metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.237D6, C4.64(C2×D12), (C2×C4).47D12, C4.24(D6⋊C4), (C2×Dic6)⋊11C4, (C2×C12).473D4, C12.144(C2×D4), (C22×C6).77D4, C6.SD16⋊28C2, C42⋊C2.9S3, C12.66(C22×C4), Dic6.27(C2×C4), (C22×C4).128D6, C12.48(C22⋊C4), (C2×C12).331C23, C22.24(D6⋊C4), C2.2(Q8.14D6), C23.62(C3⋊D4), C3⋊2(C23.38D4), C6.106(C8.C22), (C22×Dic6).12C2, (C22×C12).153C22, (C2×Dic6).263C22, C4.53(S3×C2×C4), (C2×C4).45(C4×S3), C2.19(C2×D6⋊C4), (C2×C12).93(C2×C4), (C2×C6).460(C2×D4), C6.46(C2×C22⋊C4), (C2×C3⋊C8).88C22, C22.74(C2×C3⋊D4), (C2×C4).242(C3⋊D4), (C3×C4⋊C4).268C22, (C2×C6).16(C22⋊C4), (C2×C4).431(C22×S3), (C2×C4.Dic3).19C2, (C3×C42⋊C2).10C2, SmallGroup(192,563)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.237D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab-1, dcd-1=c-1 >
Subgroups: 360 in 150 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], Q8 [×10], C23, Dic3 [×4], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4, C2×Q8 [×9], C3⋊C8 [×2], Dic6 [×4], Dic6 [×6], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C22×C6, Q8⋊C4 [×4], C42⋊C2, C2×M4(2), C22×Q8, C2×C3⋊C8 [×2], C4.Dic3 [×2], C4×C12, C3×C22⋊C4, C3×C4⋊C4 [×2], C2×Dic6 [×6], C2×Dic6 [×3], C22×Dic3, C22×C12, C23.38D4, C6.SD16 [×4], C2×C4.Dic3, C3×C42⋊C2, C22×Dic6, C4⋊C4.237D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C8.C22 [×2], D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C23.38D4, C2×D6⋊C4, Q8.14D6 [×2], C4⋊C4.237D6
►On 96 points
(1 19 4 22)(2 20 5 23)(3 21 6 24)(7 16 10 13)(8 17 11 14)(9 18 12 15)(25 36 39 32)(26 34 37 33)(27 35 38 31)(28 46 44 42)(29 47 45 40)(30 48 43 41)(49 96 52 93)(50 91 53 94)(51 92 54 95)(55 87 58 90)(56 88 59 85)(57 89 60 86)(61 69 64 72)(62 70 65 67)(63 71 66 68)(73 82 76 79)(74 83 77 80)(75 84 78 81)
(1 91 16 70)(2 95 17 68)(3 93 18 72)(4 94 13 67)(5 92 14 71)(6 96 15 69)(7 65 22 53)(8 63 23 51)(9 61 24 49)(10 62 19 50)(11 66 20 54)(12 64 21 52)(25 58 40 81)(26 56 41 79)(27 60 42 83)(28 74 35 89)(29 78 36 87)(30 76 34 85)(31 86 44 77)(32 90 45 75)(33 88 43 73)(37 59 48 82)(38 57 46 80)(39 55 47 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 4 34)(2 32 5 36)(3 31 6 35)(7 41 10 48)(8 40 11 47)(9 42 12 46)(13 30 16 43)(14 29 17 45)(15 28 18 44)(19 37 22 26)(20 39 23 25)(21 38 24 27)(49 74 52 77)(50 73 53 76)(51 78 54 75)(55 68 58 71)(56 67 59 70)(57 72 60 69)(61 89 64 86)(62 88 65 85)(63 87 66 90)(79 94 82 91)(80 93 83 96)(81 92 84 95)
G:=sub<Sym(96)| (1,19,4,22)(2,20,5,23)(3,21,6,24)(7,16,10,13)(8,17,11,14)(9,18,12,15)(25,36,39,32)(26,34,37,33)(27,35,38,31)(28,46,44,42)(29,47,45,40)(30,48,43,41)(49,96,52,93)(50,91,53,94)(51,92,54,95)(55,87,58,90)(56,88,59,85)(57,89,60,86)(61,69,64,72)(62,70,65,67)(63,71,66,68)(73,82,76,79)(74,83,77,80)(75,84,78,81), (1,91,16,70)(2,95,17,68)(3,93,18,72)(4,94,13,67)(5,92,14,71)(6,96,15,69)(7,65,22,53)(8,63,23,51)(9,61,24,49)(10,62,19,50)(11,66,20,54)(12,64,21,52)(25,58,40,81)(26,56,41,79)(27,60,42,83)(28,74,35,89)(29,78,36,87)(30,76,34,85)(31,86,44,77)(32,90,45,75)(33,88,43,73)(37,59,48,82)(38,57,46,80)(39,55,47,84), (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), (1,33,4,34)(2,32,5,36)(3,31,6,35)(7,41,10,48)(8,40,11,47)(9,42,12,46)(13,30,16,43)(14,29,17,45)(15,28,18,44)(19,37,22,26)(20,39,23,25)(21,38,24,27)(49,74,52,77)(50,73,53,76)(51,78,54,75)(55,68,58,71)(56,67,59,70)(57,72,60,69)(61,89,64,86)(62,88,65,85)(63,87,66,90)(79,94,82,91)(80,93,83,96)(81,92,84,95)>;
