metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.178D6, C6.402- (1+4), C6.842+ (1+4), C4⋊Q8⋊16S3, C4⋊C4.126D6, D6⋊3Q8⋊38C2, C4.D12⋊46C2, (C2×Q8).112D6, Dic3⋊5D4⋊44C2, C42⋊2S3⋊26C2, C42⋊7S3⋊27C2, 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), (C4×Dic3).166C22, (C2×Dic6).192C22, (C2×Dic3).274C23, 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
Subgroups: 560 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×11], C22, C22 [×9], S3 [×3], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], D4 [×4], Q8 [×4], C23 [×3], Dic3 [×5], C12 [×2], C12 [×6], D6 [×9], C2×C6, C42, C42 [×3], C22⋊C4 [×12], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4 [×3], C2×D4 [×3], C2×Q8 [×2], C2×Q8, Dic6 [×2], C4×S3 [×4], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×2], C22×S3, C22×S3 [×2], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C42⋊2C2 [×2], C4⋊Q8, C4×Dic3, C4×Dic3 [×2], Dic3⋊C4 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×2], D6⋊C4 [×10], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4, S3×C2×C4 [×2], C2×D12, C2×D12 [×2], C6×Q8 [×2], C22.36C24, C42⋊2S3, C42⋊7S3, Dic6⋊C4, Dic3⋊5D4, D6.D4 [×2], C12⋊D4, C4.D12, C4⋊C4⋊S3 [×2], D6⋊3Q8 [×2], C12.23D4 [×2], C3×C4⋊Q8, C42.178D6
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), Q8⋊3S3 [×2], S3×C23, C22.36C24, D4⋊6D6, C2×Q8⋊3S3, Q8.15D6, C42.178D6
Generators and relations
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 >
►On 96 points
(1 33 7 27)(2 28 8 34)(3 35 9 29)(4 30 10 36)(5 25 11 31)(6 32 12 26)(13 67 19 61)(14 62 20 68)(15 69 21 63)(16 64 22 70)(17 71 23 65)(18 66 24 72)(37 53 43 59)(38 60 44 54)(39 55 45 49)(40 50 46 56)(41 57 47 51)(42 52 48 58)(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 41 79 65)(2 66 80 42)(3 43 81 67)(4 68 82 44)(5 45 83 69)(6 70 84 46)(7 47 73 71)(8 72 74 48)(9 37 75 61)(10 62 76 38)(11 39 77 63)(12 64 78 40)(13 29 53 93)(14 94 54 30)(15 31 55 95)(16 96 56 32)(17 33 57 85)(18 86 58 34)(19 35 59 87)(20 88 60 36)(21 25 49 89)(22 90 50 26)(23 27 51 91)(24 92 52 28)
(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 61 7 67)(2 66 8 72)(3 71 9 65)(4 64 10 70)(5 69 11 63)(6 62 12 68)(13 85 19 91)(14 90 20 96)(15 95 21 89)(16 88 22 94)(17 93 23 87)(18 86 24 92)(25 55 31 49)(26 60 32 54)(27 53 33 59)(28 58 34 52)(29 51 35 57)(30 56 36 50)(37 73 43 79)(38 78 44 84)(39 83 45 77)(40 76 46 82)(41 81 47 75)(42 74 48 80)
G:=sub<Sym(96)| (1,33,7,27)(2,28,8,34)(3,35,9,29)(4,30,10,36)(5,25,11,31)(6,32,12,26)(13,67,19,61)(14,62,20,68)(15,69,21,63)(16,64,22,70)(17,71,23,65)(18,66,24,72)(37,53,43,59)(38,60,44,54)(39,55,45,49)(40,50,46,56)(41,57,47,51)(42,52,48,58)(73,91,79,85)(74,86,80,92)(75,93,81,87)(76,88,82,94)(77,95,83,89)(78,90,84,96), (1,41,79,65)(2,66,80,42)(3,43,81,67)(4,68,82,44)(5,45,83,69)(6,70,84,46)(7,47,73,71)(8,72,74,48)(9,37