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G = C16.19D4order 128 = 27

5th non-split extension by C16 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C16.19D4, C221Q32, C23.27D8, (C2×Q32)⋊5C2, C163C43C2, (C2×C4).60D8, C2.6(C2×Q32), (C2×C8).235D4, C8.100(C2×D4), C2.Q321C2, C8.45(C4○D4), C4.18(C4○D8), C2.11(C4○D16), C2.19(C87D4), C4.91(C4⋊D4), (C2×C8).521C23, C2.D8.6C22, (C2×C16).82C22, (C22×C16).11C2, C8.18D4.4C2, C22.107(C2×D8), (C22×C4).591D4, (C2×Q16).6C22, (C22×C8).531C22, (C2×C4).789(C2×D4), SmallGroup(128,948)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C16.19D4
C1C2C4C2×C4C2×C8C22×C8C22×C16 — C16.19D4
C1C2C4C2×C8 — C16.19D4
C1C22C22×C4C22×C8 — C16.19D4
C1C2C2C2C2C4C4C2×C8 — C16.19D4

Generators and relations for C16.19D4
 G = < a,b,c | a16=b4=1, c2=a8, bab-1=cac-1=a-1, cbc-1=a8b-1 >

Subgroups: 168 in 77 conjugacy classes, 34 normal (22 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C8 [×2], C8, C2×C4 [×2], C2×C4 [×6], Q8 [×4], C23, C16 [×2], C16, C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×2], Q16 [×4], C22×C4, C2×Q8 [×2], Q8⋊C4 [×2], C2.D8 [×2], C2×C16 [×2], C2×C16 [×2], Q32 [×2], C22⋊Q8 [×2], C22×C8, C2×Q16 [×2], C2.Q32 [×2], C163C4, C8.18D4 [×2], C22×C16, C2×Q32, C16.19D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, Q32 [×2], C4⋊D4, C2×D8, C4○D8, C87D4, C2×Q32, C4○D16, C16.19D4

Smallest permutation representation of C16.19D4
On 64 points
Generators in S64
(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)
(1 27 45 62)(2 26 46 61)(3 25 47 60)(4 24 48 59)(5 23 33 58)(6 22 34 57)(7 21 35 56)(8 20 36 55)(9 19 37 54)(10 18 38 53)(11 17 39 52)(12 32 40 51)(13 31 41 50)(14 30 42 49)(15 29 43 64)(16 28 44 63)
(1 54 9 62)(2 53 10 61)(3 52 11 60)(4 51 12 59)(5 50 13 58)(6 49 14 57)(7 64 15 56)(8 63 16 55)(17 39 25 47)(18 38 26 46)(19 37 27 45)(20 36 28 44)(21 35 29 43)(22 34 30 42)(23 33 31 41)(24 48 32 40)

G:=sub<Sym(64)| (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), (1,27,45,62)(2,26,46,61)(3,25,47,60)(4,24,48,59)(5,23,33,58)(6,22,34,57)(7,21,35,56)(8,20,36,55)(9,19,37,54)(10,18,38,53)(11,17,39,52)(12,32,40,51)(13,31,41,50)(14,30,42,49)(15,29,43,64)(16,28,44,63), (1,54,9,62)(2,53,10,61)(3,52,11,60)(4,51,12,59)(5,50,13,58)(6,49,14,57)(7,64,15,56)(8,63,16,55)(17,39,25,47)(18,38,26,46)(19,37,27,45)(20,36,28,44)(21,35,29,43)(22,34,30,42)(23,33,31,41)(24,48,32,40)>;

G:=Group( (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), (1,27,45,62)(2,26,46,61)(3,25,47,60)(4,24,48,59)(5,23,33,58)(6,22,34,57)(7,21,35,56)(8,20,36,55)(9,19,37,54)(10,18,38,53)(11,17,39,52)(12,32,40,51)(13,31,41,50)(14,30,42,49)(15,29,43,64)(16,28,44,63), (1,54,9,62)(2,53,10,61)(3,52,11,60)(4,51,12,59)(5,50,13,58)(6,49,14,57)(7,64,15,56)(8,63,16,55)(17,39,25,47)(18,38,26,46)(19,37,27,45)(20,36,28,44)(21,35,29,43)(22,34,30,42)(23,33,31,41)(24,48,32,40) );

G=PermutationGroup([(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)], [(1,27,45,62),(2,26,46,61),(3,25,47,60),(4,24,48,59),(5,23,33,58),(6,22,34,57),(7,21,35,56),(8,20,36,55),(9,19,37,54),(10,18,38,53),(11,17,39,52),(12,32,40,51),(13,31,41,50),(14,30,42,49),(15,29,43,64),(16,28,44,63)], [(1,54,9,62),(2,53,10,61),(3,52,11,60),(4,51,12,59),(5,50,13,58),(6,49,14,57),(7,64,15,56),(8,63,16,55),(17,39,25,47),(18,38,26,46),(19,37,27,45),(20,36,28,44),(21,35,29,43),(22,34,30,42),(23,33,31,41),(24,48,32,40)])

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H8A···8H16A···16P
order122222444444448···816···16
size1111222222161616162···22···2

38 irreducible representations

dim111111222222222
type+++++++++++-
imageC1C2C2C2C2C2D4D4D4C4○D4D8D8C4○D8Q32C4○D16
kernelC16.19D4C2.Q32C163C4C8.18D4C22×C16C2×Q32C16C2×C8C22×C4C8C2×C4C23C4C22C2
# reps121211211222488

Matrix representation of C16.19D4 in GL4(𝔽17) generated by

16000
01600
0030
0006
,
0100
16000
0001
0010
,
0100
1000
0001
00160
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,3,0,0,0,0,6],[0,16,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,16,0,0,1,0] >;

C16.19D4 in GAP, Magma, Sage, TeX

C_{16}._{19}D_4
% in TeX

G:=Group("C16.19D4");
// GroupNames label

G:=SmallGroup(128,948);
// by ID

G=gap.SmallGroup(128,948);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,288,422,1123,360,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^4=1,c^2=a^8,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^8*b^-1>;
// generators/relations

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