metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.178D6, C6.842+ 1+4, C6.402- 1+4, C4⋊Q8⋊16S3, C4⋊C4.126D6, C4.D12⋊46C2, D6⋊3Q8⋊38C2, (C2×Q8).112D6, Dic3⋊5D4⋊44C2, C42⋊7S3⋊27C2, C42⋊2S3⋊26C2, D6.D4⋊47C2, C12⋊D4.13C2, (C2×C6).277C24, Dic6⋊C4⋊43C2, C12.140(C4○D4), C2.88(D4⋊6D6), C12.23D4⋊27C2, (C4×C12).218C22, (C2×C12).639C23, D6⋊C4.156C22, C4.23(Q8⋊3S3), (C6×Q8).144C22, (C2×D12).173C22, C4⋊Dic3.255C22, C22.298(S3×C23), Dic3⋊C4.169C22, (C22×S3).122C23, C2.41(Q8.15D6), (C2×Dic3).274C23, (C2×Dic6).192C22, (C4×Dic3).166C22, C3⋊11(C22.36C24), (C3×C4⋊Q8)⋊19C2, C4⋊C4⋊S3⋊47C2, C6.124(C2×C4○D4), (S3×C2×C4).150C22, C2.32(C2×Q8⋊3S3), (C3×C4⋊C4).220C22, (C2×C4).602(C22×S3), SmallGroup(192,1292)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.178D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c5 >
Subgroups: 560 in 216 conjugacy classes, 95 normal (43 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, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C6×Q8, C22.36C24, C42⋊2S3, C42⋊7S3, Dic6⋊C4, Dic3⋊5D4, D6.D4, C12⋊D4, C4.D12, C4⋊C4⋊S3, D6⋊3Q8, C12.23D4, C3×C4⋊Q8, C42.178D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, Q8⋊3S3, S3×C23, C22.36C24, D4⋊6D6, C2×Q8⋊3S3, Q8.15D6, C42.178D6
►On 96 points
(1 42 7 48)(2 37 8 43)(3 44 9 38)(4 39 10 45)(5 46 11 40)(6 41 12 47)(13 53 19 59)(14 60 20 54)(15 55 21 49)(16 50 22 56)(17 57 23 51)(18 52 24 58)(25 66 31 72)(26 61 32 67)(27 68 33 62)(28 63 34 69)(29 70 35 64)(30 65 36 71)(73 91 79 85)(74 86 80 92)(75 93 81 87)(76 88 82 94)(77 95 83 89)(78 90 84 96)
(1 28 18 74)(2 75 19 29)(3 30 20 76)(4 77 21 31)(5 32 22 78)(6 79 23 33)(7 34 24 80)(8 81 13 35)(9 36 14 82)(10 83 15 25)(11 26 16 84)(12 73 17 27)(37 93 59 70)(38 71 60 94)(39 95 49 72)(40 61 50 96)(41 85 51 62)(42 63 52 86)(43 87 53 64)(44 65 54 88)(45 89 55 66)(46 67 56 90)(47 91 57 68)(48 69 58 92)
(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 82 7 76)(2 75 8 81)(3 80 9 74)(4 73 10 79)(5 78 11 84)(6 83 12 77)(13 35 19 29)(14 28 20 34)(15 33 21 27)(16 26 22 32)(17 31 23 25)(18 36 24 30)(37 64 43 70)(38 69 44 63)(39 62 45 68)(40 67 46 61)(41 72 47 66)(42 65 48 71)(49 85 55 91)(50 90 56 96)(51 95 57 89)(52 88 58 94)(53 93 59 87)(54 86 60 92)
G:=sub<Sym(96)| (1,42,7,48)(2,37,8,43)(3,44,9,38)(4,39,10,45)(5,46,11,40)(6,41,12,47)(13,53,19,59)(14,60,20,54)(15,55,21,49)(16,50,22,56)(17,57,23,51)(18,52,24,58)(25,66,31,72)(26,61,32,67)(27,68,33,62)(28,63,34,69)(29,70,35,64)(30,65,36,71)(73,91,79,85)(74,86,80,92)(75,93,81,87)(76,88,82,94)(77,95,83,89)(78,90,84,96), (1,28,18,74)(2,75,19,29)(3,30,20,76)(4,77,21,31)(5,32,22,78)(6,79,23,33)(7,34,24,80)(8,81,13,35)(9,36,14,82)(10,83,15,25)(11,26,16,84)(12,73,17,27)(37,93,59,70)(38,71,60,94)(39,95,49,72)(40,61,50,96)(41,85,51,62)(42,63,52,86)(43,87,53,64)(44,65,54,88)(45,89,55,66)(46,67,56,90)(47,91,57,68)(48,69,58,92), (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), (1,82,7,76)(2,75,8,81)(3,80,9,74)(4,73,10,79)(5,78,11,84)(6,83,12,77)(13,35,19,29)(14,28,20,34)(15,33,21,27)(16,26,22,32)(17,31,23,25)(18,36,24,30)(37,64,43,70)(38,69,44,63)(39,62,45,68)(40,67,46,61)(41,72,47,66)(42,65,48,71)(49,85,55,91)(50,90,56,96)(51,95,57,89)(52,88,58,94)(53,93,59,87)(54,86,60,92)>;
