metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.232D6, (C4×S3)⋊6Q8, C12⋊Q8⋊48C2, (C4×Q8)⋊15S3, D6.1(C2×Q8), C4.59(S3×Q8), C4⋊C4.297D6, (Q8×C12)⋊13C2, (C4×Dic6)⋊39C2, (S3×C42).6C2, (C2×Q8).201D6, D6⋊Q8.5C2, C12.117(C2×Q8), Dic3.2(C2×Q8), C4.47(C4○D12), C6.30(C22×Q8), Dic3.Q8⋊46C2, (C2×C6).122C24, C42⋊2S3.4C2, D6⋊3Q8.14C2, C4.D12.14C2, Dic3⋊Q8⋊33C2, C12.117(C4○D4), (C2×C12).499C23, (C4×C12).174C22, D6⋊C4.102C22, Dic3.4(C4○D4), (C6×Q8).222C22, C4⋊Dic3.307C22, C22.143(S3×C23), Dic3⋊C4.155C22, (C22×S3).179C23, C3⋊3(C23.37C23), (C2×Dic6).291C22, (C4×Dic3).253C22, (C2×Dic3).216C23, C2.13(C2×S3×Q8), C2.30(S3×C4○D4), C6.54(C2×C4○D4), C2.61(C2×C4○D12), (S3×C2×C4).295C22, (C3×C4⋊C4).350C22, (C2×C4).584(C22×S3), SmallGroup(192,1137)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 472 in 222 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×4], C4 [×14], C22, C22 [×4], S3 [×2], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×15], Q8 [×8], C23, Dic3 [×4], Dic3 [×5], C12 [×4], C12 [×5], D6 [×2], D6 [×2], C2×C6, C42, C42 [×2], C42 [×5], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×13], C22×C4 [×3], C2×Q8, C2×Q8 [×3], Dic6 [×6], C4×S3 [×4], C4×S3 [×4], C2×Dic3 [×3], C2×Dic3 [×4], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×2], C22×S3, C2×C42, C42⋊C2 [×2], C4×Q8, C4×Q8 [×3], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C4×Dic3 [×3], C4×Dic3 [×2], Dic3⋊C4 [×10], C4⋊Dic3, C4⋊Dic3 [×2], D6⋊C4 [×4], C4×C12, C4×C12 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C2×Dic6, C2×Dic6 [×2], S3×C2×C4 [×3], C6×Q8, C23.37C23, C4×Dic6, C4×Dic6 [×2], S3×C42, C42⋊2S3 [×2], C12⋊Q8, Dic3.Q8 [×2], D6⋊Q8 [×2], C4.D12, Dic3⋊Q8, D6⋊3Q8, Q8×C12, C42.232D6
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×4], C24, C22×S3 [×7], C22×Q8, C2×C4○D4 [×2], C4○D12 [×2], S3×Q8 [×2], S3×C23, C23.37C23, C2×C4○D12, C2×S3×Q8, S3×C4○D4, C42.232D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b2c5 >
►On 96 points
(1 66 7 72)(2 61 8 67)(3 68 9 62)(4 63 10 69)(5 70 11 64)(6 65 12 71)(13 76 19 82)(14 83 20 77)(15 78 21 84)(16 73 22 79)(17 80 23 74)(18 75 24 81)(25 88 31 94)(26 95 32 89)(27 90 33 96)(28 85 34 91)(29 92 35 86)(30 87 36 93)(37 49 43 55)(38 56 44 50)(39 51 45 57)(40 58 46 52)(41 53 47 59)(42 60 48 54)
(1 43 94 16)(2 44 95 17)(3 45 96 18)(4 46 85 19)(5 47 86 20)(6 48 87 21)(7 37 88 22)(8 38 89 23)(9 39 90 24)(10 40 91 13)(11 41 92 14)(12 42 93 15)(25 73 66 55)(26 74 67 56)(27 75 68 57)(28 76 69 58)(29 77 70 59)(30 78 71 60)(31 79 72 49)(32 80 61 50)(33 81 62 51)(34 82 63 52)(35 83 64 53)(36 84 65 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 6 88 93)(2 92 89 5)(3 4 90 91)(7 12 94 87)(8 86 95 11)(9 10 96 85)(13 45 46 24)(14 23 47 44)(15 43 48 22)(16 21 37 42)(17 41 38 20)(18 19 39 40)(25 30 72 65)(26 64 61 29)(27 28 62 63)(31 36 66 71)(32 70 67 35)(33 34 68 69)(49 54 73 78)(50 77 74 53)(51 52 75 76)(55 60 79 84)(56 83 80 59)(57 58 81 82)
G:=sub<Sym(96)| (1,66,7,72)(2,61,8,67)(3,68,9,62)(4,63,10,69)(5,70,11,64)(6,65,12,71)(13,76,19,82)(14,83,20,77)(15,78,21,84)(16,73,22,79)(17,80,23,74)(18,75,24,81)(25,88,31,94)(26,95,32,89)(27,90,33,96)(28,85,34,91)(29,92,35,86)(30,87,36,93)(37,49,43,55)(38,56,44,50)(39,51,45,57)(40,58,46,52)(41,53,47,59)(42,60,48,54), (1,43,94,16)(2,44,95,17)(3,45,96,18)(4,46,85,19)(5,47,86,20)(6,48,87,21)(7,37,88,22)(8,38,89,23)(9,39,90,24)(10,40,91,13)(11,41,92,14)(12,42,93,15)(25,73,66,55)(26,74,67,56)(27,75,68,57)(28,76,69,58)(29,77,70,59)(30,78,71,60)(31,79,72,49)(32,80,61,50)(33,81,62,51)(34,82,63,52)(35,83,64,53)(36,84,65,54), (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), (1,6,88,93)(2,92,89,5)(3,4,90,91)(7,12,94,87)(8,86,95,11)(9,10,96,85)(13,45,46,24)(14,23,47,44)(15,43,48,22)(16,21,37,42)(17,41,38,20)(18,19,39,40)(25,30,72,65)(26,64,61,29)(27,28,62,63)(31,36,66,71)(32,70,67,35)(33,34,68,69)(49,54,73,78)(50,77,74,53)(51,52,75,76)(55,60,79,84)(56,83,80,59)(57,58,81,82)>;