,75,61)(10,62,76,38)(11,39,77,63)(12,64,78,40)(13,29,53,93)(14,94,54,30)(15,31,55,95)(16,96,56,32)(17,33,57,85)(18,86,58,34)(19,35,59,87)(20,88,60,36)(21,25,49,89)(22,90,50,26)(23,27,51,91)(24,92,52,28), (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), (1,61,7,67)(2,66,8,72)(3,71,9,65)(4,64,10,70)(5,69,11,63)(6,62,12,68)(13,85,19,91)(14,90,20,96)(15,95,21,89)(16,88,22,94)(17,93,23,87)(18,86,24,92)(25,55,31,49)(26,60,32,54)(27,53,33,59)(28,58,34,52)(29,51,35,57)(30,56,36,50)(37,73,43,79)(38,78,44,84)(39,83,45,77)(40,76,46,82)(41,81,47,75)(42,74,48,80)>;
G:=Group( (1,33,7,27)(2,28,8,34)(3,35,9,29)(4,30,10,36)(5,25,11,31)(6,32,12,26)(13,67,19,61)(14,62,20,68)(15,69,21,63)(16,64,22,70)(17,71,23,65)(18,66,24,72)(37,53,43,59)(38,60,44,54)(39,55,45,49)(40,50,46,56)(41,57,47,51)(42,52,48,58)(73,91,79,85)(74,86,80,92)(75,93,81,87)(76,88,82,94)(77,95,83,89)(78,90,84,96), (1,41,79,65)(2,66,80,42)(3,43,81,67)(4,68,82,44)(5,45,83,69)(6,70,84,46)(7,47,73,71)(8,72,74,48)(9,37,75,61)(10,62,76,38)(11,39,77,63)(12,64,78,40)(13,29,53,93)(14,94,54,30)(15,31,55,95)(16,96,56,32)(17,33,57,85)(18,86,58,34)(19,35,59,87)(20,88,60,36)(21,25,49,89)(22,90,50,26)(23,27,51,91)(24,92,52,28), (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), (1,61,7,67)(2,66,8,72)(3,71,9,65)(4,64,10,70)(5,69,11,63)(6,62,12,68)(13,85,19,91)(14,90,20,96)(15,95,21,89)(16,88,22,94)(17,93,23,87)(18,86,24,92)(25,55,31,49)(26,60,32,54)(27,53,33,59)(28,58,34,52)(29,51,35,57)(30,56,36,50)(37,73,43,79)(38,78,44,84)(39,83,45,77)(40,76,46,82)(41,81,47,75)(42,74,48,80) );
G=PermutationGroup([(1,33,7,27),(2,28,8,34),(3,35,9,29),(4,30,10,36),(5,25,11,31),(6,32,12,26),(13,67,19,61),(14,62,20,68),(15,69,21,63),(16,64,22,70),(17,71,23,65),(18,66,24,72),(37,53,43,59),(38,60,44,54),(39,55,45,49),(40,50,46,56),(41,57,47,51),(42,52,48,58),(73,91,79,85),(74,86,80,92),(75,93,81,87),(76,88,82,94),(77,95,83,89),(78,90,84,96)], [(1,41,79,65),(2,66,80,42),(3,43,81,67),(4,68,82,44),(5,45,83,69),(6,70,84,46),(7,47,73,71),(8,72,74,48),(9,37,75,61),(10,62,76,38),(11,39,77,63),(12,64,78,40),(13,29,53,93),(14,94,54,30),(15,31,55,95),(16,96,56,32),(17,33,57,85),(18,86,58,34),(19,35,59,87),(20,88,60,36),(21,25,49,89),(22,90,50,26),(23,27,51,91),(24,92,52,28)], [(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)], [(1,61,7,67),(2,66,8,72),(3,71,9,65),(4,64,10,70),(5,69,11,63),(6,62,12,68),(13,85,19,91),(14,90,20,96),(15,95,21,89),(16,88,22,94),(17,93,23,87),(18,86,24,92),(25,55,31,49),(26,60,32,54),(27,53,33,59),(28,58,34,52),(29,51,35,57),(30,56,36,50),(37,73,43,79),(38,78,44,84),(39,83,45,77),(40,76,46,82),(41,81,47,75),(42,74,48,80)])
Matrix representation ►G ⊆ 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] >;
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 |
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