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G = C4⋊C4.236D6order 192 = 26·3

14th non-split extension by C4⋊C4 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.236D6, C42⋊C23S3, C6.86(C4○D8), C6.D828C2, (C2×C12).450D4, C6.Q1629C2, (C22×C6).76D4, C4.91(C4○D12), C127D4.11C2, C12.55D46C2, C2.6(D4⋊D6), C12.Q829C2, (C22×C4).127D6, C12.179(C4○D4), C6.106(C8⋊C22), (C2×C12).330C23, C2.9(Q8.13D6), (C2×D12).92C22, C36(C23.19D4), C23.27(C3⋊D4), C4⋊Dic3.135C22, (C22×C12).152C22, C6.66(C22.D4), C2.17(C23.28D6), (C2×C6).459(C2×D4), (C2×C3⋊C8).87C22, (C3×C42⋊C2)⋊3C2, (C2×C4).215(C3⋊D4), (C3×C4⋊C4).267C22, (C2×C4).430(C22×S3), C22.145(C2×C3⋊D4), SmallGroup(192,562)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C4⋊C4.236D6
C1C3C6C12C2×C12C2×D12C127D4 — C4⋊C4.236D6
C3C6C2×C12 — C4⋊C4.236D6
C1C22C22×C4C42⋊C2

Generators and relations for C4⋊C4.236D6
 G = < a,b,c,d | a4=b4=1, c6=a2, d2=b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2b2c5 >

Subgroups: 328 in 106 conjugacy classes, 39 normal (all characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3, C6 [×3], C6, C8 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×4], C23, C23, Dic3, C12 [×2], C12 [×3], D6 [×3], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C22×C4, C2×D4 [×2], C3⋊C8 [×2], D12 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×C6, C22⋊C8, D4⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4 [×2], C2×D12, C2×C3⋊D4, C22×C12, C23.19D4, C6.Q16, C12.Q8, C6.D8 [×2], C12.55D4, C127D4, C3×C42⋊C2, C4⋊C4.236D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C22.D4, C4○D8, C8⋊C22, C4○D12 [×2], C2×C3⋊D4, C23.19D4, C23.28D6, D4⋊D6, Q8.13D6, C4⋊C4.236D6

Smallest permutation representation of C4⋊C4.236D6
On 96 points
Generators in S96
(1 22 7 16)(2 23 8 17)(3 24 9 18)(4 13 10 19)(5 14 11 20)(6 15 12 21)(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 59 43 53)(38 60 44 54)(39 49 45 55)(40 50 46 56)(41 51 47 57)(42 52 48 58)(61 87 67 93)(62 88 68 94)(63 89 69 95)(64 90 70 96)(65 91 71 85)(66 92 72 86)
(1 94 80 47)(2 95 81 48)(3 96 82 37)(4 85 83 38)(5 86 84 39)(6 87 73 40)(7 88 74 41)(8 89 75 42)(9 90 76 43)(10 91 77 44)(11 92 78 45)(12 93 79 46)(13 71 26 54)(14 72 27 55)(15 61 28 56)(16 62 29 57)(17 63 30 58)(18 64 31 59)(19 65 32 60)(20 66 33 49)(21 67 34 50)(22 68 35 51)(23 69 36 52)(24 70 25 53)
(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 6 80 73)(2 84 81 5)(3 4 82 83)(7 12 74 79)(8 78 75 11)(9 10 76 77)(13 31 26 18)(14 17 27 30)(15 29 28 16)(19 25 32 24)(20 23 33 36)(21 35 34 22)(37 54 96 71)(38 70 85 53)(39 52 86 69)(40 68 87 51)(41 50 88 67)(42 66 89 49)(43 60 90 65)(44 64 91 59)(45 58 92 63)(46 62 93 57)(47 56 94 61)(48 72 95 55)

G:=sub<Sym(96)| (1,22,7,16)(2,23,8,17)(3,24,9,18)(4,13,10,19)(5,14,11,20)(6,15,12,21)(25,76,31,82)(26,77,32,83)(27,78,33,84)(28,79,34,73)(29,80,35,74)(30,81,36,75)(37,59,43,53)(38,60,44,54)(39,49,45,55)(40,50,46,56)(41,51,47,57)(42,52,48,58)(61,87,67,93)(62,88,68,94)(63,89,69,95)(64,90,70,96)(65,91,71,85)(66,92,72,86), (1,94,80,47)(2,95,81,48)(3,96,82,37)(4,85,83,38)(5,86,84,39)(6,87,73,40)(7,88,74,41)(8,89,75,42)(9,90,76,43)(10,91,77,44)(11,92,78,45)(12,93,79,46)(13,71,26,54)(14,72,27,55)(15,61,28,56)(16,62,29,57)(17,63,30,58)(18,64,31,59)(19,65,32,60)(20,66,33,49)(21,67,34,50)(22,68,35,51)(23,69,36,52)(24,70,25,53), (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), (1,6,80,73)(2,84,81,5)(3,4,82,83)(7,12,74,79)(8,78,75,11)(9,10,76,77)(13,31,26,18)(14,17,27,30)(15,29,28,16)(19,25,32,24)(20,23,33,36)(21,35,34,22)(37,54,96,71)(38,70,85,53)(39,52,86,69)(40,68,87,51)(41,50,88,67)(42,66,89,49)(43,60,90,65)(44,64,91,59)(45,58,92,63)(46,62,93,57)(47,56,94,61)(48,72,95,55)>;

