metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.156D6, C6.972- (1+4), C4⋊C4.211D6, C12⋊2Q8⋊33C2, C42.C2⋊12S3, (C2×C12).92C23, C42⋊2S3.7C2, (C2×C6).242C24, C4.D12.12C2, C2.60(Q8○D12), C12.3Q8⋊37C2, Dic6⋊C4⋊38C2, C12.132(C4○D4), (C4×C12).201C22, D6⋊C4.113C22, C4.21(Q8⋊3S3), C4⋊Dic3.244C22, C22.263(S3×C23), Dic3⋊C4.125C22, (C22×S3).107C23, C3⋊5(C22.35C24), (C2×Dic3).262C23, (C2×Dic6).182C22, (C4×Dic3).147C22, C4⋊C4⋊S3.3C2, C6.119(C2×C4○D4), (S3×C2×C4).132C22, (C3×C42.C2)⋊15C2, C2.26(C2×Q8⋊3S3), (C3×C4⋊C4).197C22, (C2×C4).206(C22×S3), SmallGroup(192,1257)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 416 in 192 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×2], C2, C3, C4 [×2], C4 [×13], C22, C22 [×3], S3, C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×9], Q8 [×4], C23, Dic3 [×7], C12 [×2], C12 [×6], D6 [×3], C2×C6, C42, C42 [×5], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×14], C22×C4, C2×Q8 [×2], Dic6 [×4], C4×S3 [×2], C2×Dic3, C2×Dic3 [×6], C2×C12, C2×C12 [×6], C22×S3, C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2, C42.C2 [×4], C42⋊2C2 [×4], C4⋊Q8, C4×Dic3, C4×Dic3 [×4], Dic3⋊C4 [×6], C4⋊Dic3 [×8], D6⋊C4 [×6], C4×C12, C3×C4⋊C4 [×6], C2×Dic6 [×2], S3×C2×C4, C22.35C24, C12⋊2Q8, C42⋊2S3, Dic6⋊C4 [×2], C12.3Q8 [×4], C4.D12 [×2], C4⋊C4⋊S3 [×4], C3×C42.C2, C42.156D6
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], Q8⋊3S3 [×2], S3×C23, C22.35C24, C2×Q8⋊3S3, Q8○D12 [×2], C42.156D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=a2b2c5 >
►On 96 points
(1 68 18 45)(2 46 19 69)(3 70 20 47)(4 48 21 71)(5 72 22 37)(6 38 23 61)(7 62 24 39)(8 40 13 63)(9 64 14 41)(10 42 15 65)(11 66 16 43)(12 44 17 67)(25 96 74 53)(26 54 75 85)(27 86 76 55)(28 56 77 87)(29 88 78 57)(30 58 79 89)(31 90 80 59)(32 60 81 91)(33 92 82 49)(34 50 83 93)(35 94 84 51)(36 52 73 95)
(1 82 7 76)(2 28 8 34)(3 84 9 78)(4 30 10 36)(5 74 11 80)(6 32 12 26)(13 83 19 77)(14 29 20 35)(15 73 21 79)(16 31 22 25)(17 75 23 81)(18 33 24 27)(37 96 43 90)(38 60 44 54)(39 86 45 92)(40 50 46 56)(41 88 47 94)(42 52 48 58)(49 62 55 68)(51 64 57 70)(53 66 59 72)(61 91 67 85)(63 93 69 87)(65 95 71 89)
(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 18 23)(2 22 19 5)(3 4 20 21)(7 12 24 17)(8 16 13 11)(9 10 14 15)(25 83 74 34)(26 33 75 82)(27 81 76 32)(28 31 77 80)(29 79 78 30)(35 73 84 36)(37 46 72 69)(38 68 61 45)(39 44 62 67)(40 66 63 43)(41 42 64 65)(47 48 70 71)(49 85 92 54)(50 53 93 96)(51 95 94 52)(55 91 86 60)(56 59 87 90)(57 89 88 58)
G:=sub<Sym(96)| (1,68,18,45)(2,46,19,69)(3,70,20,47)(4,48,21,71)(5,72,22,37)(6,38,23,61)(7,62,24,39)(8,40,13,63)(9,64,14,41)(10,42,15,65)(11,66,16,43)(12,44,17,67)(25,96,74,53)(26,54,75,85)(27,86,76,55)(28,56,77,87)(29,88,78,57)(30,58,79,89)(31,90,80,59)(32,60,81,91)(33,92,82,49)(34,50,83,93)(35,94,84,51)(36,52,73,95), (1,82,7,76)(2,28,8,34)(3,84,9,78)(4,30,10,36)(5,74,11,80)(6,32,12,26)(13,83,19,77)(14,29,20,35)(15,73,21,79)(16,31,22,25)(17,75,23,81)(18,33,24,27)(37,96,43,90)(38,60,44,54)(39,86,45,92)(40,50,46,56)(41,88,47,94)(42,52,48,58)(49,62,55,68)(51,64,57,70)(53,66,59,72)(61,91,67,85)(63,93,69,87)(65,95,71,89), (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), (1,6,18,23)(2,22,19,5)(3,4,20,21)(7,12,24,17)(8,16,13,11)(9,10,14,15)(25,83,74,34)(26,33,75,82)(27,81,76,32)(28,31,77,80)(29,79,78,30)(35,73,84,36)(37,46,72,69)(38,68,61,45)(39,44,62,67)(40,66,63,43)(41,42,64,65)(47,48,70,71)(49,85,92,54)(50,53,93,96)(51,95,94,52)(55,91,86,60)(56,59,87,90)(57,89,88,58)>;
