metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.88D6, (C2×C12)⋊4Q8, C12⋊Q8⋊46C2, (C2×C4)⋊7Dic6, C4⋊C4.266D6, (C4×Dic6)⋊6C2, C12.68(C2×Q8), (C2×C6).61C24, C22⋊C4.89D6, C4.33(C2×Dic6), C6.10(C22×Q8), (C4×C12).21C22, C4.Dic6⋊45C2, (C22×C4).378D6, C22.6(C2×Dic6), C42⋊C2.11S3, (C2×C12).140C23, Dic3.1(C4○D4), C22.94(S3×C23), C2.12(C22×Dic6), C4⋊Dic3.360C22, (C22×C6).131C23, C23.160(C22×S3), Dic3.D4.5C2, Dic3⋊C4.105C22, (C22×C12).222C22, C3⋊2(C23.37C23), (C4×Dic3).249C22, (C2×Dic3).193C23, (C2×Dic6).228C22, C6.D4.91C22, C23.26D6.22C2, (C22×Dic3).214C22, C2.8(S3×C4○D4), (C2×C6).12(C2×Q8), C6.130(C2×C4○D4), (C2×C4×Dic3).14C2, (C3×C4⋊C4).302C22, (C2×C4).574(C22×S3), (C3×C42⋊C2).12C2, (C3×C22⋊C4).98C22, SmallGroup(192,1076)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.88D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 456 in 222 conjugacy classes, 115 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C42, C42⋊C2, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×Dic3, C22×C12, C23.37C23, C4×Dic6, Dic3.D4, C12⋊Q8, C4.Dic6, C2×C4×Dic3, C23.26D6, C3×C42⋊C2, C42.88D6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C24, Dic6, C22×S3, C22×Q8, C2×C4○D4, C2×Dic6, S3×C23, C23.37C23, C22×Dic6, S3×C4○D4, C42.88D6
►On 96 points
(1 75 15 84)(2 73 13 82)(3 77 14 80)(4 52 7 90)(5 50 8 88)(6 54 9 86)(10 87 37 49)(11 85 38 53)(12 89 39 51)(16 93 41 71)(17 91 42 69)(18 95 40 67)(19 78 24 81)(20 76 22 79)(21 74 23 83)(25 61 34 57)(26 65 35 55)(27 63 36 59)(28 64 31 60)(29 62 32 58)(30 66 33 56)(43 72 48 94)(44 70 46 92)(45 68 47 96)
(1 43 19 42)(2 44 20 40)(3 45 21 41)(4 25 37 28)(5 26 38 29)(6 27 39 30)(7 34 10 31)(8 35 11 32)(9 36 12 33)(13 46 22 18)(14 47 23 16)(15 48 24 17)(49 64 52 61)(50 65 53 62)(51 66 54 63)(55 85 58 88)(56 86 59 89)(57 87 60 90)(67 73 70 76)(68 74 71 77)(69 75 72 78)(79 95 82 92)(80 96 83 93)(81 91 84 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 38 24 8)(2 37 22 7)(3 39 23 9)(4 13 10 20)(5 15 11 19)(6 14 12 21)(16 36 45 30)(17 35 43 29)(18 34 44 28)(25 46 31 40)(26 48 32 42)(27 47 33 41)(49 76 90 82)(50 75 85 81)(51 74 86 80)(52 73 87 79)(53 78 88 84)(54 77 89 83)(55 94 62 69)(56 93 63 68)(57 92 64 67)(58 91 65 72)(59 96 66 71)(60 95 61 70)
G:=sub<Sym(96)| (1,75,15,84)(2,73,13,82)(3,77,14,80)(4,52,7,90)(5,50,8,88)(6,54,9,86)(10,87,37,49)(11,85,38,53)(12,89,39,51)(16,93,41,71)(17,91,42,69)(18,95,40,67)(19,78,24,81)(20,76,22,79)(21,74,23,83)(25,61,34,57)(26,65,35,55)(27,63,36,59)(28,64,31,60)(29,62,32,58)(30,66,33,56)(43,72,48,94)(44,70,46,92)(45,68,47,96), (1,43,19,42)(2,44,20,40)(3,45,21,41)(4,25,37,28)(5,26,38,29)(6,27,39,30)(7,34,10,31)(8,35,11,32)(9,36,12,33)(13,46,22,18)(14,47,23,16)(15,48,24,17)(49,64,52,61)(50,65,53,62)(51,66,54,63)(55,85,58,88)(56,86,59,89)(57,87,60,90)(67,73,70,76)(68,74,71,77)(69,75,72,78)(79,95,82,92)(80,96,83,93)(81,91,84,94), (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), (1,38,24,8)(2,37,22,7)(3,39,23,9)(4,13,10,20)(5,15,11,19)(6,14,12,21)(16,36,45,30)(17,35,43,29)(18,34,44,28)(25,46,31,40)(26,48,32,42)(27,47,33,41)(49,76,90,82)(50,75,85,81)(51,74,86,80)(52,73,87,79)(53,78,88,84)(54,77,89,83)(55,94,62,69)(56,93,63,68)(57,92,64,67)(58,91,65,72)(59,96,66,71)(60,95,61,70)>;
