metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.232D6, C4.Dic3⋊8C4, C12.33(C4⋊C4), C12.64(C2×Q8), (C2×C12).21Q8, C6.Q16⋊27C2, (C2×C12).471D4, (C2×C4).14Dic6, C4.29(C2×Dic6), (C22×C6).71D4, C2.1(D4⋊D6), C12.62(C22×C4), C42⋊C2.4S3, C12.Q8⋊27C2, (C22×C4).123D6, C6.104(C8⋊C22), C3⋊4(M4(2)⋊C4), C4.25(Dic3⋊C4), (C2×C12).325C23, C2.1(Q8.14D6), C23.60(C3⋊D4), C6.104(C8.C22), C22.8(Dic3⋊C4), C4⋊Dic3.324C22, (C22×C12).145C22, C3⋊C8⋊7(C2×C4), C4.87(S3×C2×C4), C6.39(C2×C4⋊C4), (C2×C4).43(C4×S3), (C2×C6).11(C4⋊C4), (C2×C12).87(C2×C4), (C2×C6).454(C2×D4), (C2×C3⋊C8).85C22, (C2×C4⋊Dic3).36C2, C2.13(C2×Dic3⋊C4), C22.70(C2×C3⋊D4), (C2×C4).240(C3⋊D4), (C3×C4⋊C4).263C22, (C3×C42⋊C2).4C2, (C2×C4).425(C22×S3), (C2×C4.Dic3).16C2, SmallGroup(192,554)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.232D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab-1, dcd-1=c-1 >
Subgroups: 264 in 118 conjugacy classes, 63 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C2×C12, C22×C6, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C2×C3⋊C8, C4.Dic3, C4⋊Dic3, C4⋊Dic3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, M4(2)⋊C4, C6.Q16, C12.Q8, C2×C4.Dic3, C2×C4⋊Dic3, C3×C42⋊C2, C4⋊C4.232D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C3⋊D4, C22×S3, C2×C4⋊C4, C8⋊C22, C8.C22, Dic3⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, M4(2)⋊C4, C2×Dic3⋊C4, D4⋊D6, Q8.14D6, C4⋊C4.232D6
►On 96 points
(1 22 4 19)(2 23 5 20)(3 24 6 21)(7 37 10 13)(8 38 11 14)(9 39 12 15)(16 25 36 41)(17 26 34 42)(18 27 35 40)(28 31 44 48)(29 32 45 46)(30 33 43 47)(49 96 52 93)(50 91 53 94)(51 92 54 95)(55 66 58 63)(56 61 59 64)(57 62 60 65)(67 89 70 86)(68 90 71 87)(69 85 72 88)(73 82 76 79)(74 83 77 80)(75 84 78 81)
(1 51 7 61)(2 49 8 65)(3 53 9 63)(4 54 10 64)(5 52 11 62)(6 50 12 66)(13 59 19 92)(14 57 20 96)(15 55 21 94)(16 87 32 81)(17 85 33 79)(18 89 31 83)(22 95 37 56)(23 93 38 60)(24 91 39 58)(25 71 45 78)(26 69 43 76)(27 67 44 74)(28 77 40 70)(29 75 41 68)(30 73 42 72)(34 88 47 82)(35 86 48 80)(36 90 46 84)
(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 33 7 17)(2 32 8 16)(3 31 9 18)(4 47 10 34)(5 46 11 36)(6 48 12 35)(13 26 19 43)(14 25 20 45)(15 27 21 44)(22 30 37 42)(23 29 38 41)(24 28 39 40)(49 71 65 78)(50 70 66 77)(51 69 61 76)(52 68 62 75)(53 67 63 74)(54 72 64 73)(55 80 94 86)(56 79 95 85)(57 84 96 90)(58 83 91 89)(59 82 92 88)(60 81 93 87)
G:=sub<Sym(96)| (1,22,4,19)(2,23,5,20)(3,24,6,21)(7,37,10,13)(8,38,11,14)(9,39,12,15)(16,25,36,41)(17,26,34,42)(18,27,35,40)(28,31,44,48)(29,32,45,46)(30,33,43,47)(49,96,52,93)(50,91,53,94)(51,92,54,95)(55,66,58,63)(56,61,59,64)(57,62,60,65)(67,89,70,86)(68,90,71,87)(69,85,72,88)(73,82,76,79)(74,83,77,80)(75,84,78,81), (1,51,7,61)(2,49,8,65)(3,53,9,63)(4,54,10,64)(5,52,11,62)(6,50,12,66)(13,59,19,92)(14,57,20,96)(15,55,21,94)(16,87,32,81)(17,85,33,79)(18,89,31,83)(22,95,37,56)(23,93,38,60)(24,91,39,58)(25,71,45,78)(26,69,43,76)(27,67,44,74)(28,77,40,70)(29,75,41,68)(30,73,42,72)(34,88,47,82)(35,86,48,80)(36,90,46,84), (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), (1,33,7,17)(2,32,8,16)(3,31,9,18)(4,47,10,34)(5,46,11,36)(6,48,12,35)(13,26,19,43)(14,25,20,45)(15,27,21,44)(22,30,37,42)(23,29,38,41)(24,28,39,40)(49,71,65,78)(50,70,66,77)(51,69,61,76)(52,68,62,75)(53,67,63,74)(54,72,64,73)(55,80,94,86)(56,79,95,85)(57,84,96,90)(58,83,91,89)(59,82,92,88)(60,81,93,87)>;
