direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C4○D12, C42.275D6, (C2×C42)⋊9S3, (C4×D12)⋊53C2, D12⋊25(C2×C4), C6.5(C23×C4), C12⋊12(C4○D4), (S3×C42)⋊15C2, Dic6⋊24(C2×C4), (C4×Dic6)⋊55C2, (C2×C6).18C24, D6.1(C22×C4), C42⋊2S3⋊38C2, (C22×C4).453D6, (C2×C12).876C23, C12.118(C22×C4), (C4×C12).333C22, D6⋊C4.162C22, C22.15(S3×C23), Dic3.2(C22×C4), (C2×D12).285C22, C23.26D6⋊40C2, C4⋊Dic3.395C22, (C22×C6).380C23, C23.227(C22×S3), Dic3⋊C4.174C22, (C22×S3).147C23, (C22×C12).564C22, (C2×Dic3).174C23, (C4×Dic3).288C22, (C2×Dic6).314C22, C6.D4.139C22, C3⋊1(C4×C4○D4), (C2×C4×C12)⋊12C2, (C4×S3)⋊9(C2×C4), (C2×C4)⋊13(C4×S3), C3⋊D4⋊8(C2×C4), C4.117(S3×C2×C4), C6.6(C2×C4○D4), (C2×C12)⋊30(C2×C4), C22.9(S3×C2×C4), C2.7(S3×C22×C4), (C4×C3⋊D4)⋊62C2, C2.4(C2×C4○D12), (C2×C4○D12).26C2, (S3×C2×C4).287C22, (C2×C4).818(C22×S3), (C2×C6).148(C22×C4), (C2×C3⋊D4).145C22, SmallGroup(192,1033)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 632 in 310 conjugacy classes, 159 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×8], C4 [×10], C22, C22 [×2], C22 [×10], S3 [×4], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×26], D4 [×12], Q8 [×4], C23, C23 [×2], Dic3 [×4], Dic3 [×4], C12 [×8], C12 [×2], D6 [×4], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C42 [×2], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×6], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic6 [×4], C4×S3 [×8], C4×S3 [×8], D12 [×4], C2×Dic3 [×6], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×8], C2×C12 [×4], C22×S3 [×2], C22×C6, C2×C42, C2×C42 [×2], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C4×Dic3 [×6], Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4 [×2], C4×C12 [×2], C4×C12 [×2], C2×Dic6, S3×C2×C4 [×6], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, C22×C12 [×2], C4×C4○D4, C4×Dic6 [×2], S3×C42 [×2], C42⋊2S3 [×2], C4×D12 [×2], C23.26D6, C4×C3⋊D4 [×4], C2×C4×C12, C2×C4○D12, C4×C4○D12
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C4○D4 [×4], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, C2×C4○D4 [×2], S3×C2×C4 [×6], C4○D12 [×4], S3×C23, C4×C4○D4, S3×C22×C4, C2×C4○D12 [×2], C4×C4○D12
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >
►On 96 points
(1 13 26 71)(2 14 27 72)(3 15 28 61)(4 16 29 62)(5 17 30 63)(6 18 31 64)(7 19 32 65)(8 20 33 66)(9 21 34 67)(10 22 35 68)(11 23 36 69)(12 24 25 70)(37 76 91 56)(38 77 92 57)(39 78 93 58)(40 79 94 59)(41 80 95 60)(42 81 96 49)(43 82 85 50)(44 83 86 51)(45 84 87 52)(46 73 88 53)(47 74 89 54)(48 75 90 55)
(1 55 7 49)(2 56 8 50)(3 57 9 51)(4 58 10 52)(5 59 11 53)(6 60 12 54)(13 48 19 42)(14 37 20 43)(15 38 21 44)(16 39 22 45)(17 40 23 46)(18 41 24 47)(25 74 31 80)(26 75 32 81)(27 76 33 82)(28 77 34 83)(29 78 35 84)(30 79 36 73)(61 92 67 86)(62 93 68 87)(63 94 69 88)(64 95 70 89)(65 96 71 90)(66 85 72 91)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 29)(26 28)(30 36)(31 35)(32 34)(38 48)(39 47)(40 46)(41 45)(42 44)(49 51)(52 60)(53 59)(54 58)(55 57)(61 71)(62 70)(63 69)(64 68)(65 67)(73 79)(74 78)(75 77)(80 84)(81 83)(86 96)(87 95)(88 94)(89 93)(90 92)