G:=Group( (1,68,18,45)(2,46,19,69)(3,70,20,47)(4,48,21,71)(5,72,22,37)(6,38,23,61)(7,62,24,39)(8,40,13,63)(9,64,14,41)(10,42,15,65)(11,66,16,43)(12,44,17,67)(25,96,74,53)(26,54,75,85)(27,86,76,55)(28,56,77,87)(29,88,78,57)(30,58,79,89)(31,90,80,59)(32,60,81,91)(33,92,82,49)(34,50,83,93)(35,94,84,51)(36,52,73,95), (1,82,7,76)(2,28,8,34)(3,84,9,78)(4,30,10,36)(5,74,11,80)(6,32,12,26)(13,83,19,77)(14,29,20,35)(15,73,21,79)(16,31,22,25)(17,75,23,81)(18,33,24,27)(37,96,43,90)(38,60,44,54)(39,86,45,92)(40,50,46,56)(41,88,47,94)(42,52,48,58)(49,62,55,68)(51,64,57,70)(53,66,59,72)(61,91,67,85)(63,93,69,87)(65,95,71,89), (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), (1,6,18,23)(2,22,19,5)(3,4,20,21)(7,12,24,17)(8,16,13,11)(9,10,14,15)(25,83,74,34)(26,33,75,82)(27,81,76,32)(28,31,77,80)(29,79,78,30)(35,73,84,36)(37,46,72,69)(38,68,61,45)(39,44,62,67)(40,66,63,43)(41,42,64,65)(47,48,70,71)(49,85,92,54)(50,53,93,96)(51,95,94,52)(55,91,86,60)(56,59,87,90)(57,89,88,58) );
G=PermutationGroup([(1,68,18,45),(2,46,19,69),(3,70,20,47),(4,48,21,71),(5,72,22,37),(6,38,23,61),(7,62,24,39),(8,40,13,63),(9,64,14,41),(10,42,15,65),(11,66,16,43),(12,44,17,67),(25,96,74,53),(26,54,75,85),(27,86,76,55),(28,56,77,87),(29,88,78,57),(30,58,79,89),(31,90,80,59),(32,60,81,91),(33,92,82,49),(34,50,83,93),(35,94,84,51),(36,52,73,95)], [(1,82,7,76),(2,28,8,34),(3,84,9,78),(4,30,10,36),(5,74,11,80),(6,32,12,26),(13,83,19,77),(14,29,20,35),(15,73,21,79),(16,31,22,25),(17,75,23,81),(18,33,24,27),(37,96,43,90),(38,60,44,54),(39,86,45,92),(40,50,46,56),(41,88,47,94),(42,52,48,58),(49,62,55,68),(51,64,57,70),(53,66,59,72),(61,91,67,85),(63,93,69,87),(65,95,71,89)], [(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)], [(1,6,18,23),(2,22,19,5),(3,4,20,21),(7,12,24,17),(8,16,13,11),(9,10,14,15),(25,83,74,34),(26,33,75,82),(27,81,76,32),(28,31,77,80),(29,79,78,30),(35,73,84,36),(37,46,72,69),(38,68,61,45),(39,44,62,67),(40,66,63,43),(41,42,64,65),(47,48,70,71),(49,85,92,54),(50,53,93,96),(51,95,94,52),(55,91,86,60),(56,59,87,90),(57,89,88,58)])
Matrix representation ►G ⊆ GL8(𝔽13)
| 12 | 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 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 6 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 11 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 | 7 | 12 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 8 | 2 |
| 0 | 0 | 0 | 0 | 8 | 1 | 0 | 5 |
| 8 | 5 | 6 | 7 | 0 | 0 | 0 | 0 |
| 8 | 3 | 6 | 12 | 0 | 0 | 0 | 0 |
| 6 | 7 | 5 | 8 | 0 | 0 | 0 | 0 |
| 6 | 12 | 5 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 | 0 | 3 |
| 0 | 0 | 0 | 0 | 8 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 8 | 8 | 1 |
| 5 | 8 | 7 | 6 | 0 | 0 | 0 | 0 |
| 3 | 8 | 12 | 6 | 0 | 0 | 0 | 0 |
| 7 | 6 | 8 | 5 | 0 | 0 | 0 | 0 |
| 12 | 6 | 10 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 | 2 | 10 |
| 0 | 0 | 0 | 0 | 8 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 5 | 0 | 12 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,6,11,1,0,0,0,0,0,1,0,11,0,0,0,0,1,0,12,7,0,0,0,0,0,1,0,12],[0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,8,0,12,8,0,0,0,0,2,5,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,0,2,5],[8,8,6,6,0,0,0,0,5,3,7,12,0,0,0,0,6,6,5,5,0,0,0,0,7,12,8,10,0,0,0,0,0,0,0,0,12,5,8,12,0,0,0,0,0,12,0,8,0,0,0,0,3,0,1,8,0,0,0,0,0,3,0,1],[5,3,7,12,0,0,0,0,8,8,6,6,0,0,0,0,7,12,8,10,0,0,0,0,6,6,5,5,0,0,0,0,0,0,0,0,12,3,8,1,0,0,0,0,0,1,0,5,0,0,0,0,3,2,1,0,0,0,0,0,0,10,0,12] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4Q | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 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 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | 2- (1+4) | Q8⋊3S3 | Q8○D12 |
| kernel | C42.156D6 | C12⋊2Q8 | C42⋊2S3 | Dic6⋊C4 | C12.3Q8 | C4.D12 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 1 | 1 | 1 | 6 | 4 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{156}D_6 % in TeX
G:=Group("C4^2.156D6"); // GroupNames label
G:=SmallGroup(192,1257);
// by ID
G=gap.SmallGroup(192,1257);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,268,675,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^5>;
// generators/relations