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G = C3xC8:3D4order 192 = 26·3

Direct product of C3 and C8:3D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C3xC8:3D4, C24:18D4, C8:3(C3xD4), (C2xD8):9C6, C8:C4:4C6, C4.5(C6xD4), (C6xD8):23C2, C4:1D4:6C6, (C2xSD16):3C6, C4.4D4:5C6, (C6xSD16):14C2, C12.312(C2xD4), (C2xC12).343D4, C42.29(C2xC6), C6.46(C4:1D4), C22.117(C6xD4), C6.147(C8:C22), (C2xC24).203C22, (C2xC12).952C23, (C4xC12).271C22, (C6xD4).205C22, (C6xQ8).179C22, (C3xC8:C4):9C2, (C2xC8).27(C2xC6), (C2xC4).44(C3xD4), C2.9(C3xC4:1D4), (C3xC4:1D4):14C2, (C2xD4).28(C2xC6), (C2xC6).673(C2xD4), C2.22(C3xC8:C22), (C2xQ8).24(C2xC6), (C3xC4.4D4):25C2, (C2xC4).127(C22xC6), SmallGroup(192,929)

Series: Derived Chief Lower central Upper central

C1C2xC4 — C3xC8:3D4
C1C2C22C2xC4C2xC12C6xD4C6xSD16 — C3xC8:3D4
C1C2C2xC4 — C3xC8:3D4
C1C2xC6C4xC12 — C3xC8:3D4

Generators and relations for C3xC8:3D4
 G = < a,b,c,d | a3=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 322 in 144 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C12, C12, C2xC6, C2xC6, C42, C22:C4, C2xC8, D8, SD16, C2xD4, C2xD4, C2xD4, C2xQ8, C24, C2xC12, C2xC12, C2xC12, C3xD4, C3xQ8, C22xC6, C8:C4, C4.4D4, C4:1D4, C2xD8, C2xSD16, C4xC12, C3xC22:C4, C2xC24, C3xD8, C3xSD16, C6xD4, C6xD4, C6xD4, C6xQ8, C8:3D4, C3xC8:C4, C3xC4.4D4, C3xC4:1D4, C6xD8, C6xSD16, C3xC8:3D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, C2xD4, C3xD4, C22xC6, C4:1D4, C8:C22, C6xD4, C8:3D4, C3xC4:1D4, C3xC8:C22, C3xC8:3D4

Smallest permutation representation of C3xC8:3D4
On 96 points
Generators in S96
(1 65 17)(2 66 18)(3 67 19)(4 68 20)(5 69 21)(6 70 22)(7 71 23)(8 72 24)(9 32 62)(10 25 63)(11 26 64)(12 27 57)(13 28 58)(14 29 59)(15 30 60)(16 31 61)(33 73 81)(34 74 82)(35 75 83)(36 76 84)(37 77 85)(38 78 86)(39 79 87)(40 80 88)(41 55 89)(42 56 90)(43 49 91)(44 50 92)(45 51 93)(46 52 94)(47 53 95)(48 54 96)
(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 79 25)(2 44 80 30)(3 41 73 27)(4 46 74 32)(5 43 75 29)(6 48 76 26)(7 45 77 31)(8 42 78 28)(9 20 94 34)(10 17 95 39)(11 22 96 36)(12 19 89 33)(13 24 90 38)(14 21 91 35)(15 18 92 40)(16 23 93 37)(49 83 59 69)(50 88 60 66)(51 85 61 71)(52 82 62 68)(53 87 63 65)(54 84 64 70)(55 81 57 67)(56 86 58 72)
(1 74)(2 73)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 10)(11 16)(12 15)(13 14)(17 34)(18 33)(19 40)(20 39)(21 38)(22 37)(23 36)(24 35)(25 32)(26 31)(27 30)(28 29)(41 44)(42 43)(45 48)(46 47)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 64)(62 63)(65 82)(66 81)(67 88)(68 87)(69 86)(70 85)(71 84)(72 83)(89 92)(90 91)(93 96)(94 95)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F3A3B4A4B4C4D4E6A···6F6G···6L8A8B8C8D12A12B12C12D12E12F12G12H12I12J24A···24H
order122222233444446···66···688881212121212121212121224···24
size111188811224481···18···8444422224444884···4

48 irreducible representations

dim111111111111222244
type+++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4C3xD4C3xD4C8:C22C3xC8:C22
kernelC3xC8:3D4C3xC8:C4C3xC4.4D4C3xC4:1D4C6xD8C6xSD16C8:3D4C8:C4C4.4D4C4:1D4C2xD8C2xSD16C24C2xC12C8C2xC4C6C2
# reps111122222244428424

Matrix representation of C3xC8:3D4 in GL8(F73)

80000000
08000000
00100000
00010000
00001000
00000100
00000010
00000001
,
01000000
720000000
001710000
001720000
00005621954
0000712054
00002171558
0000170710
,
720000000
072000000
001710000
001720000
00000010
0000721171
000072000
0000721072
,
01000000
10000000
001710000
000720000
00005621954
0000271019
00002171558
0000215714

G:=sub<GL(8,GF(73))| [8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,56,71,2,17,0,0,0,0,2,2,17,0,0,0,0,0,19,0,15,71,0,0,0,0,54,54,58,0],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,0,72,72,72,0,0,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,71,0,72],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,56,2,2,2,0,0,0,0,2,71,17,15,0,0,0,0,19,0,15,71,0,0,0,0,54,19,58,4] >;

C3xC8:3D4 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes_3D_4
% in TeX

G:=Group("C3xC8:3D4");
// GroupNames label

G:=SmallGroup(192,929);
// by ID

G=gap.SmallGroup(192,929);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,176,1094,1059,268,4204,172]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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