16),(12,74,44,23),(25,69,59,89),(26,64,60,96),(27,71,49,91),(28,66,50,86),(29,61,51,93),(30,68,52,88),(31,63,53,95),(32,70,54,90),(33,65,55,85),(34,72,56,92),(35,67,57,87),(36,62,58,94)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,12),(14,24),(15,23),(16,22),(17,21),(18,20),(25,49),(26,60),(27,59),(28,58),(29,57),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(38,48),(39,47),(40,46),(41,45),(42,44),(61,93),(62,92),(63,91),(64,90),(65,89),(66,88),(67,87),(68,86),(69,85),(70,96),(71,95),(72,94),(73,79),(74,78),(75,77),(80,84),(81,83)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 2 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [0,8,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,11,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,2,1,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | C4○D4 | C4○D4 | Q8⋊3S3 | S3×C4○D4 |
| kernel | C42.237D6 | S3×C42 | C42⋊7S3 | Dic6⋊C4 | C4⋊C4⋊7S3 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | C4.D12 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | Dic3 | C12 | D6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 6 | 4 | 4 | 4 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{237}D_6 % in TeX
G:=Group("C4^2.237D6"); // GroupNames label
G:=SmallGroup(192,1250);
// by ID
G=gap.SmallGroup(192,1250);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,1123,346,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^6=a^2,a*b=b*a,c*a*c^-1=d*a*d=a*b^2,c*b*c^-1=d*b*d=a^2*b,d*c*d=a^2*c^5>;
// generators/relations