G:=sub<Sym(96)| (1,13,26,71)(2,14,27,72)(3,15,28,61)(4,16,29,62)(5,17,30,63)(6,18,31,64)(7,19,32,65)(8,20,33,66)(9,21,34,67)(10,22,35,68)(11,23,36,69)(12,24,25,70)(37,76,91,56)(38,77,92,57)(39,78,93,58)(40,79,94,59)(41,80,95,60)(42,81,96,49)(43,82,85,50)(44,83,86,51)(45,84,87,52)(46,73,88,53)(47,74,89,54)(48,75,90,55), (1,55,7,49)(2,56,8,50)(3,57,9,51)(4,58,10,52)(5,59,11,53)(6,60,12,54)(13,48,19,42)(14,37,20,43)(15,38,21,44)(16,39,22,45)(17,40,23,46)(18,41,24,47)(25,74,31,80)(26,75,32,81)(27,76,33,82)(28,77,34,83)(29,78,35,84)(30,79,36,73)(61,92,67,86)(62,93,68,87)(63,94,69,88)(64,95,70,89)(65,96,71,90)(66,85,72,91), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,15)(16,24)(17,23)(18,22)(19,21)(25,29)(26,28)(30,36)(31,35)(32,34)(38,48)(39,47)(40,46)(41,45)(42,44)(49,51)(52,60)(53,59)(54,58)(55,57)(61,71)(62,70)(63,69)(64,68)(65,67)(73,79)(74,78)(75,77)(80,84)(81,83)(86,96)(87,95)(88,94)(89,93)(90,92)>;
G:=Group( (1,13,26,71)(2,14,27,72)(3,15,28,61)(4,16,29,62)(5,17,30,63)(6,18,31,64)(7,19,32,65)(8,20,33,66)(9,21,34,67)(10,22,35,68)(11,23,36,69)(12,24,25,70)(37,76,91,56)(38,77,92,57)(39,78,93,58)(40,79,94,59)(41,80,95,60)(42,81,96,49)(43,82,85,50)(44,83,86,51)(45,84,87,52)(46,73,88,53)(47,74,89,54)(48,75,90,55), (1,55,7,49)(2,56,8,50)(3,57,9,51)(4,58,10,52)(5,59,11,53)(6,60,12,54)(13,48,19,42)(14,37,20,43)(15,38,21,44)(16,39,22,45)(17,40,23,46)(18,41,24,47)(25,74,31,80)(26,75,32,81)(27,76,33,82)(28,77,34,83)(29,78,35,84)(30,79,36,73)(61,92,67,86)(62,93,68,87)(63,94,69,88)(64,95,70,89)(65,96,71,90)(66,85,72,91), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,15)(16,24)(17,23)(18,22)(19,21)(25,29)(26,28)(30,36)(31,35)(32,34)(38,48)(39,47)(40,46)(41,45)(42,44)(49,51)(52,60)(53,59)(54,58)(55,57)(61,71)(62,70)(63,69)(64,68)(65,67)(73,79)(74,78)(75,77)(80,84)(81,83)(86,96)(87,95)(88,94)(89,93)(90,92) );
G=PermutationGroup([(1,13,26,71),(2,14,27,72),(3,15,28,61),(4,16,29,62),(5,17,30,63),(6,18,31,64),(7,19,32,65),(8,20,33,66),(9,21,34,67),(10,22,35,68),(11,23,36,69),(12,24,25,70),(37,76,91,56),(38,77,92,57),(39,78,93,58),(40,79,94,59),(41,80,95,60),(42,81,96,49),(43,82,85,50),(44,83,86,51),(45,84,87,52),(46,73,88,53),(47,74,89,54),(48,75,90,55)], [(1,55,7,49),(2,56,8,50),(3,57,9,51),(4,58,10,52),(5,59,11,53),(6,60,12,54),(13,48,19,42),(14,37,20,43),(15,38,21,44),(16,39,22,45),(17,40,23,46),(18,41,24,47),(25,74,31,80),(26,75,32,81),(27,76,33,82),(28,77,34,83),(29,78,35,84),(30,79,36,73),(61,92,67,86),(62,93,68,87),(63,94,69,88),(64,95,70,89),(65,96,71,90),(66,85,72,91)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,15),(16,24),(17,23),(18,22),(19,21),(25,29),(26,28),(30,36),(31,35),(32,34),(38,48),(39,47),(40,46),(41,45),(42,44),(49,51),(52,60),(53,59),(54,58),(55,57),(61,71),(62,70),(63,69),(64,68),(65,67),(73,79),(74,78),(75,77),(80,84),(81,83),(86,96),(87,95),(88,94),(89,93),(90,92)])
Matrix representation ►G ⊆ GL3(𝔽13) generated by
| 8 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 6 | 3 |
| 0 | 10 | 3 |
| 12 | 0 | 0 |
| 0 | 0 | 12 |
| 0 | 12 | 0 |
G:=sub<GL(3,GF(13))| [8,0,0,0,12,0,0,0,12],[1,0,0,0,8,0,0,0,8],[1,0,0,0,6,10,0,3,3],[12,0,0,0,0,12,0,12,0] >;
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | ··· | 4L | 4M | ··· | 4R | 4S | ··· | 4AD | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | C4○D4 | C4×S3 | C4○D12 |
| kernel | C4×C4○D12 | C4×Dic6 | S3×C42 | C42⋊2S3 | C4×D12 | C23.26D6 | C4×C3⋊D4 | C2×C4×C12 | C2×C4○D12 | C4○D12 | C2×C42 | C42 | C22×C4 | C12 | C2×C4 | C4 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 4 | 1 | 1 | 16 | 1 | 4 | 3 | 8 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_4\times C_4\circ D_{12} % in TeX
G:=Group("C4xC4oD12"); // GroupNames label
G:=SmallGroup(192,1033);
// by ID
G=gap.SmallGroup(192,1033);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^5>;
// generators/relations