metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D12⋊3C4, D12⋊20(C2×C4), C4⋊C4.226D6, (C2×C4).43D12, C4.60(C2×D12), C4.37(D6⋊C4), Dic6⋊18(C2×C4), C12.140(C2×D4), (C2×C12).134D4, C6.D8⋊24C2, C6.SD16⋊24C2, C6.83(C8⋊C22), C12.60(C22×C4), C22.4(D6⋊C4), (C22×C6).184D4, (C22×C4).109D6, C12.20(C22⋊C4), (C2×C12).319C23, C2.2(D12⋊6C22), C6.83(C8.C22), C23.83(C3⋊D4), C3⋊3(C23.36D4), C2.2(Q8.11D6), (C2×D12).233C22, (C22×C12).134C22, (C2×Dic6).260C22, (C6×C4⋊C4)⋊2C2, (C2×C4⋊C4)⋊2S3, C4.49(S3×C2×C4), (C2×C4).39(C4×S3), C2.12(C2×D6⋊C4), (C2×C12).78(C2×C4), (C2×C4○D12).6C2, (C2×C6).439(C2×D4), (C2×C3⋊C8).80C22, C6.39(C2×C22⋊C4), (C2×C4.Dic3)⋊8C2, C22.58(C2×C3⋊D4), (C2×C4).182(C3⋊D4), (C3×C4⋊C4).257C22, (C2×C6).58(C22⋊C4), (C2×C4).419(C22×S3), SmallGroup(192,525)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D12⋊C4
G = < a,b,c,d | a4=c2=d4=1, b6=a2, ab=ba, ac=ca, dad-1=a-1, cbc=a2b5, dbd-1=a2b, dcd-1=b3c >
Subgroups: 440 in 162 conjugacy classes, 63 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C2×C6, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, C22×C12, C23.36D4, C6.D8, C6.SD16, C2×C4.Dic3, C6×C4⋊C4, C2×C4○D12, C4○D12⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C8⋊C22, C8.C22, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C23.36D4, C2×D6⋊C4, D12⋊6C22, Q8.11D6, C4○D12⋊C4
►On 96 points
(1 60 7 54)(2 49 8 55)(3 50 9 56)(4 51 10 57)(5 52 11 58)(6 53 12 59)(13 70 19 64)(14 71 20 65)(15 72 21 66)(16 61 22 67)(17 62 23 68)(18 63 24 69)(25 76 31 82)(26 77 32 83)(27 78 33 84)(28 79 34 73)(29 80 35 74)(30 81 36 75)(37 93 43 87)(38 94 44 88)(39 95 45 89)(40 96 46 90)(41 85 47 91)(42 86 48 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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(26 36)(27 35)(28 34)(29 33)(30 32)(38 48)(39 47)(40 46)(41 45)(42 44)(49 58)(50 57)(51 56)(52 55)(53 54)(59 60)(61 66)(62 65)(63 64)(67 72)(68 71)(69 70)(73 79)(74 78)(75 77)(80 84)(81 83)(85 89)(86 88)(90 96)(91 95)(92 94)
(1 39 13 27)(2 46 14 34)(3 41 15 29)(4 48 16 36)(5 43 17 31)(6 38 18 26)(7 45 19 33)(8 40 20 28)(9 47 21 35)(10 42 22 30)(11 37 23 25)(12 44 24 32)(49 96 71 79)(50 91 72 74)(51 86 61 81)(52 93 62 76)(53 88 63 83)(54 95 64 78)(55 90 65 73)(56 85 66 80)(57 92 67 75)(58 87 68 82)(59 94 69 77)(60 89 70 84)
G:=sub<Sym(96)| (1,60,7,54)(2,49,8,55)(3,50,9,56)(4,51,10,57)(5,52,11,58)(6,53,12,59)(13,70,19,64)(14,71,20,65)(15,72,21,66)(16,61,22,67)(17,62,23,68)(18,63,24,69)(25,76,31,82)(26,77,32,83)(27,78,33,84)(28,79,34,73)(29,80,35,74)(30,81,36,75)(37,93,43,87)(38,94,44,88)(39,95,45,89)(40,96,46,90)(41,85,47,91)(42,86,48,92), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(26,36)(27,35)(28,34)(29,33)(30,32)(38,48)(39,47)(40,46)(41,45)(42,44)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,66)(62,65)(63,64)(67,72)(68,71)(69,70)(73,79)(74,78)(75,77)(80,84)(81,83)(85,89)(86,88)(90,96)(91,95)(92,94), (1,39,13,27)(2,46,14,34)(3,41,15,29)(4,48,16,36)(5,43,17,31)(6,38,18,26)(7,45,19,33)(8,40,20,28)(9,47,21,35)(10,42,22,30)(11,37,23,25)(12,44,24,32)(49,96,71,79)(50,91,72,74)(51,86,61,81)(52,93,62,76)(53,88,63,83)(54,95,64,78)(55,90,65,73)(56,85,66,80)(57,92,67,75)(58,87,68,82)(59,94,69,77)(60,89,70,84)>;
