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G = C3⋊C823D4order 192 = 26·3

5th semidirect product of C3⋊C8 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3⋊C823D4, C34(C88D4), C4⋊C4.61D6, (C2×C6)⋊4SD16, (C2×D4).41D6, C4⋊D4.6S3, C4.172(S3×D4), C6.98(C4○D8), (C2×C12).264D4, C12.150(C2×D4), C6.56(C2×SD16), (C22×C6).87D4, D4⋊Dic317C2, C6.SD1635C2, C6.95(C4⋊D4), C222(D4.S3), C12.Q836C2, (C6×D4).57C22, (C22×C4).357D6, C12.185(C4○D4), C12.48D424C2, C4.61(D42S3), (C2×C12).360C23, C23.47(C3⋊D4), C2.17(Q8.13D6), C4⋊Dic3.144C22, C2.16(C23.14D6), (C22×C12).164C22, (C2×Dic6).103C22, (C22×C3⋊C8)⋊4C2, (C2×D4.S3)⋊10C2, (C3×C4⋊D4).5C2, (C2×C6).491(C2×D4), C2.10(C2×D4.S3), (C2×C3⋊C8).249C22, (C2×C4).106(C3⋊D4), (C3×C4⋊C4).108C22, (C2×C4).460(C22×S3), C22.166(C2×C3⋊D4), SmallGroup(192,600)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C3⋊C823D4
C1C3C6C12C2×C12C2×Dic6C12.48D4 — C3⋊C823D4
C3C6C2×C12 — C3⋊C823D4
C1C22C22×C4C4⋊D4

Generators and relations for C3⋊C823D4
 G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=b3, bd=db, dcd=c-1 >

Subgroups: 320 in 124 conjugacy classes, 45 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C6 [×3], C6 [×3], C8 [×3], C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8 [×4], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, C3⋊C8 [×2], C3⋊C8, Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C22×C6, C22×C6, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C2×C3⋊C8 [×2], C2×C3⋊C8 [×2], Dic3⋊C4, C4⋊Dic3, D4.S3 [×2], C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×D4, C6×D4, C88D4, C12.Q8, C6.SD16, D4⋊Dic3, C22×C3⋊C8, C12.48D4, C2×D4.S3, C3×C4⋊D4, C3⋊C823D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], SD16 [×2], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C2×SD16, C4○D8, D4.S3 [×2], S3×D4, D42S3, C2×C3⋊D4, C88D4, C2×D4.S3, C23.14D6, Q8.13D6, C3⋊C823D4

Smallest permutation representation of C3⋊C823D4
On 96 points
Generators in S96
(1 46 61)(2 62 47)(3 48 63)(4 64 41)(5 42 57)(6 58 43)(7 44 59)(8 60 45)(9 70 33)(10 34 71)(11 72 35)(12 36 65)(13 66 37)(14 38 67)(15 68 39)(16 40 69)(17 73 93)(18 94 74)(19 75 95)(20 96 76)(21 77 89)(22 90 78)(23 79 91)(24 92 80)(25 52 84)(26 85 53)(27 54 86)(28 87 55)(29 56 88)(30 81 49)(31 50 82)(32 83 51)
(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 33 91 50)(2 36 92 53)(3 39 93 56)(4 34 94 51)(5 37 95 54)(6 40 96 49)(7 35 89 52)(8 38 90 55)(9 23 82 46)(10 18 83 41)(11 21 84 44)(12 24 85 47)(13 19 86 42)(14 22 87 45)(15 17 88 48)(16 20 81 43)(25 59 72 77)(26 62 65 80)(27 57 66 75)(28 60 67 78)(29 63 68 73)(30 58 69 76)(31 61 70 79)(32 64 71 74)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 49)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 89)(40 90)(41 85)(42 86)(43 87)(44 88)(45 81)(46 82)(47 83)(48 84)(65 74)(66 75)(67 76)(68 77)(69 78)(70 79)(71 80)(72 73)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G6A6B6C6D6E6F6G8A···8H12A12B12C12D12E12F
order12222223444444466666668···8121212121212
size1111228222228242422244886···6444488

36 irreducible representations

dim111111112222222222224444
type++++++++++++++++--
imageC1C2C2C2C2C2C2C2S3D4D4D4D6D6D6C4○D4SD16C3⋊D4C3⋊D4C4○D8S3×D4D42S3D4.S3Q8.13D6
kernelC3⋊C823D4C12.Q8C6.SD16D4⋊Dic3C22×C3⋊C8C12.48D4C2×D4.S3C3×C4⋊D4C4⋊D4C3⋊C8C2×C12C22×C6C4⋊C4C22×C4C2×D4C12C2×C6C2×C4C23C6C4C4C22C2
# reps111111111211111242241122

Matrix representation of C3⋊C823D4 in GL6(𝔽73)

100000
010000
001000
000100
0000640
0000208
,
6760000
67670000
0013600
0047200
000094
00001664
,
100000
0720000
00272300
0004600
0000720
0000411
,
100000
010000
00272300
00354600
0000720
0000072

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,20,0,0,0,0,0,8],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,1,4,0,0,0,0,36,72,0,0,0,0,0,0,9,16,0,0,0,0,4,64],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,23,46,0,0,0,0,0,0,72,41,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,35,0,0,0,0,23,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

C3⋊C823D4 in GAP, Magma, Sage, TeX

C_3\rtimes C_8\rtimes_{23}D_4
% in TeX

G:=Group("C3:C8:23D4");
// GroupNames label

G:=SmallGroup(192,600);
// by ID

G=gap.SmallGroup(192,600);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,219,1123,297,136,6278]);
// 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^3,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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