G:=Group( (1,42,7,48)(2,37,8,43)(3,44,9,38)(4,39,10,45)(5,46,11,40)(6,41,12,47)(13,53,19,59)(14,60,20,54)(15,55,21,49)(16,50,22,56)(17,57,23,51)(18,52,24,58)(25,66,31,72)(26,61,32,67)(27,68,33,62)(28,63,34,69)(29,70,35,64)(30,65,36,71)(73,91,79,85)(74,86,80,92)(75,93,81,87)(76,88,82,94)(77,95,83,89)(78,90,84,96), (1,28,18,74)(2,75,19,29)(3,30,20,76)(4,77,21,31)(5,32,22,78)(6,79,23,33)(7,34,24,80)(8,81,13,35)(9,36,14,82)(10,83,15,25)(11,26,16,84)(12,73,17,27)(37,93,59,70)(38,71,60,94)(39,95,49,72)(40,61,50,96)(41,85,51,62)(42,63,52,86)(43,87,53,64)(44,65,54,88)(45,89,55,66)(46,67,56,90)(47,91,57,68)(48,69,58,92), (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), (1,82,7,76)(2,75,8,81)(3,80,9,74)(4,73,10,79)(5,78,11,84)(6,83,12,77)(13,35,19,29)(14,28,20,34)(15,33,21,27)(16,26,22,32)(17,31,23,25)(18,36,24,30)(37,64,43,70)(38,69,44,63)(39,62,45,68)(40,67,46,61)(41,72,47,66)(42,65,48,71)(49,85,55,91)(50,90,56,96)(51,95,57,89)(52,88,58,94)(53,93,59,87)(54,86,60,92) );
G=PermutationGroup([[(1,42,7,48),(2,37,8,43),(3,44,9,38),(4,39,10,45),(5,46,11,40),(6,41,12,47),(13,53,19,59),(14,60,20,54),(15,55,21,49),(16,50,22,56),(17,57,23,51),(18,52,24,58),(25,66,31,72),(26,61,32,67),(27,68,33,62),(28,63,34,69),(29,70,35,64),(30,65,36,71),(73,91,79,85),(74,86,80,92),(75,93,81,87),(76,88,82,94),(77,95,83,89),(78,90,84,96)], [(1,28,18,74),(2,75,19,29),(3,30,20,76),(4,77,21,31),(5,32,22,78),(6,79,23,33),(7,34,24,80),(8,81,13,35),(9,36,14,82),(10,83,15,25),(11,26,16,84),(12,73,17,27),(37,93,59,70),(38,71,60,94),(39,95,49,72),(40,61,50,96),(41,85,51,62),(42,63,52,86),(43,87,53,64),(44,65,54,88),(45,89,55,66),(46,67,56,90),(47,91,57,68),(48,69,58,92)], [(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)], [(1,82,7,76),(2,75,8,81),(3,80,9,74),(4,73,10,79),(5,78,11,84),(6,83,12,77),(13,35,19,29),(14,28,20,34),(15,33,21,27),(16,26,22,32),(17,31,23,25),(18,36,24,30),(37,64,43,70),(38,69,44,63),(39,62,45,68),(40,67,46,61),(41,72,47,66),(42,65,48,71),(49,85,55,91),(50,90,56,96),(51,95,57,89),(52,88,58,94),(53,93,59,87),(54,86,60,92)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 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 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | 2- 1+4 | Q8⋊3S3 | D4⋊6D6 | Q8.15D6 |
| kernel | C42.178D6 | C42⋊2S3 | C42⋊7S3 | Dic6⋊C4 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | C4.D12 | C4⋊C4⋊S3 | D6⋊3Q8 | C12.23D4 | C3×C4⋊Q8 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C42.178D6 ►in GL8(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 12 | 11 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 12 |
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 12 | 11 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 8 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 | 0 | 8 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 12 | 11 |
| 0 | 0 | 0 | 0 | 0 | 12 | 1 | 1 |
G:=sub<GL(8,GF(13))| [5,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,12,1,0,0,0,0,0,1,0,1,0,0,0,0,1,12,0,0,0,0,0,0,0,11,0,12],[5,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,11,1],[8,0,0,0,0,0,0,0,11,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,5,0,0,5,0,0,0,0,0,5,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,1,12,0,0,0,0,0,0,12,1,0,0,0,0,0,0,11,1] >;
C42.178D6 in GAP, Magma, Sage, TeX
C_4^2._{178}D_6 % in TeX
G:=Group("C4^2.178D6"); // GroupNames label
G:=SmallGroup(192,1292);
// by ID
G=gap.SmallGroup(192,1292);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,100,675,570,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations