metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.99D6, C6.542- (1+4), C6.992+ (1+4), (C4×D12)⋊11C2, C4⋊C4.274D6, D6⋊Q8⋊6C2, C12⋊2Q8⋊8C2, C42⋊7S3⋊5C2, (C4×Dic6)⋊12C2, C23.9D6⋊5C2, (C2×C6).78C24, D6⋊C4.4C22, C42⋊C2⋊18S3, C2.11(D4○D12), C12⋊7D4.18C2, (C4×C12).29C22, C2.12(Q8○D12), C22⋊C4.102D6, (C22×C4).215D6, C4.120(C4○D12), C12.236(C4○D4), C12.48D4⋊42C2, (C2×C12).151C23, C23.11D6⋊5C2, C4⋊Dic3.36C22, C23.99(C22×S3), (C2×D12).208C22, Dic3⋊C4.75C22, (C22×S3).26C23, C22.107(S3×C23), (C22×C6).148C23, (C2×Dic3).31C23, C6.D4.6C22, (C22×C12).308C22, C3⋊1(C22.36C24), (C4×Dic3).200C22, (C2×Dic6).233C22, C4⋊C4⋊S3⋊6C2, C6.34(C2×C4○D4), C2.37(C2×C4○D12), (S3×C2×C4).195C22, (C3×C42⋊C2)⋊20C2, (C3×C4⋊C4).314C22, (C2×C4).151(C22×S3), (C2×C3⋊D4).11C22, (C3×C22⋊C4).117C22, SmallGroup(192,1093)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 552 in 216 conjugacy classes, 95 normal (51 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×11], C22, C22 [×9], S3 [×2], C6 [×3], C6, C2×C4 [×6], C2×C4 [×10], D4 [×4], Q8 [×4], C23, C23 [×2], Dic3 [×6], C12 [×2], C12 [×5], D6 [×6], C2×C6, C2×C6 [×3], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8 [×3], Dic6 [×4], C4×S3 [×2], D12 [×2], C2×Dic3 [×6], C3⋊D4 [×2], C2×C12 [×6], C2×C12 [×2], C22×S3 [×2], C22×C6, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C42⋊2C2 [×2], C4⋊Q8, C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C4⋊Dic3 [×2], D6⋊C4 [×8], C6.D4 [×2], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, C2×Dic6 [×2], S3×C2×C4 [×2], C2×D12, C2×C3⋊D4 [×2], C22×C12, C22.36C24, C4×Dic6, C12⋊2Q8, C4×D12, C42⋊7S3, C23.9D6 [×2], C23.11D6 [×2], D6⋊Q8 [×2], C4⋊C4⋊S3 [×2], C12.48D4, C12⋊7D4, C3×C42⋊C2, C42.99D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C4○D12 [×2], S3×C23, C22.36C24, C2×C4○D12, D4○D12, Q8○D12, C42.99D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c-1 >
►On 96 points
(1 67 19 62)(2 68 20 63)(3 69 21 64)(4 70 22 65)(5 71 23 66)(6 72 24 61)(7 75 27 55)(8 76 28 56)(9 77 29 57)(10 78 30 58)(11 73 25 59)(12 74 26 60)(13 51 81 33)(14 52 82 34)(15 53 83 35)(16 54 84 36)(17 49 79 31)(18 50 80 32)(37 94 46 85)(38 95 47 86)(39 96 48 87)(40 91 43 88)(41 92 44 89)(42 93 45 90)
(1 53 12 43)(2 36 7 41)(3 49 8 45)(4 32 9 37)(5 51 10 47)(6 34 11 39)(13 58 95 66)(14 73 96 72)(15 60 91 62)(16 75 92 68)(17 56 93 64)(18 77 94 70)(19 35 26 40)(20 54 27 44)(21 31 28 42)(22 50 29 46)(23 33 30 38)(24 52 25 48)(55 89 63 84)(57 85 65 80)(59 87 61 82)(67 83 74 88)(69 79 76 90)(71 81 78 86)
(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 12 11)(2 10 7 5)(3 4 8 9)(13 84 95 89)(14 88 96 83)(15 82 91 87)(16 86 92 81)(17 80 93 85)(18 90 94 79)(19 24 26 25)(20 30 27 23)(21 22 28 29)(31 32 42 37)(33 36 38 41)(34 40 39 35)(43 48 53 52)(44 51 54 47)(45 46 49 50)(55 71 63 78)(56 77 64 70)(57 69 65 76)(58 75 66 68)(59 67 61 74)(60 73 62 72)
G:=sub<Sym(96)| (1,67,19,62)(2,68,20,63)(3,69,21,64)(4,70,22,65)(5,71,23,66)(6,72,24,61)(7,75,27,55)(8,76,28,56)(9,77,29,57)(10,78,30,58)(11,73,25,59)(12,74,26,60)(13,51,81,33)(14,52,82,34)(15,53,83,35)(16,54,84,36)(17,49,79,31)(18,50,80,32)(37,94,46,85)(38,95,47,86)(39,96,48,87)(40,91,43,88)(41,92,44,89)(42,93,45,90), (1,53,12,43)(2,36,7,41)(3,49,8,45)(4,32,9,37)(5,51,10,47)(6,34,11,39)(13,58,95,66)(14,73,96,72)(15,60,91,62)(16,75,92,68)(17,56,93,64)(18,77,94,70)(19,35,26,40)(20,54,27,44)(21,31,28,42)(22,50,29,46)(23,33,30,38)(24,52,25,48)(55,89,63,84)(57,85,65,80)(59,87,61,82)(67,83,74,88)(69,79,76,90)(71,81,78,86), (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), (1,6,12,11)(2,10,7,5)(3,4,8,9)(13,84,95,89)(14,88,96,83)(15,82,91,87)(16,86,92,81)(17,80,93,85)(18,90,94,79)(19,24,26,25)(20,30,27,23)(21,22,28,29)(31,32,42,37)(33,36,38,41)(34,40,39,35)(43,48,53,52)(44,51,54,47)(45,46,49,50)(55,71,63,78)(56,77,64,70)(57,69,65,76)(58,75,66,68)(59,67,61,74)(60,73,62,72)>;
