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G = C3⋊C8.29D4order 192 = 26·3

6th non-split extension by C3⋊C8 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3⋊C8.29D4, (C2×C6)⋊3Q16, C4⋊C4.70D6, C4.176(S3×D4), (C2×Q8).56D6, C6.39(C2×Q16), C22⋊Q8.6S3, C12.157(C2×D4), C6.Q1641C2, (C2×C12).268D4, C33(C8.18D4), (C22×C6).97D4, C6.102(C4○D8), Q82Dic317C2, C6.SD1639C2, C6.99(C4⋊D4), C222(C3⋊Q16), (C22×C4).359D6, (C6×Q8).50C22, C12.192(C4○D4), C4.65(D42S3), (C2×C12).370C23, C23.49(C3⋊D4), C12.48D4.12C2, C2.21(Q8.13D6), C4⋊Dic3.148C22, C2.20(C23.14D6), (C22×C12).174C22, (C2×Dic6).105C22, (C22×C3⋊C8).9C2, (C2×C3⋊Q16)⋊10C2, (C2×C6).501(C2×D4), C2.10(C2×C3⋊Q16), (C3×C22⋊Q8).5C2, (C2×C3⋊C8).251C22, (C2×C4).108(C3⋊D4), (C3×C4⋊C4).117C22, (C2×C4).470(C22×S3), C22.176(C2×C3⋊D4), SmallGroup(192,610)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C3⋊C8.29D4
C1C3C6C12C2×C12C2×Dic6C12.48D4 — C3⋊C8.29D4
C3C6C2×C12 — C3⋊C8.29D4
C1C22C22×C4C22⋊Q8

Generators and relations for C3⋊C8.29D4
 G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b-1, bd=db, dcd=b4c-1 >

Subgroups: 272 in 114 conjugacy classes, 45 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×6], Q8 [×4], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8 [×4], Q16 [×2], C22×C4, C2×Q8, C2×Q8, C3⋊C8 [×2], C3⋊C8, Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], C22×C6, Q8⋊C4 [×2], C2.D8, C22⋊Q8, C22⋊Q8, C22×C8, C2×Q16, C2×C3⋊C8 [×2], C2×C3⋊C8 [×2], Dic3⋊C4, C4⋊Dic3, C3⋊Q16 [×2], C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×Q8, C8.18D4, C6.Q16, C6.SD16, Q82Dic3, C22×C3⋊C8, C12.48D4, C2×C3⋊Q16, C3×C22⋊Q8, C3⋊C8.29D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], Q16 [×2], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C2×Q16, C4○D8, C3⋊Q16 [×2], S3×D4, D42S3, C2×C3⋊D4, C8.18D4, C23.14D6, C2×C3⋊Q16, Q8.13D6, C3⋊C8.29D4

Smallest permutation representation of C3⋊C8.29D4
On 96 points
Generators in S96
(1 83 94)(2 95 84)(3 85 96)(4 89 86)(5 87 90)(6 91 88)(7 81 92)(8 93 82)(9 30 49)(10 50 31)(11 32 51)(12 52 25)(13 26 53)(14 54 27)(15 28 55)(16 56 29)(17 64 33)(18 34 57)(19 58 35)(20 36 59)(21 60 37)(22 38 61)(23 62 39)(24 40 63)(41 75 70)(42 71 76)(43 77 72)(44 65 78)(45 79 66)(46 67 80)(47 73 68)(48 69 74)
(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 47 12 24)(2 46 13 23)(3 45 14 22)(4 44 15 21)(5 43 16 20)(6 42 9 19)(7 41 10 18)(8 48 11 17)(25 40 94 73)(26 39 95 80)(27 38 96 79)(28 37 89 78)(29 36 90 77)(30 35 91 76)(31 34 92 75)(32 33 93 74)(49 58 88 71)(50 57 81 70)(51 64 82 69)(52 63 83 68)(53 62 84 67)(54 61 85 66)(55 60 86 65)(56 59 87 72)
(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(33 78)(34 79)(35 80)(36 73)(37 74)(38 75)(39 76)(40 77)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(64 65)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H6A6B6C6D6E8A···8H12A12B12C12D12E12F12G12H
order122222344444444666668···81212121212121212
size11112222222882424222446···644448888

36 irreducible representations

dim111111112222222222224444
type+++++++++++++++-+--
imageC1C2C2C2C2C2C2C2S3D4D4D4D6D6D6C4○D4Q16C3⋊D4C3⋊D4C4○D8S3×D4D42S3C3⋊Q16Q8.13D6
kernelC3⋊C8.29D4C6.Q16C6.SD16Q82Dic3C22×C3⋊C8C12.48D4C2×C3⋊Q16C3×C22⋊Q8C22⋊Q8C3⋊C8C2×C12C22×C6C4⋊C4C22×C4C2×Q8C12C2×C6C2×C4C23C6C4C4C22C2
# reps111111111211111242241122

Matrix representation of C3⋊C8.29D4 in GL6(𝔽73)

100000
010000
00727200
001000
000010
000001
,
6300000
5510000
0066700
0014700
0000270
0000046
,
6470000
7290000
0076600
00596600
0000027
0000270
,
100000
13720000
001000
000100
000010
0000072

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[63,5,0,0,0,0,0,51,0,0,0,0,0,0,66,14,0,0,0,0,7,7,0,0,0,0,0,0,27,0,0,0,0,0,0,46],[64,72,0,0,0,0,7,9,0,0,0,0,0,0,7,59,0,0,0,0,66,66,0,0,0,0,0,0,0,27,0,0,0,0,27,0],[1,13,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;

C3⋊C8.29D4 in GAP, Magma, Sage, TeX

C_3\rtimes C_8._{29}D_4
% in TeX

G:=Group("C3:C8.29D4");
// GroupNames label

G:=SmallGroup(192,610);
// by ID

G=gap.SmallGroup(192,610);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,219,184,1123,297,136,6278]);
// 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^-1,b*d=d*b,d*c*d=b^4*c^-1>;
// generators/relations

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