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G = D6⋊C47C4order 192 = 26·3

7th semidirect product of D6⋊C4 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6⋊C47C4, C6.60(C4×D4), (C2×C12).252D4, (C22×C4).59D6, C6.91(C4⋊D4), C22.113(S3×D4), C6.C4242C2, (C2×Dic3).178D4, C2.7(D6.D4), C6.52(C4.4D4), C2.20(Dic35D4), C6.38(C42⋊C2), C6.28(C422C2), C2.6(C23.14D6), C2.3(C12.23D4), C22.59(C4○D12), (S3×C23).19C22, (C22×C6).354C23, (C22×C12).30C22, C23.308(C22×S3), C37(C24.C22), C22.60(D42S3), C22.30(Q83S3), C6.52(C22.D4), (C22×Dic3).193C22, (C2×C4⋊C4)⋊8S3, (C6×C4⋊C4)⋊25C2, (C2×C4).42(C4×S3), (C2×C4×Dic3)⋊25C2, C2.14(C4×C3⋊D4), (C2×D6⋊C4).13C2, (C2×C6).335(C2×D4), C22.139(S3×C2×C4), C2.7(C4⋊C4⋊S3), (C2×C12).186(C2×C4), C2.14(C4⋊C47S3), C22.69(C2×C3⋊D4), (C2×C6).157(C4○D4), (C2×C4).100(C3⋊D4), (C22×S3).22(C2×C4), (C2×C6).122(C22×C4), (C2×Dic3).64(C2×C4), SmallGroup(192,549)

Series: Derived Chief Lower central Upper central

C1C2×C6 — D6⋊C47C4
C1C3C6C2×C6C22×C6S3×C23C2×D6⋊C4 — D6⋊C47C4
C3C2×C6 — D6⋊C47C4
C1C23C2×C4⋊C4

Generators and relations for D6⋊C47C4
 G = < a,b,c,d | a6=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, cbc-1=a3b, dbd-1=bc2, dcd-1=c-1 >

Subgroups: 536 in 190 conjugacy classes, 69 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C3, C4 [×10], C22 [×7], C22 [×10], S3 [×2], C6 [×7], C2×C4 [×4], C2×C4 [×18], C23, C23 [×8], Dic3 [×5], C12 [×5], D6 [×10], C2×C6 [×7], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×3], C22×C4 [×3], C24, C2×Dic3 [×4], C2×Dic3 [×7], C2×C12 [×4], C2×C12 [×7], C22×S3 [×2], C22×S3 [×6], C22×C6, C2.C42 [×2], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4, C4×Dic3 [×2], D6⋊C4 [×4], D6⋊C4 [×4], C3×C4⋊C4 [×2], C22×Dic3 [×3], C22×C12 [×3], S3×C23, C24.C22, C6.C42 [×2], C2×C4×Dic3, C2×D6⋊C4 [×3], C6×C4⋊C4, D6⋊C47C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22×C4, C2×D4 [×2], C4○D4 [×4], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, S3×C2×C4, C4○D12, S3×D4, D42S3, Q83S3 [×2], C2×C3⋊D4, C24.C22, C4⋊C47S3, Dic35D4, D6.D4, C4⋊C4⋊S3, C4×C3⋊D4, C23.14D6, C12.23D4, D6⋊C47C4

Smallest permutation representation of D6⋊C47C4
On 96 points
Generators in S96
(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)(2 5)(3 4)(7 93)(8 92)(9 91)(10 96)(11 95)(12 94)(13 18)(14 17)(15 16)(19 22)(20 21)(23 24)(25 30)(26 29)(27 28)(31 38)(32 37)(33 42)(34 41)(35 40)(36 39)(43 50)(44 49)(45 54)(46 53)(47 52)(48 51)(56 60)(57 59)(61 63)(64 66)(68 72)(69 71)(73 75)(76 78)(79 89)(80 88)(81 87)(82 86)(83 85)(84 90)
(1 64 16 60)(2 65 17 55)(3 66 18 56)(4 61 13 57)(5 62 14 58)(6 63 15 59)(7 51 95 47)(8 52 96 48)(9 53 91 43)(10 54 92 44)(11 49 93 45)(12 50 94 46)(19 77 29 67)(20 78 30 68)(21 73 25 69)(22 74 26 70)(23 75 27 71)(24 76 28 72)(31 89 41 79)(32 90 42 80)(33 85 37 81)(34 86 38 82)(35 87 39 83)(36 88 40 84)
(1 48 24 36)(2 43 19 31)(3 44 20 32)(4 45 21 33)(5 46 22 34)(6 47 23 35)(7 71 87 59)(8 72 88 60)(9 67 89 55)(10 68 90 56)(11 69 85 57)(12 70 86 58)(13 49 25 37)(14 50 26 38)(15 51 27 39)(16 52 28 40)(17 53 29 41)(18 54 30 42)(61 93 73 81)(62 94 74 82)(63 95 75 83)(64 96 76 84)(65 91 77 79)(66 92 78 80)

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

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

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

48 conjugacy classes

class 1 2A···2G2H2I 3 4A4B4C4D4E4F4G4H4I···4P4Q4R6A···6G12A···12L
order12···2223444444444···4446···612···12
size11···112122222244446···612122···24···4

48 irreducible representations

dim11111122222222444
type++++++++++-+
imageC1C2C2C2C2C4S3D4D4D6C4○D4C4×S3C3⋊D4C4○D12S3×D4D42S3Q83S3
kernelD6⋊C47C4C6.C42C2×C4×Dic3C2×D6⋊C4C6×C4⋊C4D6⋊C4C2×C4⋊C4C2×Dic3C2×C12C22×C4C2×C6C2×C4C2×C4C22C22C22C22
# reps12131812238444112

Matrix representation of D6⋊C47C4 in GL6(𝔽13)

1120000
100000
0001200
001100
0000120
0000012
,
100000
1120000
0001200
0012000
0000120
000001
,
1140000
920000
0011900
004200
000001
0000120
,
100000
010000
008000
000800
000001
000010

G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[11,9,0,0,0,0,4,2,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D6⋊C47C4 in GAP, Magma, Sage, TeX

D_6\rtimes C_4\rtimes_7C_4
% in TeX

G:=Group("D6:C4:7C4");
// GroupNames label

G:=SmallGroup(192,549);
// by ID

G=gap.SmallGroup(192,549);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,758,387,58,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^4=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,d*b*d^-1=b*c^2,d*c*d^-1=c^-1>;
// generators/relations

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