G:=Group( (1,19,4,22)(2,20,5,23)(3,21,6,24)(7,16,10,13)(8,17,11,14)(9,18,12,15)(25,36,39,32)(26,34,37,33)(27,35,38,31)(28,46,44,42)(29,47,45,40)(30,48,43,41)(49,96,52,93)(50,91,53,94)(51,92,54,95)(55,87,58,90)(56,88,59,85)(57,89,60,86)(61,69,64,72)(62,70,65,67)(63,71,66,68)(73,82,76,79)(74,83,77,80)(75,84,78,81), (1,91,16,70)(2,95,17,68)(3,93,18,72)(4,94,13,67)(5,92,14,71)(6,96,15,69)(7,65,22,53)(8,63,23,51)(9,61,24,49)(10,62,19,50)(11,66,20,54)(12,64,21,52)(25,58,40,81)(26,56,41,79)(27,60,42,83)(28,74,35,89)(29,78,36,87)(30,76,34,85)(31,86,44,77)(32,90,45,75)(33,88,43,73)(37,59,48,82)(38,57,46,80)(39,55,47,84), (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), (1,33,4,34)(2,32,5,36)(3,31,6,35)(7,41,10,48)(8,40,11,47)(9,42,12,46)(13,30,16,43)(14,29,17,45)(15,28,18,44)(19,37,22,26)(20,39,23,25)(21,38,24,27)(49,74,52,77)(50,73,53,76)(51,78,54,75)(55,68,58,71)(56,67,59,70)(57,72,60,69)(61,89,64,86)(62,88,65,85)(63,87,66,90)(79,94,82,91)(80,93,83,96)(81,92,84,95) );
G=PermutationGroup([(1,19,4,22),(2,20,5,23),(3,21,6,24),(7,16,10,13),(8,17,11,14),(9,18,12,15),(25,36,39,32),(26,34,37,33),(27,35,38,31),(28,46,44,42),(29,47,45,40),(30,48,43,41),(49,96,52,93),(50,91,53,94),(51,92,54,95),(55,87,58,90),(56,88,59,85),(57,89,60,86),(61,69,64,72),(62,70,65,67),(63,71,66,68),(73,82,76,79),(74,83,77,80),(75,84,78,81)], [(1,91,16,70),(2,95,17,68),(3,93,18,72),(4,94,13,67),(5,92,14,71),(6,96,15,69),(7,65,22,53),(8,63,23,51),(9,61,24,49),(10,62,19,50),(11,66,20,54),(12,64,21,52),(25,58,40,81),(26,56,41,79),(27,60,42,83),(28,74,35,89),(29,78,36,87),(30,76,34,85),(31,86,44,77),(32,90,45,75),(33,88,43,73),(37,59,48,82),(38,57,46,80),(39,55,47,84)], [(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)], [(1,33,4,34),(2,32,5,36),(3,31,6,35),(7,41,10,48),(8,40,11,47),(9,42,12,46),(13,30,16,43),(14,29,17,45),(15,28,18,44),(19,37,22,26),(20,39,23,25),(21,38,24,27),(49,74,52,77),(50,73,53,76),(51,78,54,75),(55,68,58,71),(56,67,59,70),(57,72,60,69),(61,89,64,86),(62,88,65,85),(63,87,66,90),(79,94,82,91),(80,93,83,96),(81,92,84,95)])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | C3⋊D4 | C8.C22 | Q8.14D6 |
| kernel | C4⋊C4.237D6 | C6.SD16 | C2×C4.Dic3 | C3×C42⋊C2 | C22×Dic6 | C2×Dic6 | C42⋊C2 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 3 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of C4⋊C4.237D6 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 14 | 0 | 0 |
| 0 | 0 | 59 | 66 | 0 | 0 |
| 0 | 0 | 57 | 12 | 7 | 59 |
| 0 | 0 | 45 | 57 | 14 | 66 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 50 | 48 | 48 |
| 0 | 0 | 23 | 34 | 50 | 48 |
| 0 | 0 | 8 | 9 | 39 | 23 |
| 0 | 0 | 72 | 8 | 50 | 62 |
| 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 | 28 | 12 | 0 | 1 |
| 0 | 0 | 16 | 28 | 72 | 1 |
| 3 | 28 | 0 | 0 | 0 | 0 |
| 31 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 43 | 0 | 0 |
| 0 | 0 | 13 | 43 | 0 | 0 |
| 0 | 0 | 67 | 26 | 46 | 27 |
| 0 | 0 | 26 | 32 | 0 | 27 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,7,59,57,45,0,0,14,66,12,57,0,0,0,0,7,14,0,0,0,0,59,66],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,11,23,8,72,0,0,50,34,9,8,0,0,48,50,39,50,0,0,48,48,23,62],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,1,28,16,0,0,72,0,12,28,0,0,0,0,0,72,0,0,0,0,1,1],[3,31,0,0,0,0,28,70,0,0,0,0,0,0,30,13,67,26,0,0,43,43,26,32,0,0,0,0,46,0,0,0,0,0,27,27] >;
C4⋊C4.237D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{237}D_6 % in TeX
G:=Group("C4:C4.237D6"); // GroupNames label
G:=SmallGroup(192,563);
// by ID
G=gap.SmallGroup(192,563);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,422,387,58,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^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