metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊C4⋊7C4, C6.60(C4×D4), (C2×C12).252D4, (C22×C4).59D6, C6.91(C4⋊D4), C22.113(S3×D4), C6.C42⋊42C2, (C2×Dic3).178D4, C2.7(D6.D4), C6.52(C4.4D4), C2.20(Dic3⋊5D4), C6.38(C42⋊C2), C6.28(C42⋊2C2), C2.6(C23.14D6), C2.3(C12.23D4), C22.59(C4○D12), (S3×C23).19C22, (C22×C6).354C23, (C22×C12).30C22, C23.308(C22×S3), C3⋊7(C24.C22), C22.60(D4⋊2S3), C22.30(Q8⋊3S3), C6.52(C22.D4), (C22×Dic3).193C22, (C2×C4⋊C4)⋊8S3, (C6×C4⋊C4)⋊25C2, (C2×C4).42(C4×S3), (C2×C4×Dic3)⋊25C2, C2.14(C4×C3⋊D4), (C2×D6⋊C4).13C2, (C2×C6).335(C2×D4), C22.139(S3×C2×C4), C2.7(C4⋊C4⋊S3), (C2×C12).186(C2×C4), C2.14(C4⋊C4⋊7S3), C22.69(C2×C3⋊D4), (C2×C6).157(C4○D4), (C2×C4).100(C3⋊D4), (C22×S3).22(C2×C4), (C2×C6).122(C22×C4), (C2×Dic3).64(C2×C4), SmallGroup(192,549)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊C4⋊7C4
G = < a,b,c,d | a6=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, cbc-1=a3b, dbd-1=bc2, dcd-1=c-1 >
Subgroups: 536 in 190 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×Dic3, D6⋊C4, D6⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, S3×C23, C24.C22, C6.C42, C2×C4×Dic3, C2×D6⋊C4, C6×C4⋊C4, D6⋊C4⋊7C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C3⋊D4, C22×S3, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, Q8⋊3S3, C2×C3⋊D4, C24.C22, C4⋊C4⋊7S3, Dic3⋊5D4, D6.D4, C4⋊C4⋊S3, C4×C3⋊D4, C23.14D6, C12.23D4, D6⋊C4⋊7C4
►On 96 points
(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)(2 5)(3 4)(7 95)(8 94)(9 93)(10 92)(11 91)(12 96)(13 18)(14 17)(15 16)(19 20)(21 24)(22 23)(25 30)(26 29)(27 28)(31 40)(32 39)(33 38)(34 37)(35 42)(36 41)(43 52)(44 51)(45 50)(46 49)(47 54)(48 53)(55 59)(56 58)(61 63)(64 66)(67 71)(68 70)(73 75)(76 78)(79 85)(80 90)(81 89)(82 88)(83 87)(84 86)
(1 64 16 56)(2 65 17 57)(3 66 18 58)(4 61 13 59)(5 62 14 60)(6 63 15 55)(7 51 91 43)(8 52 92 44)(9 53 93 45)(10 54 94 46)(11 49 95 47)(12 50 96 48)(19 75 27 67)(20 76 28 68)(21 77 29 69)(22 78 30 70)(23 73 25 71)(24 74 26 72)(31 87 39 79)(32 88 40 80)(33 89 41 81)(34 90 42 82)(35 85 37 83)(36 86 38 84)
(1 44 20 32)(2 45 21 33)(3 46 22 34)(4 47 23 35)(5 48 24 36)(6 43 19 31)(7 67 87 55)(8 68 88 56)(9 69 89 57)(10 70 90 58)(11 71 85 59)(12 72 86 60)(13 49 25 37)(14 50 26 38)(15 51 27 39)(16 52 28 40)(17 53 29 41)(18 54 30 42)(61 95 73 83)(62 96 74 84)(63 91 75 79)(64 92 76 80)(65 93 77 81)(66 94 78 82)
G:=sub<Sym(96)| (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)(2,5)(3,4)(7,95)(8,94)(9,93)(10,92)(11,91)(12,96)(13,18)(14,17)(15,16)(19,20)(21,24)(22,23)(25,30)(26,29)(27,28)(31,40)(32,39)(33,38)(34,37)(35,42)(36,41)(43,52)(44,51)(45,50)(46,49)(47,54)(48,53)(55,59)(56,58)(61,63)(64,66)(67,71)(68,70)(73,75)(76,78)(79,85)(80,90)(81,89)(82,88)(83,87)(84,86), (1,64,16,56)(2,65,17,57)(3,66,18,58)(4,61,13,59)(5,62,14,60)(6,63,15,55)(7,51,91,43)(8,52,92,44)(9,53,93,45)(10,54,94,46)(11,49,95,47)(12,50,96,48)(19,75,27,67)(20,76,28,68)(21,77,29,69)(22,78,30,70)(23,73,25,71)(24,74,26,72)(31,87,39,79)(32,88,40,80)(33,89,41,81)(34,90,42,82)(35,85,37,83)(36,86,38,84), (1,44,20,32)(2,45,21,33)(3,46,22,34)(4,47,23,35)(5,48,24,36)(6,43,19,31)(7,67,87,55)(8,68,88,56)(9,69,89,57)(10,70,90,58)(11,71,85,59)(12,72,86,60)(13,49,25,37)(14,50,26,38)(15,51,27,39)(16,52,28,40)(17,53,29,41)(18,54,30,42)(61,95,73,83)(62,96,74,84)(63,91,75,79)(64,92,76,80)(65,93,77,81)(66,94,78,82)>;