G:=Group( (1,22,7,16)(2,23,8,17)(3,24,9,18)(4,13,10,19)(5,14,11,20)(6,15,12,21)(25,76,31,82)(26,77,32,83)(27,78,33,84)(28,79,34,73)(29,80,35,74)(30,81,36,75)(37,59,43,53)(38,60,44,54)(39,49,45,55)(40,50,46,56)(41,51,47,57)(42,52,48,58)(61,87,67,93)(62,88,68,94)(63,89,69,95)(64,90,70,96)(65,91,71,85)(66,92,72,86), (1,94,80,47)(2,95,81,48)(3,96,82,37)(4,85,83,38)(5,86,84,39)(6,87,73,40)(7,88,74,41)(8,89,75,42)(9,90,76,43)(10,91,77,44)(11,92,78,45)(12,93,79,46)(13,71,26,54)(14,72,27,55)(15,61,28,56)(16,62,29,57)(17,63,30,58)(18,64,31,59)(19,65,32,60)(20,66,33,49)(21,67,34,50)(22,68,35,51)(23,69,36,52)(24,70,25,53), (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), (1,6,80,73)(2,84,81,5)(3,4,82,83)(7,12,74,79)(8,78,75,11)(9,10,76,77)(13,31,26,18)(14,17,27,30)(15,29,28,16)(19,25,32,24)(20,23,33,36)(21,35,34,22)(37,54,96,71)(38,70,85,53)(39,52,86,69)(40,68,87,51)(41,50,88,67)(42,66,89,49)(43,60,90,65)(44,64,91,59)(45,58,92,63)(46,62,93,57)(47,56,94,61)(48,72,95,55) );

G=PermutationGroup([(1,22,7,16),(2,23,8,17),(3,24,9,18),(4,13,10,19),(5,14,11,20),(6,15,12,21),(25,76,31,82),(26,77,32,83),(27,78,33,84),(28,79,34,73),(29,80,35,74),(30,81,36,75),(37,59,43,53),(38,60,44,54),(39,49,45,55),(40,50,46,56),(41,51,47,57),(42,52,48,58),(61,87,67,93),(62,88,68,94),(63,89,69,95),(64,90,70,96),(65,91,71,85),(66,92,72,86)], [(1,94,80,47),(2,95,81,48),(3,96,82,37),(4,85,83,38),(5,86,84,39),(6,87,73,40),(7,88,74,41),(8,89,75,42),(9,90,76,43),(10,91,77,44),(11,92,78,45),(12,93,79,46),(13,71,26,54),(14,72,27,55),(15,61,28,56),(16,62,29,57),(17,63,30,58),(18,64,31,59),(19,65,32,60),(20,66,33,49),(21,67,34,50),(22,68,35,51),(23,69,36,52),(24,70,25,53)], [(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)], [(1,6,80,73),(2,84,81,5),(3,4,82,83),(7,12,74,79),(8,78,75,11),(9,10,76,77),(13,31,26,18),(14,17,27,30),(15,29,28,16),(19,25,32,24),(20,23,33,36),(21,35,34,22),(37,54,96,71),(38,70,85,53),(39,52,86,69),(40,68,87,51),(41,50,88,67),(42,66,89,49),(43,60,90,65),(44,64,91,59),(45,58,92,63),(46,62,93,57),(47,56,94,61),(48,72,95,55)])

39 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E8A8B8C8D12A12B12C12D12E···12N
order12222234444444446666688881212121212···12
size111142422222444424222441212121222224···4

39 irreducible representations

dim11111112222222222444
type++++++++++++++
imageC1C2C2C2C2C2C2S3D4D4D6D6C4○D4C3⋊D4C3⋊D4C4○D8C4○D12C8⋊C22D4⋊D6Q8.13D6
kernelC4⋊C4.236D6C6.Q16C12.Q8C6.D8C12.55D4C127D4C3×C42⋊C2C42⋊C2C2×C12C22×C6C4⋊C4C22×C4C12C2×C4C23C6C4C6C2C2
# reps11121111112142248122

Matrix representation of C4⋊C4.236D6 in GL4(𝔽73) generated by

72000
07200
007270
00251
,
71400
596600
006155
00412
,
304300
306000
00270
00027
,
433000
603000
00270
005546
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,25,0,0,70,1],[7,59,0,0,14,66,0,0,0,0,61,4,0,0,55,12],[30,30,0,0,43,60,0,0,0,0,27,0,0,0,0,27],[43,60,0,0,30,30,0,0,0,0,27,55,0,0,0,46] >;

C4⋊C4.236D6 in GAP, Magma, Sage, TeX

C_4\rtimes C_4._{236}D_6
% in TeX

G:=Group("C4:C4.236D6");
// GroupNames label

G:=SmallGroup(192,562);
// by ID

G=gap.SmallGroup(192,562);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,254,100,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2,d^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*b,d*c*d^-1=a^2*b^2*c^5>;
// generators/relations

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