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G = C3⋊C822D4order 192 = 26·3

4th semidirect product of C3⋊C8 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3⋊C822D4, (C2×C6)⋊3D8, C4⋊D43S3, C33(C87D4), C4⋊C4.58D6, C6.55(C2×D8), (C2×D4).38D6, C4.170(S3×D4), C127D423C2, C222(D4⋊S3), C6.97(C4○D8), C12.147(C2×D4), C6.D835C2, C6.Q1636C2, (C2×C12).263D4, (C22×C6).84D4, D4⋊Dic315C2, C6.93(C4⋊D4), (C6×D4).54C22, (C22×C4).356D6, C12.183(C4○D4), C4.59(D42S3), (C2×C12).357C23, (C2×D12).97C22, C23.46(C3⋊D4), C2.16(Q8.13D6), C4⋊Dic3.142C22, C2.14(C23.14D6), (C22×C12).161C22, (C22×C3⋊C8)⋊3C2, (C2×D4⋊S3)⋊10C2, (C3×C4⋊D4)⋊3C2, C2.10(C2×D4⋊S3), (C2×C6).488(C2×D4), (C2×C3⋊C8).248C22, (C2×C4).105(C3⋊D4), (C3×C4⋊C4).105C22, (C2×C4).457(C22×S3), C22.163(C2×C3⋊D4), SmallGroup(192,597)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C3⋊C822D4
C1C3C6C12C2×C12C2×D12C127D4 — C3⋊C822D4
C3C6C2×C12 — C3⋊C822D4
C1C22C22×C4C4⋊D4

Generators and relations for C3⋊C822D4
 G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 416 in 134 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C3⋊C8, C3⋊C8, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, D4⋊C4, C2.D8, C4⋊D4, C4⋊D4, C22×C8, C2×D8, C2×C3⋊C8, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, D4⋊S3, C3×C22⋊C4, C3×C4⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C87D4, C6.Q16, C6.D8, D4⋊Dic3, C22×C3⋊C8, C127D4, C2×D4⋊S3, C3×C4⋊D4, C3⋊C822D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C2×D8, C4○D8, D4⋊S3, S3×D4, D42S3, C2×C3⋊D4, C87D4, C2×D4⋊S3, C23.14D6, Q8.13D6, C3⋊C822D4

Smallest permutation representation of C3⋊C822D4
On 96 points
Generators in S96
(1 81 61)(2 62 82)(3 83 63)(4 64 84)(5 85 57)(6 58 86)(7 87 59)(8 60 88)(9 35 52)(10 53 36)(11 37 54)(12 55 38)(13 39 56)(14 49 40)(15 33 50)(16 51 34)(17 76 68)(18 69 77)(19 78 70)(20 71 79)(21 80 72)(22 65 73)(23 74 66)(24 67 75)(25 95 42)(26 43 96)(27 89 44)(28 45 90)(29 91 46)(30 47 92)(31 93 48)(32 41 94)
(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 91 70 11)(2 90 71 10)(3 89 72 9)(4 96 65 16)(5 95 66 15)(6 94 67 14)(7 93 68 13)(8 92 69 12)(17 39 87 48)(18 38 88 47)(19 37 81 46)(20 36 82 45)(21 35 83 44)(22 34 84 43)(23 33 85 42)(24 40 86 41)(25 74 50 57)(26 73 51 64)(27 80 52 63)(28 79 53 62)(29 78 54 61)(30 77 55 60)(31 76 56 59)(32 75 49 58)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 89)(9 69)(10 70)(11 71)(12 72)(13 65)(14 66)(15 67)(16 68)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(25 86)(26 87)(27 88)(28 81)(29 82)(30 83)(31 84)(32 85)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 73)(40 74)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F6A6B6C6D6E6F6G8A···8H12A12B12C12D12E12F
order12222222344444466666668···8121212121212
size1111228242222282422244886···6444488

36 irreducible representations

dim111111112222222222224444
type+++++++++++++++++-+
imageC1C2C2C2C2C2C2C2S3D4D4D4D6D6D6C4○D4D8C3⋊D4C3⋊D4C4○D8S3×D4D42S3D4⋊S3Q8.13D6
kernelC3⋊C822D4C6.Q16C6.D8D4⋊Dic3C22×C3⋊C8C127D4C2×D4⋊S3C3×C4⋊D4C4⋊D4C3⋊C8C2×C12C22×C6C4⋊C4C22×C4C2×D4C12C2×C6C2×C4C23C6C4C4C22C2
# reps111111111211111242241122

Matrix representation of C3⋊C822D4 in GL6(𝔽73)

0720000
1720000
001000
000100
000010
000001
,
13300000
43600000
00165700
00161600
000001
0000720
,
30130000
60430000
0067600
006600
0000270
0000046
,
100000
010000
0002700
0046000
0000046
0000270

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,43,0,0,0,0,30,60,0,0,0,0,0,0,16,16,0,0,0,0,57,16,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[30,60,0,0,0,0,13,43,0,0,0,0,0,0,67,6,0,0,0,0,6,6,0,0,0,0,0,0,27,0,0,0,0,0,0,46],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,46,0] >;

C3⋊C822D4 in GAP, Magma, Sage, TeX

C_3\rtimes C_8\rtimes_{22}D_4
% in TeX

G:=Group("C3:C8:22D4");
// GroupNames label

G:=SmallGroup(192,597);
// by ID

G=gap.SmallGroup(192,597);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,219,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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