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, C12.3Q8⋊45C2, (C22×C4).378D6, C22.6(C2×Dic6), (C2×C12).140C23, C42⋊C2.11S3, 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
Subgroups: 456 in 222 conjugacy classes, 115 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C4 [×14], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], Q8 [×8], C23, Dic3 [×4], Dic3 [×6], C12 [×4], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C42 [×6], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic6 [×8], C2×Dic3 [×8], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×8], C22×C6, C2×C42, C42⋊C2, C42⋊C2, C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C4×Dic3 [×2], C4×Dic3 [×4], Dic3⋊C4 [×8], C4⋊Dic3 [×6], C6.D4 [×2], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6 [×4], C22×Dic3 [×2], C22×C12, C23.37C23, C4×Dic6 [×4], Dic3.D4 [×4], C12⋊Q8 [×2], C12.3Q8 [×2], C2×C4×Dic3, C23.26D6, C3×C42⋊C2, C42.88D6
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×4], C24, Dic6 [×4], C22×S3 [×7], C22×Q8, C2×C4○D4 [×2], C2×Dic6 [×6], S3×C23, C23.37C23, C22×Dic6, S3×C4○D4 [×2], C42.88D6
Generators and relations
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 >
►On 96 points
(1 50 10 92)(2 54 11 96)(3 52 12 94)(4 53 7 95)(5 51 8 93)(6 49 9 91)(13 86 40 71)(14 90 41 69)(15 88 42 67)(16 82 29 62)(17 80 30 66)(18 84 28 64)(19 59 24 74)(20 57 22 78)(21 55 23 76)(25 61 34 81)(26 65 35 79)(27 63 36 83)(31 58 38 73)(32 56 39 77)(33 60 37 75)(43 70 48 85)(44 68 46 89)(45 72 47 87)
(1 28 4 25)(2 29 5 26)(3 30 6 27)(7 34 10 18)(8 35 11 16)(9 36 12 17)(13 38 46 23)(14 39 47 24)(15 37 48 22)(19 41 32 45)(20 42 33 43)(21 40 31 44)(49 63 52 66)(50 64 53 61)(51 65 54 62)(55 71 58 68)(56 72 59 69)(57 67 60 70)(73 89 76 86)(74 90 77 87)(75 85 78 88)(79 96 82 93)(80 91 83 94)(81 92 84 95)
(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 22 7 33)(2 24 8 32)(3 23 9 31)(4 37 10 20)(5 39 11 19)(6 38 12 21)(13 36 44 30)(14 35 45 29)(15 34 43 28)(16 41 26 47)(17 40 27 46)(18 42 25 48)(49 58 94 76)(50 57 95 75)(51 56 96 74)(52 55 91 73)(53 60 92 78)(54 59 93 77)(61 70 84 88)(62 69 79 87)(63 68 80 86)(64 67 81 85)(65 72 82 90)(66 71 83 89)
G:=sub<Sym(96)| (1,50,10,92)(2,54,11,96)(3,52,12,94)(4,53,7,95)(5,51,8,93)(6,49,9,91)(13,86,40,71)(14,90,41,69)(15,88,42,67)(16,82,29,62)(17,80,30,66)(18,84,28,64)(19,59,24,74)(20,57,22,78)(21,55,23,76)(25,61,34,81)(26,65,35,79)(27,63,36,83)(31,58,38,73)(32,56,39,77)(33,60,37,75)(43,70,48,85)(44,68,46,89)(45,72,47,87), (1,28,4,25)(2,29,5,26)(3,30,6,27)(7,34,10,18)(8,35,11,16)(9,36,12,17)(13,38,46,23)(14,39,47,24)(15,37,48,22)(19,41,32,45)(20,42,33,43)(21,40,31,44)(49,63,52,66)(50,64,53,61)(51,65,54,62)(55,71,58,68)(56,72,59,69)(57,67,60,70)(73,89,76,86)(74,90