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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 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], S3, C6 [×3], C6 [×3], C8 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×2], Dic3, C12 [×2], C12 [×2], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8 [×4], D8 [×2], C22×C4, C2×D4, C2×D4 [×3], C3⋊C8 [×2], C3⋊C8, D12 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C22×S3, C22×C6, C22×C6, D4⋊C4 [×2], C2.D8, C4⋊D4, C4⋊D4, C22×C8, C2×D8, C2×C3⋊C8 [×2], C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, D4⋊S3 [×2], 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 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], D8 [×2], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C2×D8, C4○D8, D4⋊S3 [×2], 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 70 61)(2 62 71)(3 72 63)(4 64 65)(5 66 57)(6 58 67)(7 68 59)(8 60 69)(9 25 95)(10 96 26)(11 27 89)(12 90 28)(13 29 91)(14 92 30)(15 31 93)(16 94 32)(17 84 73)(18 74 85)(19 86 75)(20 76 87)(21 88 77)(22 78 81)(23 82 79)(24 80 83)(33 50 48)(34 41 51)(35 52 42)(36 43 53)(37 54 44)(38 45 55)(39 56 46)(40 47 49)
(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 81 44)(2 90 82 43)(3 89 83 42)(4 96 84 41)(5 95 85 48)(6 94 86 47)(7 93 87 46)(8 92 88 45)(9 18 33 66)(10 17 34 65)(11 24 35 72)(12 23 36 71)(13 22 37 70)(14 21 38 69)(15 20 39 68)(16 19 40 67)(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 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 57)(17 56)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 65)(32 66)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 73)(40 74)(41 87)(42 88)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)

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

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

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

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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