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G = C6×SD32order 192 = 26·3

Direct product of C6 and SD32

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

Aliases: C6×SD32, C24.69D4, C12.45D8, C4811C22, C24.64C23, (C2×C16)⋊7C6, C163(C2×C6), C4.8(C6×D4), C4.7(C3×D8), (C2×C48)⋊14C2, Q161(C2×C6), (C2×Q16)⋊6C6, D8.1(C2×C6), (C2×D8).4C6, (C2×C6).56D8, C2.13(C6×D8), C8.10(C3×D4), C6.85(C2×D8), (C6×Q16)⋊20C2, (C6×D8).11C2, C8.4(C22×C6), (C2×C12).427D4, C12.315(C2×D4), C22.15(C3×D8), (C3×Q16)⋊15C22, (C3×D8).11C22, (C2×C24).405C22, (C2×C8).85(C2×C6), (C2×C4).83(C3×D4), SmallGroup(192,939)

Series: Derived Chief Lower central Upper central

C1C8 — C6×SD32
C1C2C4C8C24C3×Q16C3×SD32 — C6×SD32
C1C2C4C8 — C6×SD32
C1C2×C6C2×C12C2×C24 — C6×SD32

Generators and relations for C6×SD32
 G = < a,b,c | a6=b16=c2=1, ab=ba, ac=ca, cbc=b7 >

Subgroups: 210 in 90 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C16, C2×C8, D8, D8, Q16, Q16, C2×D4, C2×Q8, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C16, SD32, C2×D8, C2×Q16, C48, C2×C24, C3×D8, C3×D8, C3×Q16, C3×Q16, C6×D4, C6×Q8, C2×SD32, C2×C48, C3×SD32, C6×D8, C6×Q16, C6×SD32
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C3×D4, C22×C6, SD32, C2×D8, C3×D8, C6×D4, C2×SD32, C3×SD32, C6×D8, C6×SD32

Smallest permutation representation of C6×SD32
On 96 points
Generators in S96
(1 56 21 46 92 66)(2 57 22 47 93 67)(3 58 23 48 94 68)(4 59 24 33 95 69)(5 60 25 34 96 70)(6 61 26 35 81 71)(7 62 27 36 82 72)(8 63 28 37 83 73)(9 64 29 38 84 74)(10 49 30 39 85 75)(11 50 31 40 86 76)(12 51 32 41 87 77)(13 52 17 42 88 78)(14 53 18 43 89 79)(15 54 19 44 90 80)(16 55 20 45 91 65)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 25)(18 32)(19 23)(20 30)(22 28)(24 26)(27 31)(33 35)(34 42)(36 40)(37 47)(39 45)(41 43)(44 48)(49 55)(50 62)(51 53)(52 60)(54 58)(57 63)(59 61)(65 75)(67 73)(68 80)(69 71)(70 78)(72 76)(77 79)(81 95)(82 86)(83 93)(85 91)(87 89)(88 96)(90 94)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D6A···6F6G6H6I6J8A8B8C8D12A12B12C12D12E12F12G12H16A···16H24A···24H48A···48P
order1222223344446···666668888121212121212121216···1624···2448···48
size1111881122881···188882222222288882···22···22···2

66 irreducible representations

dim11111111112222222222
type+++++++++
imageC1C2C2C2C2C3C6C6C6C6D4D4D8D8C3×D4C3×D4SD32C3×D8C3×D8C3×SD32
kernelC6×SD32C2×C48C3×SD32C6×D8C6×Q16C2×SD32C2×C16SD32C2×D8C2×Q16C24C2×C12C12C2×C6C8C2×C4C6C4C22C2
# reps114112282211222284416

Matrix representation of C6×SD32 in GL3(𝔽97) generated by

9600
0620
0062
,
9600
06377
01043
,
9600
010
0196
G:=sub<GL(3,GF(97))| [96,0,0,0,62,0,0,0,62],[96,0,0,0,63,10,0,77,43],[96,0,0,0,1,1,0,0,96] >;

C6×SD32 in GAP, Magma, Sage, TeX

C_6\times {\rm SD}_{32}
% in TeX

G:=Group("C6xSD32");
// GroupNames label

G:=SmallGroup(192,939);
// by ID

G=gap.SmallGroup(192,939);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,2524,1271,242,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c|a^6=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
// generators/relations

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