,77,87)(75,85,78,88)(79,96,82,93)(80,91,83,94)(81,92,84,95), (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), (1,22,7,33)(2,24,8,32)(3,23,9,31)(4,37,10,20)(5,39,11,19)(6,38,12,21)(13,36,44,30)(14,35,45,29)(15,34,43,28)(16,41,26,47)(17,40,27,46)(18,42,25,48)(49,58,94,76)(50,57,95,75)(51,56,96,74)(52,55,91,73)(53,60,92,78)(54,59,93,77)(61,70,84,88)(62,69,79,87)(63,68,80,86)(64,67,81,85)(65,72,82,90)(66,71,83,89)>;
G:=Group( (1,50,10,92)(2,54,11,96)(3,52,12,94)(4,53,7,95)(5,51,8,93)(6,49,9,91)(13,86,40,71)(14,90,41,69)(15,88,42,67)(16,82,29,62)(17,80,30,66)(18,84,28,64)(19,59,24,74)(20,57,22,78)(21,55,23,76)(25,61,34,81)(26,65,35,79)(27,63,36,83)(31,58,38,73)(32,56,39,77)(33,60,37,75)(43,70,48,85)(44,68,46,89)(45,72,47,87), (1,28,4,25)(2,29,5,26)(3,30,6,27)(7,34,10,18)(8,35,11,16)(9,36,12,17)(13,38,46,23)(14,39,47,24)(15,37,48,22)(19,41,32,45)(20,42,33,43)(21,40,31,44)(49,63,52,66)(50,64,53,61)(51,65,54,62)(55,71,58,68)(56,72,59,69)(57,67,60,70)(73,89,76,86)(74,90,77,87)(75,85,78,88)(79,96,82,93)(80,91,83,94)(81,92,84,95), (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), (1,22,7,33)(2,24,8,32)(3,23,9,31)(4,37,10,20)(5,39,11,19)(6,38,12,21)(13,36,44,30)(14,35,45,29)(15,34,43,28)(16,41,26,47)(17,40,27,46)(18,42,25,48)(49,58,94,76)(50,57,95,75)(51,56,96,74)(52,55,91,73)(53,60,92,78)(54,59,93,77)(61,70,84,88)(62,69,79,87)(63,68,80,86)(64,67,81,85)(65,72,82,90)(66,71,83,89) );
G=PermutationGroup([(1,50,10,92),(2,54,11,96),(3,52,12,94),(4,53,7,95),(5,51,8,93),(6,49,9,91),(13,86,40,71),(14,90,41,69),(15,88,42,67),(16,82,29,62),(17,80,30,66),(18,84,28,64),(19,59,24,74),(20,57,22,78),(21,55,23,76),(25,61,34,81),(26,65,35,79),(27,63,36,83),(31,58,38,73),(32,56,39,77),(33,60,37,75),(43,70,48,85),(44,68,46,89),(45,72,47,87)], [(1,28,4,25),(2,29,5,26),(3,30,6,27),(7,34,10,18),(8,35,11,16),(9,36,12,17),(13,38,46,23),(14,39,47,24),(15,37,48,22),(19,41,32,45),(20,42,33,43),(21,40,31,44),(49,63,52,66),(50,64,53,61),(51,65,54,62),(55,71,58,68),(56,72,59,69),(57,67,60,70),(73,89,76,86),(74,90,77,87),(75,85,78,88),(79,96,82,93),(80,91,83,94),(81,92,84,95)], [(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)], [(1,22,7,33),(2,24,8,32),(3,23,9,31),(4,37,10,20),(5,39,11,19),(6,38,12,21),(13,36,44,30),(14,35,45,29),(15,34,43,28),(16,41,26,47),(17,40,27,46),(18,42,25,48),(49,58,94,76),(50,57,95,75),(51,56,96,74),(52,55,91,73),(53,60,92,78),(54,59,93,77),(61,70,84,88),(62,69,79,87),(63,68,80,86),(64,67,81,85),(65,72,82,90),(66,71,83,89)])
Matrix representation ►G ⊆ 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] >;
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 | C12.3Q8 | 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 |
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