G:=Group( (1,22,4,19)(2,23,5,20)(3,24,6,21)(7,37,10,13)(8,38,11,14)(9,39,12,15)(16,25,36,41)(17,26,34,42)(18,27,35,40)(28,31,44,48)(29,32,45,46)(30,33,43,47)(49,96,52,93)(50,91,53,94)(51,92,54,95)(55,66,58,63)(56,61,59,64)(57,62,60,65)(67,89,70,86)(68,90,71,87)(69,85,72,88)(73,82,76,79)(74,83,77,80)(75,84,78,81), (1,51,7,61)(2,49,8,65)(3,53,9,63)(4,54,10,64)(5,52,11,62)(6,50,12,66)(13,59,19,92)(14,57,20,96)(15,55,21,94)(16,87,32,81)(17,85,33,79)(18,89,31,83)(22,95,37,56)(23,93,38,60)(24,91,39,58)(25,71,45,78)(26,69,43,76)(27,67,44,74)(28,77,40,70)(29,75,41,68)(30,73,42,72)(34,88,47,82)(35,86,48,80)(36,90,46,84), (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), (1,33,7,17)(2,32,8,16)(3,31,9,18)(4,47,10,34)(5,46,11,36)(6,48,12,35)(13,26,19,43)(14,25,20,45)(15,27,21,44)(22,30,37,42)(23,29,38,41)(24,28,39,40)(49,71,65,78)(50,70,66,77)(51,69,61,76)(52,68,62,75)(53,67,63,74)(54,72,64,73)(55,80,94,86)(56,79,95,85)(57,84,96,90)(58,83,91,89)(59,82,92,88)(60,81,93,87) );
G=PermutationGroup([[(1,22,4,19),(2,23,5,20),(3,24,6,21),(7,37,10,13),(8,38,11,14),(9,39,12,15),(16,25,36,41),(17,26,34,42),(18,27,35,40),(28,31,44,48),(29,32,45,46),(30,33,43,47),(49,96,52,93),(50,91,53,94),(51,92,54,95),(55,66,58,63),(56,61,59,64),(57,62,60,65),(67,89,70,86),(68,90,71,87),(69,85,72,88),(73,82,76,79),(74,83,77,80),(75,84,78,81)], [(1,51,7,61),(2,49,8,65),(3,53,9,63),(4,54,10,64),(5,52,11,62),(6,50,12,66),(13,59,19,92),(14,57,20,96),(15,55,21,94),(16,87,32,81),(17,85,33,79),(18,89,31,83),(22,95,37,56),(23,93,38,60),(24,91,39,58),(25,71,45,78),(26,69,43,76),(27,67,44,74),(28,77,40,70),(29,75,41,68),(30,73,42,72),(34,88,47,82),(35,86,48,80),(36,90,46,84)], [(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)], [(1,33,7,17),(2,32,8,16),(3,31,9,18),(4,47,10,34),(5,46,11,36),(6,48,12,35),(13,26,19,43),(14,25,20,45),(15,27,21,44),(22,30,37,42),(23,29,38,41),(24,28,39,40),(49,71,65,78),(50,70,66,77),(51,69,61,76),(52,68,62,75),(53,67,63,74),(54,72,64,73),(55,80,94,86),(56,79,95,85),(57,84,96,90),(58,83,91,89),(59,82,92,88),(60,81,93,87)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D4 | D6 | D6 | Dic6 | C4×S3 | C3⋊D4 | C3⋊D4 | C8⋊C22 | C8.C22 | D4⋊D6 | Q8.14D6 |
| kernel | C4⋊C4.232D6 | C6.Q16 | C12.Q8 | C2×C4.Dic3 | C2×C4⋊Dic3 | C3×C42⋊C2 | C4.Dic3 | C42⋊C2 | C2×C12 | 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 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 |
Matrix representation of C4⋊C4.232D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 59 | 0 | 0 |
| 0 | 0 | 14 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 14 |
| 0 | 0 | 0 | 0 | 59 | 66 |
| 7 | 59 | 0 | 0 | 0 | 0 |
| 14 | 66 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 25 | 37 | 0 | 0 | 0 | 0 |
| 62 | 48 | 0 | 0 | 0 | 0 |
| 0 | 0 | 54 | 29 | 0 | 0 |
| 0 | 0 | 48 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 42 |
| 0 | 0 | 0 | 0 | 70 | 28 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,14,0,0,0,0,59,7,0,0,0,0,0,0,7,59,0,0,0,0,14,66],[7,14,0,0,0,0,59,66,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[25,62,0,0,0,0,37,48,0,0,0,0,0,0,54,48,0,0,0,0,29,19,0,0,0,0,0,0,45,70,0,0,0,0,42,28] >;
C4⋊C4.232D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{232}D_6 % in TeX
G:=Group("C4:C4.232D6"); // GroupNames label
G:=SmallGroup(192,554);
// by ID
G=gap.SmallGroup(192,554);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,254,387,100,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a*b^-1,d*c*d^-1=c^-1>;
// generators/relations