G:=Group( (1,66,7,72)(2,61,8,67)(3,68,9,62)(4,63,10,69)(5,70,11,64)(6,65,12,71)(13,76,19,82)(14,83,20,77)(15,78,21,84)(16,73,22,79)(17,80,23,74)(18,75,24,81)(25,88,31,94)(26,95,32,89)(27,90,33,96)(28,85,34,91)(29,92,35,86)(30,87,36,93)(37,49,43,55)(38,56,44,50)(39,51,45,57)(40,58,46,52)(41,53,47,59)(42,60,48,54), (1,43,94,16)(2,44,95,17)(3,45,96,18)(4,46,85,19)(5,47,86,20)(6,48,87,21)(7,37,88,22)(8,38,89,23)(9,39,90,24)(10,40,91,13)(11,41,92,14)(12,42,93,15)(25,73,66,55)(26,74,67,56)(27,75,68,57)(28,76,69,58)(29,77,70,59)(30,78,71,60)(31,79,72,49)(32,80,61,50)(33,81,62,51)(34,82,63,52)(35,83,64,53)(36,84,65,54), (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), (1,6,88,93)(2,92,89,5)(3,4,90,91)(7,12,94,87)(8,86,95,11)(9,10,96,85)(13,45,46,24)(14,23,47,44)(15,43,48,22)(16,21,37,42)(17,41,38,20)(18,19,39,40)(25,30,72,65)(26,64,61,29)(27,28,62,63)(31,36,66,71)(32,70,67,35)(33,34,68,69)(49,54,73,78)(50,77,74,53)(51,52,75,76)(55,60,79,84)(56,83,80,59)(57,58,81,82) );
G=PermutationGroup([(1,66,7,72),(2,61,8,67),(3,68,9,62),(4,63,10,69),(5,70,11,64),(6,65,12,71),(13,76,19,82),(14,83,20,77),(15,78,21,84),(16,73,22,79),(17,80,23,74),(18,75,24,81),(25,88,31,94),(26,95,32,89),(27,90,33,96),(28,85,34,91),(29,92,35,86),(30,87,36,93),(37,49,43,55),(38,56,44,50),(39,51,45,57),(40,58,46,52),(41,53,47,59),(42,60,48,54)], [(1,43,94,16),(2,44,95,17),(3,45,96,18),(4,46,85,19),(5,47,86,20),(6,48,87,21),(7,37,88,22),(8,38,89,23),(9,39,90,24),(10,40,91,13),(11,41,92,14),(12,42,93,15),(25,73,66,55),(26,74,67,56),(27,75,68,57),(28,76,69,58),(29,77,70,59),(30,78,71,60),(31,79,72,49),(32,80,61,50),(33,81,62,51),(34,82,63,52),(35,83,64,53),(36,84,65,54)], [(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)], [(1,6,88,93),(2,92,89,5),(3,4,90,91),(7,12,94,87),(8,86,95,11),(9,10,96,85),(13,45,46,24),(14,23,47,44),(15,43,48,22),(16,21,37,42),(17,41,38,20),(18,19,39,40),(25,30,72,65),(26,64,61,29),(27,28,62,63),(31,36,66,71),(32,70,67,35),(33,34,68,69),(49,54,73,78),(50,77,74,53),(51,52,75,76),(55,60,79,84),(56,83,80,59),(57,58,81,82)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 3 |
| 0 | 0 | 3 | 4 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 2 | 0 | 0 |
| 11 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 2 | 11 | 0 | 0 |
| 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,3,0,0,3,4],[8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[11,11,0,0,2,9,0,0,0,0,0,12,0,0,1,0],[2,9,0,0,11,11,0,0,0,0,0,12,0,0,1,0] >;
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 4S | 4T | 4U | 4V | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 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 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | S3×Q8 | S3×C4○D4 |
| kernel | C42.232D6 | C4×Dic6 | S3×C42 | C42⋊2S3 | C12⋊Q8 | Dic3.Q8 | D6⋊Q8 | C4.D12 | Dic3⋊Q8 | D6⋊3Q8 | Q8×C12 | C4×Q8 | C4×S3 | C42 | C4⋊C4 | C2×Q8 | Dic3 | C12 | C4 | C4 | C2 |
| # reps | 1 | 3 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 1 | 4 | 4 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{232}D_6 % in TeX
G:=Group("C4^2.232D6"); // GroupNames label
G:=SmallGroup(192,1137);
// by ID
G=gap.SmallGroup(192,1137);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,675,570,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^5>;
// generators/relations