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G = C42.240D6order 192 = 26·3

60th non-split extension by C42 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.240D6, (C4×S3)⋊9D4, C4⋊Q819S3, D6.8(C2×D4), C4.38(S3×D4), C126(C4○D4), C4⋊C4.217D6, C12.70(C2×D4), C12⋊D439C2, C4⋊D1217C2, (S3×C42)⋊14C2, C41(Q83S3), (C2×Q8).168D6, Dic35D443C2, C6.99(C22×D4), (C2×C6).269C24, D6⋊C4.50C22, Dic3.67(C2×D4), C12.23D426C2, (C2×C12).102C23, (C4×C12).210C22, (C6×Q8).136C22, (C2×D12).172C22, C22.290(S3×C23), C36(C22.26C24), (C22×S3).119C23, (C4×Dic3).258C22, (C2×Dic3).272C23, C2.72(C2×S3×D4), (C3×C4⋊Q8)⋊11C2, C6.120(C2×C4○D4), (C2×Q83S3)⋊12C2, (S3×C2×C4).143C22, C2.27(C2×Q83S3), (C3×C4⋊C4).212C22, (C2×C4).599(C22×S3), SmallGroup(192,1284)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C42.240D6
C1C3C6C2×C6C22×S3S3×C2×C4S3×C42 — C42.240D6
C3C2×C6 — C42.240D6
C1C22C4⋊Q8

Generators and relations for C42.240D6
 G = < a,b,c,d | a4=b4=d2=1, c6=b2, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=b2c5 >

Subgroups: 848 in 310 conjugacy classes, 111 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×6], C4 [×8], C22, C22 [×16], S3 [×6], C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×19], D4 [×20], Q8 [×4], C23 [×5], Dic3 [×2], Dic3 [×2], C12 [×6], C12 [×4], D6 [×2], D6 [×14], C2×C6, C42, C42 [×3], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×7], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], C4×S3 [×4], C4×S3 [×12], D12 [×20], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3, C22×S3 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C2×C4○D4 [×2], C4×Dic3, C4×Dic3 [×2], D6⋊C4 [×8], C4×C12, C3×C4⋊C4 [×4], S3×C2×C4, S3×C2×C4 [×6], C2×D12 [×10], Q83S3 [×8], C6×Q8 [×2], C22.26C24, S3×C42, C4⋊D12, Dic35D4 [×4], C12⋊D4 [×4], C12.23D4 [×2], C3×C4⋊Q8, C2×Q83S3 [×2], C42.240D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C22×S3 [×7], C22×D4, C2×C4○D4 [×2], S3×D4 [×2], Q83S3 [×4], S3×C23, C22.26C24, C2×S3×D4, C2×Q83S3 [×2], C42.240D6

Smallest permutation representation of C42.240D6
On 96 points
Generators in S96
(1 78 18 63)(2 64 19 79)(3 80 20 65)(4 66 21 81)(5 82 22 67)(6 68 23 83)(7 84 24 69)(8 70 13 73)(9 74 14 71)(10 72 15 75)(11 76 16 61)(12 62 17 77)(25 89 60 43)(26 44 49 90)(27 91 50 45)(28 46 51 92)(29 93 52 47)(30 48 53 94)(31 95 54 37)(32 38 55 96)(33 85 56 39)(34 40 57 86)(35 87 58 41)(36 42 59 88)
(1 30 7 36)(2 25 8 31)(3 32 9 26)(4 27 10 33)(5 34 11 28)(6 29 12 35)(13 54 19 60)(14 49 20 55)(15 56 21 50)(16 51 22 57)(17 58 23 52)(18 53 24 59)(37 79 43 73)(38 74 44 80)(39 81 45 75)(40 76 46 82)(41 83 47 77)(42 78 48 84)(61 92 67 86)(62 87 68 93)(63 94 69 88)(64 89 70 95)(65 96 71 90)(66 91 72 85)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 19)(14 18)(15 17)(20 24)(21 23)(26 36)(27 35)(28 34)(29 33)(30 32)(37 95)(38 94)(39 93)(40 92)(41 91)(42 90)(43 89)(44 88)(45 87)(46 86)(47 85)(48 96)(49 59)(50 58)(51 57)(52 56)(53 55)(61 76)(62 75)(63 74)(64 73)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A···4F4G4H4I4J4K4L4M4N4O4P4Q4R6A6B6C12A···12F12G12H12I12J
order122222222234···444444444444466612···1212121212
size1111661212121222···23333444466662224···48888

42 irreducible representations

dim1111111122222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D6D6D6C4○D4S3×D4Q83S3
kernelC42.240D6S3×C42C4⋊D12Dic35D4C12⋊D4C12.23D4C3×C4⋊Q8C2×Q83S3C4⋊Q8C4×S3C42C4⋊C4C2×Q8C12C4C4
# reps1114421214142824

Matrix representation of C42.240D6 in GL6(𝔽13)

500000
380000
001000
000100
000010
000001
,
800000
1050000
0012000
0001200
000008
000080
,
110000
11120000
0001200
0011200
000001
0000120
,
12120000
010000
0011200
0001200
000010
0000012

G:=sub<GL(6,GF(13))| [5,3,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,10,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[1,11,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12] >;

C42.240D6 in GAP, Magma, Sage, TeX

C_4^2._{240}D_6
% in TeX

G:=Group("C4^2.240D6");
// GroupNames label

G:=SmallGroup(192,1284);
// by ID

G=gap.SmallGroup(192,1284);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,675,570,185,80,6278]);
// Polycyclic

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

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