G:=Group( (1,60,7,54)(2,49,8,55)(3,50,9,56)(4,51,10,57)(5,52,11,58)(6,53,12,59)(13,70,19,64)(14,71,20,65)(15,72,21,66)(16,61,22,67)(17,62,23,68)(18,63,24,69)(25,76,31,82)(26,77,32,83)(27,78,33,84)(28,79,34,73)(29,80,35,74)(30,81,36,75)(37,93,43,87)(38,94,44,88)(39,95,45,89)(40,96,46,90)(41,85,47,91)(42,86,48,92), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(26,36)(27,35)(28,34)(29,33)(30,32)(38,48)(39,47)(40,46)(41,45)(42,44)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,66)(62,65)(63,64)(67,72)(68,71)(69,70)(73,79)(74,78)(75,77)(80,84)(81,83)(85,89)(86,88)(90,96)(91,95)(92,94), (1,39,13,27)(2,46,14,34)(3,41,15,29)(4,48,16,36)(5,43,17,31)(6,38,18,26)(7,45,19,33)(8,40,20,28)(9,47,21,35)(10,42,22,30)(11,37,23,25)(12,44,24,32)(49,96,71,79)(50,91,72,74)(51,86,61,81)(52,93,62,76)(53,88,63,83)(54,95,64,78)(55,90,65,73)(56,85,66,80)(57,92,67,75)(58,87,68,82)(59,94,69,77)(60,89,70,84) );
G=PermutationGroup([[(1,60,7,54),(2,49,8,55),(3,50,9,56),(4,51,10,57),(5,52,11,58),(6,53,12,59),(13,70,19,64),(14,71,20,65),(15,72,21,66),(16,61,22,67),(17,62,23,68),(18,63,24,69),(25,76,31,82),(26,77,32,83),(27,78,33,84),(28,79,34,73),(29,80,35,74),(30,81,36,75),(37,93,43,87),(38,94,44,88),(39,95,45,89),(40,96,46,90),(41,85,47,91),(42,86,48,92)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(26,36),(27,35),(28,34),(29,33),(30,32),(38,48),(39,47),(40,46),(41,45),(42,44),(49,58),(50,57),(51,56),(52,55),(53,54),(59,60),(61,66),(62,65),(63,64),(67,72),(68,71),(69,70),(73,79),(74,78),(75,77),(80,84),(81,83),(85,89),(86,88),(90,96),(91,95),(92,94)], [(1,39,13,27),(2,46,14,34),(3,41,15,29),(4,48,16,36),(5,43,17,31),(6,38,18,26),(7,45,19,33),(8,40,20,28),(9,47,21,35),(10,42,22,30),(11,37,23,25),(12,44,24,32),(49,96,71,79),(50,91,72,74),(51,86,61,81),(52,93,62,76),(53,88,63,83),(54,95,64,78),(55,90,65,73),(56,85,66,80),(57,92,67,75),(58,87,68,82),(59,94,69,77),(60,89,70,84)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6G | 8A | 8B | 8C | 8D | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | C3⋊D4 | C8⋊C22 | C8.C22 | D12⋊6C22 | Q8.11D6 |
| kernel | C4○D12⋊C4 | C6.D8 | C6.SD16 | C2×C4.Dic3 | C6×C4⋊C4 | C2×C4○D12 | C4○D12 | C2×C4⋊C4 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 3 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 |
Matrix representation of C4○D12⋊C4 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 13 |
| 0 | 0 | 0 | 0 | 60 | 43 |
| 0 | 0 | 43 | 60 | 0 | 0 |
| 0 | 0 | 13 | 30 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 56 | 32 | 15 | 41 |
| 0 | 0 | 41 | 15 | 32 | 56 |
| 0 | 0 | 15 | 41 | 17 | 41 |
| 0 | 0 | 32 | 56 | 32 | 58 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,43,13,0,0,0,0,60,30,0,0,30,60,0,0,0,0,13,43,0,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,72,72,0,0,0,0,1,0,0,0],[72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,56,41,15,32,0,0,32,15,41,56,0,0,15,32,17,32,0,0,41,56,41,58] >;
C4○D12⋊C4 in GAP, Magma, Sage, TeX
C_4\circ D_{12}\rtimes C_4 % in TeX
G:=Group("C4oD12:C4"); // GroupNames label
G:=SmallGroup(192,525);
// by ID
G=gap.SmallGroup(192,525);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,422,387,58,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^2=d^4=1,b^6=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=a^2*b^5,d*b*d^-1=a^2*b,d*c*d^-1=b^3*c>;
// generators/relations