G:=Group( (1,67,19,62)(2,68,20,63)(3,69,21,64)(4,70,22,65)(5,71,23,66)(6,72,24,61)(7,75,27,55)(8,76,28,56)(9,77,29,57)(10,78,30,58)(11,73,25,59)(12,74,26,60)(13,51,81,33)(14,52,82,34)(15,53,83,35)(16,54,84,36)(17,49,79,31)(18,50,80,32)(37,94,46,85)(38,95,47,86)(39,96,48,87)(40,91,43,88)(41,92,44,89)(42,93,45,90), (1,53,12,43)(2,36,7,41)(3,49,8,45)(4,32,9,37)(5,51,10,47)(6,34,11,39)(13,58,95,66)(14,73,96,72)(15,60,91,62)(16,75,92,68)(17,56,93,64)(18,77,94,70)(19,35,26,40)(20,54,27,44)(21,31,28,42)(22,50,29,46)(23,33,30,38)(24,52,25,48)(55,89,63,84)(57,85,65,80)(59,87,61,82)(67,83,74,88)(69,79,76,90)(71,81,78,86), (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), (1,6,12,11)(2,10,7,5)(3,4,8,9)(13,84,95,89)(14,88,96,83)(15,82,91,87)(16,86,92,81)(17,80,93,85)(18,90,94,79)(19,24,26,25)(20,30,27,23)(21,22,28,29)(31,32,42,37)(33,36,38,41)(34,40,39,35)(43,48,53,52)(44,51,54,47)(45,46,49,50)(55,71,63,78)(56,77,64,70)(57,69,65,76)(58,75,66,68)(59,67,61,74)(60,73,62,72) );
G=PermutationGroup([(1,67,19,62),(2,68,20,63),(3,69,21,64),(4,70,22,65),(5,71,23,66),(6,72,24,61),(7,75,27,55),(8,76,28,56),(9,77,29,57),(10,78,30,58),(11,73,25,59),(12,74,26,60),(13,51,81,33),(14,52,82,34),(15,53,83,35),(16,54,84,36),(17,49,79,31),(18,50,80,32),(37,94,46,85),(38,95,47,86),(39,96,48,87),(40,91,43,88),(41,92,44,89),(42,93,45,90)], [(1,53,12,43),(2,36,7,41),(3,49,8,45),(4,32,9,37),(5,51,10,47),(6,34,11,39),(13,58,95,66),(14,73,96,72),(15,60,91,62),(16,75,92,68),(17,56,93,64),(18,77,94,70),(19,35,26,40),(20,54,27,44),(21,31,28,42),(22,50,29,46),(23,33,30,38),(24,52,25,48),(55,89,63,84),(57,85,65,80),(59,87,61,82),(67,83,74,88),(69,79,76,90),(71,81,78,86)], [(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)], [(1,6,12,11),(2,10,7,5),(3,4,8,9),(13,84,95,89),(14,88,96,83),(15,82,91,87),(16,86,92,81),(17,80,93,85),(18,90,94,79),(19,24,26,25),(20,30,27,23),(21,22,28,29),(31,32,42,37),(33,36,38,41),(34,40,39,35),(43,48,53,52),(44,51,54,47),(45,46,49,50),(55,71,63,78),(56,77,64,70),(57,69,65,76),(58,75,66,68),(59,67,61,74),(60,73,62,72)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 |
| 0 | 0 | 7 | 3 | 0 | 0 |
| 0 | 0 | 9 | 0 | 3 | 6 |
| 0 | 0 | 0 | 4 | 7 | 10 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 5 | 5 | 11 | 12 |
| 0 | 0 | 8 | 8 | 8 | 0 |
| 0 | 0 | 9 | 8 | 8 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 7 | 2 | 1 | 1 |
| 0 | 0 | 12 | 7 | 12 | 0 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 9 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 7 | 2 | 1 | 1 |
| 0 | 0 | 12 | 7 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,7,9,0,0,0,6,3,0,4,0,0,0,0,3,7,0,0,0,0,6,10],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,5,8,9,0,0,0,5,8,8,0,0,12,11,8,8,0,0,1,12,0,0],[4,2,0,0,0,0,11,2,0,0,0,0,0,0,0,1,7,12,0,0,12,12,2,7,0,0,0,0,1,12,0,0,0,0,1,0],[11,9,0,0,0,0,11,2,0,0,0,0,0,0,1,0,7,12,0,0,12,12,2,7,0,0,0,0,1,0,0,0,0,0,1,12] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | ··· | 4O | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ (1+4) | 2- (1+4) | D4○D12 | Q8○D12 |
| kernel | C42.99D6 | C4×Dic6 | C12⋊2Q8 | C4×D12 | C42⋊7S3 | C23.9D6 | C23.11D6 | D6⋊Q8 | C4⋊C4⋊S3 | C12.48D4 | C12⋊7D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C12 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 8 | 1 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{99}D_6 % in TeX
G:=Group("C4^2.99D6"); // GroupNames label
G:=SmallGroup(192,1093);
// by ID
G=gap.SmallGroup(192,1093);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,100,675,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^-1>;
// generators/relations