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G = C3⋊C824D4order 192 = 26·3

6th semidirect product of C3⋊C8 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3⋊C824D4, C35(C88D4), C4⋊C4.67D6, (C2×C6)⋊6SD16, C22⋊Q83S3, C4.174(S3×D4), (C2×Q8).53D6, C12.154(C2×D4), (C2×C12).267D4, C6.D839C2, C6.73(C2×SD16), (C22×C6).94D4, C6.101(C4○D8), C127D4.12C2, Q82Dic315C2, C6.97(C4⋊D4), C12.Q839C2, (C22×C4).358D6, C12.190(C4○D4), (C6×Q8).47C22, C4.63(D42S3), (C2×C12).367C23, C222(Q82S3), (C2×D12).99C22, C23.48(C3⋊D4), C2.20(Q8.13D6), C4⋊Dic3.146C22, C2.18(C23.14D6), (C22×C12).171C22, (C22×C3⋊C8)⋊5C2, (C3×C22⋊Q8)⋊3C2, (C2×C6).498(C2×D4), (C2×Q82S3)⋊10C2, (C2×C3⋊C8).250C22, C2.10(C2×Q82S3), (C2×C4).107(C3⋊D4), (C3×C4⋊C4).114C22, (C2×C4).467(C22×S3), C22.173(C2×C3⋊D4), SmallGroup(192,607)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C3⋊C824D4
C1C3C6C12C2×C12C2×D12C127D4 — C3⋊C824D4
C3C6C2×C12 — C3⋊C824D4
C1C22C22×C4C22⋊Q8

Generators and relations for C3⋊C824D4
 G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b3, bd=db, dcd=c-1 >

Subgroups: 368 in 124 conjugacy classes, 45 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], S3, C6 [×3], C6 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, Dic3, C12 [×2], C12 [×3], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8 [×4], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, C3⋊C8 [×2], C3⋊C8, D12 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C2×C3⋊C8 [×2], C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, Q82S3 [×2], C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C6×Q8, C88D4, C12.Q8, C6.D8, Q82Dic3, C22×C3⋊C8, C127D4, C2×Q82S3, C3×C22⋊Q8, C3⋊C824D4
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, Q82S3 [×2], S3×D4, D42S3, C2×C3⋊D4, C88D4, C23.14D6, C2×Q82S3, Q8.13D6, C3⋊C824D4

Smallest permutation representation of C3⋊C824D4
On 96 points
Generators in S96
(1 81 57)(2 58 82)(3 83 59)(4 60 84)(5 85 61)(6 62 86)(7 87 63)(8 64 88)(9 28 52)(10 53 29)(11 30 54)(12 55 31)(13 32 56)(14 49 25)(15 26 50)(16 51 27)(17 76 72)(18 65 77)(19 78 66)(20 67 79)(21 80 68)(22 69 73)(23 74 70)(24 71 75)(33 48 96)(34 89 41)(35 42 90)(36 91 43)(37 44 92)(38 93 45)(39 46 94)(40 95 47)
(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 94 70 10)(2 89 71 13)(3 92 72 16)(4 95 65 11)(5 90 66 14)(6 93 67 9)(7 96 68 12)(8 91 69 15)(17 27 83 44)(18 30 84 47)(19 25 85 42)(20 28 86 45)(21 31 87 48)(22 26 88 43)(23 29 81 46)(24 32 82 41)(33 80 55 63)(34 75 56 58)(35 78 49 61)(36 73 50 64)(37 76 51 59)(38 79 52 62)(39 74 53 57)(40 77 54 60)
(1 94)(2 95)(3 96)(4 89)(5 90)(6 91)(7 92)(8 93)(9 69)(10 70)(11 71)(12 72)(13 65)(14 66)(15 67)(16 68)(17 55)(18 56)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 78)(26 79)(27 80)(28 73)(29 74)(30 75)(31 76)(32 77)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 81)(40 82)(41 60)(42 61)(43 62)(44 63)(45 64)(46 57)(47 58)(48 59)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G6A6B6C6D6E8A···8H12A12B12C12D12E12F12G12H
order122222234444444666668···81212121212121212
size11112224222228824222446···644448888

36 irreducible representations

dim111111112222222222224444
type++++++++++++++++-+
imageC1C2C2C2C2C2C2C2S3D4D4D4D6D6D6C4○D4SD16C3⋊D4C3⋊D4C4○D8S3×D4D42S3Q82S3Q8.13D6
kernelC3⋊C824D4C12.Q8C6.D8Q82Dic3C22×C3⋊C8C127D4C2×Q82S3C3×C22⋊Q8C22⋊Q8C3⋊C8C2×C12C22×C6C4⋊C4C22×C4C2×Q8C12C2×C6C2×C4C23C6C4C4C22C2
# reps111111111211111242241122

Matrix representation of C3⋊C824D4 in GL6(𝔽73)

100000
010000
001000
000100
0000072
0000172
,
6760000
67670000
001000
000100
00001330
00004360
,
0460000
4600000
00596900
00311400
0000072
0000720
,
0270000
4600000
0014400
0065900
0000720
0000072

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,43,0,0,0,0,30,60],[0,46,0,0,0,0,46,0,0,0,0,0,0,0,59,31,0,0,0,0,69,14,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[0,46,0,0,0,0,27,0,0,0,0,0,0,0,14,6,0,0,0,0,4,59,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

C3⋊C824D4 in GAP, Magma, Sage, TeX

C_3\rtimes C_8\rtimes_{24}D_4
% in TeX

G:=Group("C3:C8:24D4");
// GroupNames label

G:=SmallGroup(192,607);
// by ID

G=gap.SmallGroup(192,607);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,219,184,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=a^-1,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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