metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.177D6, C6.392- 1+4, C4⋊Q8⋊15S3, C4⋊C4.221D6, C4.D12⋊45C2, (Q8×Dic3)⋊24C2, (C4×Dic6)⋊54C2, (C4×D12).28C2, (C2×Q8).173D6, (C2×C6).276C24, C12.139(C4○D4), C4.19(D4⋊2S3), (C2×C12).109C23, (C4×C12).217C22, D6⋊C4.155C22, C4.41(Q8⋊3S3), C12.23D4.8C2, (C6×Q8).143C22, (C2×D12).274C22, C4⋊Dic3.386C22, C22.297(S3×C23), Dic3⋊C4.168C22, (C22×S3).121C23, C2.40(Q8.15D6), C3⋊8(C22.50C24), (C2×Dic6).304C22, (C4×Dic3).165C22, (C2×Dic3).146C23, (C3×C4⋊Q8)⋊18C2, C4⋊C4⋊S3⋊46C2, C4⋊C4⋊7S3⋊43C2, C6.123(C2×C4○D4), C2.66(C2×D4⋊2S3), (S3×C2×C4).149C22, C2.31(C2×Q8⋊3S3), (C3×C4⋊C4).219C22, (C2×C4).601(C22×S3), SmallGroup(192,1291)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.177D6
G = < a,b,c,d | a4=b4=1, c6=a2b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=b2c5 >
Subgroups: 480 in 212 conjugacy classes, 99 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C6×Q8, C22.50C24, C4×Dic6, C4×D12, C4⋊C4⋊7S3, C4.D12, C4⋊C4⋊S3, Q8×Dic3, C12.23D4, C3×C4⋊Q8, C42.177D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, D4⋊2S3, Q8⋊3S3, S3×C23, C22.50C24, C2×D4⋊2S3, C2×Q8⋊3S3, Q8.15D6, C42.177D6
►On 96 points
(1 89 32 52)(2 53 33 90)(3 91 34 54)(4 55 35 92)(5 93 36 56)(6 57 25 94)(7 95 26 58)(8 59 27 96)(9 85 28 60)(10 49 29 86)(11 87 30 50)(12 51 31 88)(13 37 82 66)(14 67 83 38)(15 39 84 68)(16 69 73 40)(17 41 74 70)(18 71 75 42)(19 43 76 72)(20 61 77 44)(21 45 78 62)(22 63 79 46)(23 47 80 64)(24 65 81 48)
(1 73 26 22)(2 23 27 74)(3 75 28 24)(4 13 29 76)(5 77 30 14)(6 15 31 78)(7 79 32 16)(8 17 33 80)(9 81 34 18)(10 19 35 82)(11 83 36 20)(12 21 25 84)(37 86 72 55)(38 56 61 87)(39 88 62 57)(40 58 63 89)(41 90 64 59)(42 60 65 91)(43 92 66 49)(44 50 67 93)(45 94 68 51)(46 52 69 95)(47 96 70 53)(48 54 71 85)
(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 32 25)(2 36 33 5)(3 4 34 35)(7 12 26 31)(8 30 27 11)(9 10 28 29)(13 75 82 18)(14 17 83 74)(15 73 84 16)(19 81 76 24)(20 23 77 80)(21 79 78 22)(37 71 66 42)(38 41 67 70)(39 69 68 40)(43 65 72 48)(44 47 61 64)(45 63 62 46)(49 85 86 60)(50 59 87 96)(51 95 88 58)(52 57 89 94)(53 93 90 56)(54 55 91 92)
G:=sub<Sym(96)| (1,89,32,52)(2,53,33,90)(3,91,34,54)(4,55,35,92)(5,93,36,56)(6,57,25,94)(7,95,26,58)(8,59,27,96)(9,85,28,60)(10,49,29,86)(11,87,30,50)(12,51,31,88)(13,37,82,66)(14,67,83,38)(15,39,84,68)(16,69,73,40)(17,41,74,70)(18,71,75,42)(19,43,76,72)(20,61,77,44)(21,45,78,62)(22,63,79,46)(23,47,80,64)(24,65,81,48), (1,73,26,22)(2,23,27,74)(3,75,28,24)(4,13,29,76)(5,77,30,14)(6,15,31,78)(7,79,32,16)(8,17,33,80)(9,81,34,18)(10,19,35,82)(11,83,36,20)(12,21,25,84)(37,86,72,55)(38,56,61,87)(39,88,62,57)(40,58,63,89)(41,90,64,59)(42,60,65,91)(43,92,66,49)(44,50,67,93)(45,94,68,51)(46,52,69,95)(47,96,70,53)(48,54,71,85), (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), (1,6,32,25)(2,36,33,5)(3,4,34,35)(7,12,26,31)(8,30,27,11)(9,10,28,29)(13,75,82,18)(14,17,83,74)(15,73,84,16)(19,81,76,24)(20,23,77,80)(21,79,78,22)(37,71,66,42)(38,41,67,70)(39,69,68,40)(43,65,72,48)(44,47,61,64)(45,63,62,46)(49,85,86,60)(50,59,87,96)(51,95,88,58)(52,57,89,94)(53,93,90,56)(54,55,91,92)>;