G:=Group( (1,75,15,84)(2,73,13,82)(3,77,14,80)(4,52,7,90)(5,50,8,88)(6,54,9,86)(10,87,37,49)(11,85,38,53)(12,89,39,51)(16,93,41,71)(17,91,42,69)(18,95,40,67)(19,78,24,81)(20,76,22,79)(21,74,23,83)(25,61,34,57)(26,65,35,55)(27,63,36,59)(28,64,31,60)(29,62,32,58)(30,66,33,56)(43,72,48,94)(44,70,46,92)(45,68,47,96), (1,43,19,42)(2,44,20,40)(3,45,21,41)(4,25,37,28)(5,26,38,29)(6,27,39,30)(7,34,10,31)(8,35,11,32)(9,36,12,33)(13,46,22,18)(14,47,23,16)(15,48,24,17)(49,64,52,61)(50,65,53,62)(51,66,54,63)(55,85,58,88)(56,86,59,89)(57,87,60,90)(67,73,70,76)(68,74,71,77)(69,75,72,78)(79,95,82,92)(80,96,83,93)(81,91,84,94), (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), (1,38,24,8)(2,37,22,7)(3,39,23,9)(4,13,10,20)(5,15,11,19)(6,14,12,21)(16,36,45,30)(17,35,43,29)(18,34,44,28)(25,46,31,40)(26,48,32,42)(27,47,33,41)(49,76,90,82)(50,75,85,81)(51,74,86,80)(52,73,87,79)(53,78,88,84)(54,77,89,83)(55,94,62,69)(56,93,63,68)(57,92,64,67)(58,91,65,72)(59,96,66,71)(60,95,61,70) );
G=PermutationGroup([[(1,75,15,84),(2,73,13,82),(3,77,14,80),(4,52,7,90),(5,50,8,88),(6,54,9,86),(10,87,37,49),(11,85,38,53),(12,89,39,51),(16,93,41,71),(17,91,42,69),(18,95,40,67),(19,78,24,81),(20,76,22,79),(21,74,23,83),(25,61,34,57),(26,65,35,55),(27,63,36,59),(28,64,31,60),(29,62,32,58),(30,66,33,56),(43,72,48,94),(44,70,46,92),(45,68,47,96)], [(1,43,19,42),(2,44,20,40),(3,45,21,41),(4,25,37,28),(5,26,38,29),(6,27,39,30),(7,34,10,31),(8,35,11,32),(9,36,12,33),(13,46,22,18),(14,47,23,16),(15,48,24,17),(49,64,52,61),(50,65,53,62),(51,66,54,63),(55,85,58,88),(56,86,59,89),(57,87,60,90),(67,73,70,76),(68,74,71,77),(69,75,72,78),(79,95,82,92),(80,96,83,93),(81,91,84,94)], [(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)], [(1,38,24,8),(2,37,22,7),(3,39,23,9),(4,13,10,20),(5,15,11,19),(6,14,12,21),(16,36,45,30),(17,35,43,29),(18,34,44,28),(25,46,31,40),(26,48,32,42),(27,47,33,41),(49,76,90,82),(50,75,85,81),(51,74,86,80),(52,73,87,79),(53,78,88,84),(54,77,89,83),(55,94,62,69),(56,93,63,68),(57,92,64,67),(58,91,65,72),(59,96,66,71),(60,95,61,70)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4R | 4S | 4T | 4U | 4V | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 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 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | D6 | C4○D4 | Dic6 | S3×C4○D4 |
| kernel | C42.88D6 | C4×Dic6 | Dic3.D4 | C12⋊Q8 | C4.Dic6 | C2×C4×Dic3 | C23.26D6 | C3×C42⋊C2 | C42⋊C2 | C2×C12 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | Dic3 | C2×C4 | C2 |
| # reps | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 8 | 8 | 4 |
Matrix representation of C42.88D6 ►in GL4(𝔽13) generated by
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 |
| 0 | 0 | 7 | 10 |
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 1 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 0 | 5 | 0 |
G:=sub<GL(4,GF(13))| [0,1,0,0,1,0,0,0,0,0,3,7,0,0,6,10],[5,0,0,0,0,5,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,12,1,0,0,12,0],[5,0,0,0,0,5,0,0,0,0,0,5,0,0,5,0] >;
C42.88D6 in GAP, Magma, Sage, TeX
C_4^2._{88}D_6 % in TeX
G:=Group("C4^2.88D6"); // GroupNames label
G:=SmallGroup(192,1076);
// by ID
G=gap.SmallGroup(192,1076);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,184,675,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations