metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.231D6, (C2×C12).286D4, (C2×C6).33SD16, C6.50(C2×SD16), C4.89(C4○D12), C6.SD16⋊26C2, C12.Q8⋊26C2, (C22×C4).113D6, (C22×C6).189D4, C12.177(C4○D4), (C2×C12).324C23, C12.55D4.4C2, C6.85(C8.C22), C3⋊4(C23.47D4), C23.87(C3⋊D4), C22.8(D4.S3), C12.48D4.10C2, C2.7(Q8.11D6), C4⋊Dic3.133C22, (C2×Dic6).97C22, (C22×C12).139C22, C6.61(C22.D4), C2.11(C23.28D6), (C6×C4⋊C4).8C2, (C2×C4⋊C4).9S3, C2.5(C2×D4.S3), (C2×C6).444(C2×D4), (C2×C3⋊C8).84C22, (C2×C4).34(C3⋊D4), (C3×C4⋊C4).262C22, (C2×C4).424(C22×S3), C22.133(C2×C3⋊D4), SmallGroup(192,530)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.231D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=b2c-1 >
Subgroups: 248 in 104 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, C22⋊C8, Q8⋊C4, C4.Q8, C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C22×C12, C23.47D4, C12.Q8, C6.SD16, C12.55D4, C12.48D4, C6×C4⋊C4, C4⋊C4.231D6
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, C2×SD16, C8.C22, D4.S3, C4○D12, C2×C3⋊D4, C23.47D4, C23.28D6, C2×D4.S3, Q8.11D6, C4⋊C4.231D6
►On 96 points
Generators in S96
(1 22 4 19)(2 23 5 20)(3 24 6 21)(7 16 10 13)(8 17 11 14)(9 18 12 15)(25 43 28 46)(26 44 29 47)(27 45 30 48)(31 37 34 40)(32 38 35 41)(33 39 36 42)(49 71 64 59)(50 72 65 60)(51 67 66 55)(52 68 61 56)(53 69 62 57)(54 70 63 58)(73 83 88 95)(74 84 89 96)(75 79 90 91)(76 80 85 92)(77 81 86 93)(78 82 87 94)
(1 31 7 25)(2 32 8 26)(3 33 9 27)(4 34 10 28)(5 35 11 29)(6 36 12 30)(13 43 19 37)(14 44 20 38)(15 45 21 39)(16 46 22 40)(17 47 23 41)(18 48 24 42)(49 92 52 95)(50 93 53 96)(51 94 54 91)(55 78 58 75)(56 73 59 76)(57 74 60 77)(61 83 64 80)(62 84 65 81)(63 79 66 82)(67 87 70 90)(68 88 71 85)(69 89 72 86)
(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 63 10 51)(2 65 11 53)(3 61 12 49)(4 54 7 66)(5 50 8 62)(6 52 9 64)(13 55 22 70)(14 57 23 72)(15 59 24 68)(16 67 19 58)(17 69 20 60)(18 71 21 56)(25 87 34 75)(26 89 35 77)(27 85 36 73)(28 78 31 90)(29 74 32 86)(30 76 33 88)(37 79 46 94)(38 81 47 96)(39 83 48 92)(40 91 43 82)(41 93 44 84)(42 95 45 80)
G:=sub<Sym(96)| (1,22,4,19)(2,23,5,20)(3,24,6,21)(7,16,10,13)(8,17,11,14)(9,18,12,15)(25,43,28,46)(26,44,29,47)(27,45,30,48)(31,37,34,40)(32,38,35,41)(33,39,36,42)(49,71,64,59)(50,72,65,60)(51,67,66,55)(52,68,61,56)(53,69,62,57)(54,70,63,58)(73,83,88,95)(74,84,89,96)(75,79,90,91)(76,80,85,92)(77,81,86,93)(78,82,87,94), (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,43,19,37)(14,44,20,38)(15,45,21,39)(16,46,22,40)(17,47,23,41)(18,48,24,42)(49,92,52,95)(50,93,53,96)(51,94,54,91)(55,78,58,75)(56,73,59,76)(57,74,60,77)(61,83,64,80)(62,84,65,81)(63,79,66,82)(67,87,70,90)(68,88,71,85)(69,89,72,86), (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), (1,63,10,51)(2,65,11,53)(3,61,12,49)(4,54,7,66)(5,50,8,62)(6,52,9,64)(13,55,22,70)(14,57,23,72)(15,59,24,68)(16,67,19,58)(17,69,20,60)(18,71,21,56)(25,87,34,75)(26,89,35,77)(27,85,36,73)(28,78,31,90)(29,74,32,86)(30,76,33,88)(37,79,46,94)(38,81,47,96)(39,83,48,92)(40,91,43,82)(41,93,44,84)(42,95,45,80)>;