G:=Group( (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)(2,5)(3,4)(7,95)(8,94)(9,93)(10,92)(11,91)(12,96)(13,18)(14,17)(15,16)(19,20)(21,24)(22,23)(25,30)(26,29)(27,28)(31,40)(32,39)(33,38)(34,37)(35,42)(36,41)(43,52)(44,51)(45,50)(46,49)(47,54)(48,53)(55,59)(56,58)(61,63)(64,66)(67,71)(68,70)(73,75)(76,78)(79,85)(80,90)(81,89)(82,88)(83,87)(84,86), (1,64,16,56)(2,65,17,57)(3,66,18,58)(4,61,13,59)(5,62,14,60)(6,63,15,55)(7,51,91,43)(8,52,92,44)(9,53,93,45)(10,54,94,46)(11,49,95,47)(12,50,96,48)(19,75,27,67)(20,76,28,68)(21,77,29,69)(22,78,30,70)(23,73,25,71)(24,74,26,72)(31,87,39,79)(32,88,40,80)(33,89,41,81)(34,90,42,82)(35,85,37,83)(36,86,38,84), (1,44,20,32)(2,45,21,33)(3,46,22,34)(4,47,23,35)(5,48,24,36)(6,43,19,31)(7,67,87,55)(8,68,88,56)(9,69,89,57)(10,70,90,58)(11,71,85,59)(12,72,86,60)(13,49,25,37)(14,50,26,38)(15,51,27,39)(16,52,28,40)(17,53,29,41)(18,54,30,42)(61,95,73,83)(62,96,74,84)(63,91,75,79)(64,92,76,80)(65,93,77,81)(66,94,78,82) );
G=PermutationGroup([[(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),(2,5),(3,4),(7,95),(8,94),(9,93),(10,92),(11,91),(12,96),(13,18),(14,17),(15,16),(19,20),(21,24),(22,23),(25,30),(26,29),(27,28),(31,40),(32,39),(33,38),(34,37),(35,42),(36,41),(43,52),(44,51),(45,50),(46,49),(47,54),(48,53),(55,59),(56,58),(61,63),(64,66),(67,71),(68,70),(73,75),(76,78),(79,85),(80,90),(81,89),(82,88),(83,87),(84,86)], [(1,64,16,56),(2,65,17,57),(3,66,18,58),(4,61,13,59),(5,62,14,60),(6,63,15,55),(7,51,91,43),(8,52,92,44),(9,53,93,45),(10,54,94,46),(11,49,95,47),(12,50,96,48),(19,75,27,67),(20,76,28,68),(21,77,29,69),(22,78,30,70),(23,73,25,71),(24,74,26,72),(31,87,39,79),(32,88,40,80),(33,89,41,81),(34,90,42,82),(35,85,37,83),(36,86,38,84)], [(1,44,20,32),(2,45,21,33),(3,46,22,34),(4,47,23,35),(5,48,24,36),(6,43,19,31),(7,67,87,55),(8,68,88,56),(9,69,89,57),(10,70,90,58),(11,71,85,59),(12,72,86,60),(13,49,25,37),(14,50,26,38),(15,51,27,39),(16,52,28,40),(17,53,29,41),(18,54,30,42),(61,95,73,83),(62,96,74,84),(63,91,75,79),(64,92,76,80),(65,93,77,81),(66,94,78,82)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | C4○D4 | C4×S3 | C3⋊D4 | C4○D12 | S3×D4 | D4⋊2S3 | Q8⋊3S3 |
| kernel | D6⋊C4⋊7C4 | C6.C42 | C2×C4×Dic3 | C2×D6⋊C4 | C6×C4⋊C4 | D6⋊C4 | C2×C4⋊C4 | C2×Dic3 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 3 | 1 | 8 | 1 | 2 | 2 | 3 | 8 | 4 | 4 | 4 | 1 | 1 | 2 |
Matrix representation of D6⋊C4⋊7C4 ►in GL6(𝔽13)
| 1 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 4 | 0 | 0 | 0 | 0 |
| 9 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[11,9,0,0,0,0,4,2,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D6⋊C4⋊7C4 in GAP, Magma, Sage, TeX
D_6\rtimes C_4\rtimes_7C_4
% in TeX
G:=Group("D6:C4:7C4"); // GroupNames label
G:=SmallGroup(192,549);
// by ID
G=gap.SmallGroup(192,549);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,758,387,58,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^4=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,d*b*d^-1=b*c^2,d*c*d^-1=c^-1>;
// generators/relations