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G = C6.D16order 192 = 26·3

2nd non-split extension by C6 of D16 acting via D16/D8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D245C4, C6.7D16, C24.5D4, C8.15D12, C6.7SD32, C12.4SD16, C2.D81S3, C8.12(C4×S3), C24.9(C2×C4), (C2×C6).33D8, C31(C2.D16), C4.2(D6⋊C4), (C2×D24).8C2, (C2×C12).91D4, (C2×C8).221D6, C2.2(C3⋊D16), C6.5(D4⋊C4), C12.2(C22⋊C4), (C2×C24).73C22, C4.1(Q82S3), C2.2(C8.6D6), C2.7(C6.D8), C22.14(D4⋊S3), (C2×C3⋊C16)⋊4C2, (C3×C2.D8)⋊1C2, (C2×C4).115(C3⋊D4), SmallGroup(192,50)

Series: Derived Chief Lower central Upper central

C1C24 — C6.D16
C1C3C6C12C2×C12C2×C24C2×D24 — C6.D16
C3C6C12C24 — C6.D16
C1C22C2×C4C2×C8C2.D8

Generators and relations for C6.D16
 G = < a,b,c | a6=b16=c2=1, bab-1=cac=a-1, cbc=a3b-1 >

Subgroups: 304 in 66 conjugacy classes, 29 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], S3 [×2], C6 [×3], C8 [×2], C2×C4, C2×C4, D4 [×3], C23, C12 [×2], C12, D6 [×4], C2×C6, C16, C4⋊C4, C2×C8, D8 [×3], C2×D4, C24 [×2], D12 [×3], C2×C12, C2×C12, C22×S3, C2.D8, C2×C16, C2×D8, C3⋊C16, D24 [×2], D24, C3×C4⋊C4, C2×C24, C2×D12, C2.D16, C2×C3⋊C16, C3×C2.D8, C2×D24, C6.D16
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, D16, SD32, D6⋊C4, D4⋊S3, Q82S3, C2.D16, C6.D8, C3⋊D16, C8.6D6, C6.D16

Smallest permutation representation of C6.D16
On 96 points
Generators in S96
(1 24 60 90 75 33)(2 34 76 91 61 25)(3 26 62 92 77 35)(4 36 78 93 63 27)(5 28 64 94 79 37)(6 38 80 95 49 29)(7 30 50 96 65 39)(8 40 66 81 51 31)(9 32 52 82 67 41)(10 42 68 83 53 17)(11 18 54 84 69 43)(12 44 70 85 55 19)(13 20 56 86 71 45)(14 46 72 87 57 21)(15 22 58 88 73 47)(16 48 74 89 59 23)
(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 89)(3 15)(4 87)(5 13)(6 85)(7 11)(8 83)(10 81)(12 95)(14 93)(16 91)(17 51)(18 39)(19 49)(20 37)(21 63)(22 35)(23 61)(24 33)(25 59)(26 47)(27 57)(28 45)(29 55)(30 43)(31 53)(32 41)(34 74)(36 72)(38 70)(40 68)(42 66)(44 80)(46 78)(48 76)(50 69)(52 67)(54 65)(56 79)(58 77)(60 75)(62 73)(64 71)(84 96)(86 94)(88 92)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D6A6B6C8A8B8C8D12A12B12C12D12E12F16A···16H24A24B24C24D
order12222234444666888812121212121216···1624242424
size111124242228822222224488886···64444

36 irreducible representations

dim11111222222222224444
type+++++++++++++++
imageC1C2C2C2C4S3D4D4D6SD16D8C4×S3D12C3⋊D4D16SD32Q82S3D4⋊S3C3⋊D16C8.6D6
kernelC6.D16C2×C3⋊C16C3×C2.D8C2×D24D24C2.D8C24C2×C12C2×C8C12C2×C6C8C8C2×C4C6C6C4C22C2C2
# reps11114111122222441122

Matrix representation of C6.D16 in GL5(𝔽97)

960000
0969600
01000
000960
000096
,
750000
01000
0969600
0005435
000363
,
960000
01000
0969600
00010
0006496

G:=sub<GL(5,GF(97))| [96,0,0,0,0,0,96,1,0,0,0,96,0,0,0,0,0,0,96,0,0,0,0,0,96],[75,0,0,0,0,0,1,96,0,0,0,0,96,0,0,0,0,0,54,3,0,0,0,35,63],[96,0,0,0,0,0,1,96,0,0,0,0,96,0,0,0,0,0,1,64,0,0,0,0,96] >;

C6.D16 in GAP, Magma, Sage, TeX

C_6.D_{16}
% in TeX

G:=Group("C6.D16");
// GroupNames label

G:=SmallGroup(192,50);
// by ID

G=gap.SmallGroup(192,50);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,675,346,192,1684,851,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^6=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=a^3*b^-1>;
// generators/relations

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