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G = C3×D8.C4order 192 = 26·3

Direct product of C3 and D8.C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C3×D8.C4, D8.1C12, C12.69D8, C24.101D4, Q16.1C12, (C2×C16)⋊4C6, (C2×C48)⋊8C2, C8.9(C2×C12), C4○D8.1C6, (C3×D8).3C4, C4.18(C3×D8), C8.21(C3×D4), C8.C41C6, C24.55(C2×C4), (C3×Q16).3C4, (C2×C12).407D4, (C2×C6).17SD16, C6.39(D4⋊C4), C22.1(C3×SD16), C12.71(C22⋊C4), (C2×C24).408C22, (C2×C8).88(C2×C6), (C3×C4○D8).6C2, (C2×C4).61(C3×D4), C4.3(C3×C22⋊C4), C2.8(C3×D4⋊C4), (C3×C8.C4)⋊10C2, SmallGroup(192,165)

Series: Derived Chief Lower central Upper central

C1C8 — C3×D8.C4
C1C2C4C2×C4C2×C8C2×C24C3×C8.C4 — C3×D8.C4
C1C2C4C8 — C3×D8.C4
C1C12C2×C12C2×C24 — C3×D8.C4

Generators and relations for C3×D8.C4
 G = < a,b,c,d | a3=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=b5c >

2C2
8C2
4C4
4C22
2C6
8C6
2D4
2Q8
4D4
4C2×C4
4C8
4C2×C6
4C12
2M4(2)
2C4○D4
2C16
2SD16
2C3×D4
2C3×Q8
4C2×C12
4C24
4C3×D4
2C3×C4○D4
2C48
2C3×M4(2)
2C3×SD16

Smallest permutation representation of C3×D8.C4
On 96 points
Generators in S96
(1 46 30)(2 47 31)(3 48 32)(4 41 25)(5 42 26)(6 43 27)(7 44 28)(8 45 29)(9 84 68)(10 85 69)(11 86 70)(12 87 71)(13 88 72)(14 81 65)(15 82 66)(16 83 67)(17 55 33)(18 56 34)(19 49 35)(20 50 36)(21 51 37)(22 52 38)(23 53 39)(24 54 40)(57 89 73)(58 90 74)(59 91 75)(60 92 76)(61 93 77)(62 94 78)(63 95 79)(64 96 80)
(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 8)(2 7)(3 6)(4 5)(9 13)(10 12)(14 16)(17 20)(18 19)(21 24)(22 23)(25 26)(27 32)(28 31)(29 30)(33 36)(34 35)(37 40)(38 39)(41 42)(43 48)(44 47)(45 46)(49 56)(50 55)(51 54)(52 53)(57 61)(58 60)(62 64)(65 67)(68 72)(69 71)(73 77)(74 76)(78 80)(81 83)(84 88)(85 87)(89 93)(90 92)(94 96)
(1 71 23 60 5 67 19 64)(2 70 24 59 6 66 20 63)(3 69 17 58 7 65 21 62)(4 68 18 57 8 72 22 61)(9 56 89 45 13 52 93 41)(10 55 90 44 14 51 94 48)(11 54 91 43 15 50 95 47)(12 53 92 42 16 49 96 46)(25 84 34 73 29 88 38 77)(26 83 35 80 30 87 39 76)(27 82 36 79 31 86 40 75)(28 81 37 78 32 85 33 74)

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

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

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

66 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A6B6C6D6E6F8A8B8C8D8E8F12A12B12C12D12E12F12G12H16A···16H24A···24H24I24J24K24L48A···48P
order1222334444666666888888121212121212121216···1624···242424242448···48
size1128111128112288222288111122882···22···288882···2

66 irreducible representations

dim1111111111112222222222
type+++++++
imageC1C2C2C2C3C4C4C6C6C6C12C12D4D4D8SD16C3×D4C3×D4C3×D8C3×SD16D8.C4C3×D8.C4
kernelC3×D8.C4C3×C8.C4C2×C48C3×C4○D8D8.C4C3×D8C3×Q16C8.C4C2×C16C4○D8D8Q16C24C2×C12C12C2×C6C8C2×C4C4C22C3C1
# reps11112222224411222244816

Matrix representation of C3×D8.C4 in GL2(𝔽97) generated by

610
061
,
790
77
,
77
790
,
3994
9458
G:=sub<GL(2,GF(97))| [61,0,0,61],[7,7,90,7],[7,7,7,90],[39,94,94,58] >;

C3×D8.C4 in GAP, Magma, Sage, TeX

C_3\times D_8.C_4
% in TeX

G:=Group("C3xD8.C4");
// GroupNames label

G:=SmallGroup(192,165);
// by ID

G=gap.SmallGroup(192,165);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,850,360,172,6053,3036,124]);
// Polycyclic

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

Export

Subgroup lattice of C3×D8.C4 in TeX

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