G:=Group( (1,89,32,52)(2,53,33,90)(3,91,34,54)(4,55,35,92)(5,93,36,56)(6,57,25,94)(7,95,26,58)(8,59,27,96)(9,85,28,60)(10,49,29,86)(11,87,30,50)(12,51,31,88)(13,37,82,66)(14,67,83,38)(15,39,84,68)(16,69,73,40)(17,41,74,70)(18,71,75,42)(19,43,76,72)(20,61,77,44)(21,45,78,62)(22,63,79,46)(23,47,80,64)(24,65,81,48), (1,73,26,22)(2,23,27,74)(3,75,28,24)(4,13,29,76)(5,77,30,14)(6,15,31,78)(7,79,32,16)(8,17,33,80)(9,81,34,18)(10,19,35,82)(11,83,36,20)(12,21,25,84)(37,86,72,55)(38,56,61,87)(39,88,62,57)(40,58,63,89)(41,90,64,59)(42,60,65,91)(43,92,66,49)(44,50,67,93)(45,94,68,51)(46,52,69,95)(47,96,70,53)(48,54,71,85), (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), (1,6,32,25)(2,36,33,5)(3,4,34,35)(7,12,26,31)(8,30,27,11)(9,10,28,29)(13,75,82,18)(14,17,83,74)(15,73,84,16)(19,81,76,24)(20,23,77,80)(21,79,78,22)(37,71,66,42)(38,41,67,70)(39,69,68,40)(43,65,72,48)(44,47,61,64)(45,63,62,46)(49,85,86,60)(50,59,87,96)(51,95,88,58)(52,57,89,94)(53,93,90,56)(54,55,91,92) );
G=PermutationGroup([[(1,89,32,52),(2,53,33,90),(3,91,34,54),(4,55,35,92),(5,93,36,56),(6,57,25,94),(7,95,26,58),(8,59,27,96),(9,85,28,60),(10,49,29,86),(11,87,30,50),(12,51,31,88),(13,37,82,66),(14,67,83,38),(15,39,84,68),(16,69,73,40),(17,41,74,70),(18,71,75,42),(19,43,76,72),(20,61,77,44),(21,45,78,62),(22,63,79,46),(23,47,80,64),(24,65,81,48)], [(1,73,26,22),(2,23,27,74),(3,75,28,24),(4,13,29,76),(5,77,30,14),(6,15,31,78),(7,79,32,16),(8,17,33,80),(9,81,34,18),(10,19,35,82),(11,83,36,20),(12,21,25,84),(37,86,72,55),(38,56,61,87),(39,88,62,57),(40,58,63,89),(41,90,64,59),(42,60,65,91),(43,92,66,49),(44,50,67,93),(45,94,68,51),(46,52,69,95),(47,96,70,53),(48,54,71,85)], [(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)], [(1,6,32,25),(2,36,33,5),(3,4,34,35),(7,12,26,31),(8,30,27,11),(9,10,28,29),(13,75,82,18),(14,17,83,74),(15,73,84,16),(19,81,76,24),(20,23,77,80),(21,79,78,22),(37,71,66,42),(38,41,67,70),(39,69,68,40),(43,65,72,48),(44,47,61,64),(45,63,62,46),(49,85,86,60),(50,59,87,96),(51,95,88,58),(52,57,89,94),(53,93,90,56),(54,55,91,92)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | 2- 1+4 | D4⋊2S3 | Q8⋊3S3 | Q8.15D6 |
| kernel | C42.177D6 | C4×Dic6 | C4×D12 | C4⋊C4⋊7S3 | C4.D12 | C4⋊C4⋊S3 | Q8×Dic3 | C12.23D4 | C3×C4⋊Q8 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 8 | 1 | 2 | 2 | 2 |
Matrix representation of C42.177D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 10 | 5 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 3 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 10 | 5 |
| 7 | 2 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 6 | 11 | 0 | 0 | 0 | 0 |
| 11 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 2 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,10,0,0,0,0,0,5],[4,3,0,0,0,0,3,9,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,10,0,0,0,0,0,5],[7,1,0,0,0,0,2,6,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,1,12],[6,11,0,0,0,0,11,7,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,2,0,0,0,0,12,1] >;
C42.177D6 in GAP, Magma, Sage, TeX
C_4^2._{177}D_6 % in TeX
G:=Group("C4^2.177D6"); // GroupNames label
G:=SmallGroup(192,1291);
// by ID
G=gap.SmallGroup(192,1291);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,268,1571,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2*b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^5>;
// generators/relations