G:=Group( (1,22,4,19)(2,23,5,20)(3,24,6,21)(7,16,10,13)(8,17,11,14)(9,18,12,15)(25,43,28,46)(26,44,29,47)(27,45,30,48)(31,37,34,40)(32,38,35,41)(33,39,36,42)(49,71,64,59)(50,72,65,60)(51,67,66,55)(52,68,61,56)(53,69,62,57)(54,70,63,58)(73,83,88,95)(74,84,89,96)(75,79,90,91)(76,80,85,92)(77,81,86,93)(78,82,87,94), (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,43,19,37)(14,44,20,38)(15,45,21,39)(16,46,22,40)(17,47,23,41)(18,48,24,42)(49,92,52,95)(50,93,53,96)(51,94,54,91)(55,78,58,75)(56,73,59,76)(57,74,60,77)(61,83,64,80)(62,84,65,81)(63,79,66,82)(67,87,70,90)(68,88,71,85)(69,89,72,86), (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), (1,63,10,51)(2,65,11,53)(3,61,12,49)(4,54,7,66)(5,50,8,62)(6,52,9,64)(13,55,22,70)(14,57,23,72)(15,59,24,68)(16,67,19,58)(17,69,20,60)(18,71,21,56)(25,87,34,75)(26,89,35,77)(27,85,36,73)(28,78,31,90)(29,74,32,86)(30,76,33,88)(37,79,46,94)(38,81,47,96)(39,83,48,92)(40,91,43,82)(41,93,44,84)(42,95,45,80) );
G=PermutationGroup([[(1,22,4,19),(2,23,5,20),(3,24,6,21),(7,16,10,13),(8,17,11,14),(9,18,12,15),(25,43,28,46),(26,44,29,47),(27,45,30,48),(31,37,34,40),(32,38,35,41),(33,39,36,42),(49,71,64,59),(50,72,65,60),(51,67,66,55),(52,68,61,56),(53,69,62,57),(54,70,63,58),(73,83,88,95),(74,84,89,96),(75,79,90,91),(76,80,85,92),(77,81,86,93),(78,82,87,94)], [(1,31,7,25),(2,32,8,26),(3,33,9,27),(4,34,10,28),(5,35,11,29),(6,36,12,30),(13,43,19,37),(14,44,20,38),(15,45,21,39),(16,46,22,40),(17,47,23,41),(18,48,24,42),(49,92,52,95),(50,93,53,96),(51,94,54,91),(55,78,58,75),(56,73,59,76),(57,74,60,77),(61,83,64,80),(62,84,65,81),(63,79,66,82),(67,87,70,90),(68,88,71,85),(69,89,72,86)], [(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)], [(1,63,10,51),(2,65,11,53),(3,61,12,49),(4,54,7,66),(5,50,8,62),(6,52,9,64),(13,55,22,70),(14,57,23,72),(15,59,24,68),(16,67,19,58),(17,69,20,60),(18,71,21,56),(25,87,34,75),(26,89,35,77),(27,85,36,73),(28,78,31,90),(29,74,32,86),(30,76,33,88),(37,79,46,94),(38,81,47,96),(39,83,48,92),(40,91,43,82),(41,93,44,84),(42,95,45,80)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 6A | ··· | 6G | 8A | 8B | 8C | 8D | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 24 | 24 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | SD16 | C3⋊D4 | C3⋊D4 | C4○D12 | C8.C22 | D4.S3 | Q8.11D6 |
| kernel | C4⋊C4.231D6 | C12.Q8 | C6.SD16 | C12.55D4 | C12.48D4 | C6×C4⋊C4 | C2×C4⋊C4 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C12 | C2×C6 | C2×C4 | C23 | C4 | C6 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 8 | 1 | 2 | 2 |
Matrix representation of C4⋊C4.231D6 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 28 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 26 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 56 | 71 |
| 0 | 0 | 0 | 0 | 71 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 47 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 71 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 29 | 0 | 0 | 0 | 0 |
| 68 | 61 | 0 | 0 | 0 | 0 |
| 0 | 0 | 61 | 29 | 0 | 0 |
| 0 | 0 | 58 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 17 |
| 0 | 0 | 0 | 0 | 17 | 41 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[27,28,0,0,0,0,0,46,0,0,0,0,0,0,1,26,0,0,0,0,0,72,0,0,0,0,0,0,56,71,0,0,0,0,71,17],[1,47,0,0,0,0,0,72,0,0,0,0,0,0,64,71,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,68,0,0,0,0,29,61,0,0,0,0,0,0,61,58,0,0,0,0,29,12,0,0,0,0,0,0,32,17,0,0,0,0,17,41] >;
C4⋊C4.231D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{231}D_6 % in TeX
G:=Group("C4:C4.231D6"); // GroupNames label
G:=SmallGroup(192,530);
// by ID
G=gap.SmallGroup(192,530);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,100,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2*b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=b^2*c^-1>;
// generators/relations