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G = C4.4D16order 128 = 27

4th non-split extension by C4 of D16 acting via D16/C16=C2

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

Aliases: C4.4D16, C4.3SD32, C8.18SD16, C42.333D4, (C4×C16)⋊6C2, C2.9(C2×D16), (C2×C4).81D8, C2.D163C2, C82Q811C2, (C2×C8).250D4, C84D4.9C2, C8.50(C4○D4), C2.14(C2×SD32), C4.14(C2×SD16), (C2×C16).72C22, (C2×C8).538C23, (C4×C8).399C22, C4.7(C4.4D4), C2.9(C4.4D8), (C2×D8).12C22, C22.124(C2×D8), C2.D8.23C22, (C2×C4).806(C2×D4), SmallGroup(128,972)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C4.4D16
C1C2C4C8C2×C8C2×C16C4×C16 — C4.4D16
C1C2C4C2×C8 — C4.4D16
C1C22C42C4×C8 — C4.4D16
C1C2C2C2C2C4C4C2×C8 — C4.4D16

Generators and relations for C4.4D16
 G = < a,b,c | a4=b16=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >

Subgroups: 264 in 80 conjugacy classes, 36 normal (18 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C4 [×2], C22, C22 [×6], C8 [×4], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×2], C23 [×2], C16 [×2], C42, C4⋊C4 [×3], C2×C8 [×2], D8 [×6], C2×D4 [×4], C2×Q8, C4×C8, C2.D8 [×2], C2.D8, C2×C16 [×2], C41D4, C4⋊Q8, C2×D8 [×2], C2×D8, C4×C16, C2.D16 [×4], C84D4, C82Q8, C4.4D16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], SD16 [×2], C2×D4, C4○D4 [×2], D16 [×2], SD32 [×2], C4.4D4, C2×D8, C2×SD16, C4.4D8, C2×D16, C2×SD32, C4.4D16

Smallest permutation representation of C4.4D16
On 64 points
Generators in S64
(1 20 50 33)(2 21 51 34)(3 22 52 35)(4 23 53 36)(5 24 54 37)(6 25 55 38)(7 26 56 39)(8 27 57 40)(9 28 58 41)(10 29 59 42)(11 30 60 43)(12 31 61 44)(13 32 62 45)(14 17 63 46)(15 18 64 47)(16 19 49 48)
(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 49 50 16)(2 15 51 64)(3 63 52 14)(4 13 53 62)(5 61 54 12)(6 11 55 60)(7 59 56 10)(8 9 57 58)(17 35 46 22)(18 21 47 34)(19 33 48 20)(23 45 36 32)(24 31 37 44)(25 43 38 30)(26 29 39 42)(27 41 40 28)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H8A···8H16A···16P
order1222224···4448···816···16
size111116162···216162···22···2

38 irreducible representations

dim111112222222
type+++++++++
imageC1C2C2C2C2D4D4SD16C4○D4D8D16SD32
kernelC4.4D16C4×C16C2.D16C84D4C82Q8C42C2×C8C8C8C2×C4C4C4
# reps114111144488

Matrix representation of C4.4D16 in GL4(𝔽17) generated by

01600
1000
00160
00016
,
161000
71600
00136
001113
,
161000
10100
0064
00411
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,16,0,0,0,0,16],[16,7,0,0,10,16,0,0,0,0,13,11,0,0,6,13],[16,10,0,0,10,1,0,0,0,0,6,4,0,0,4,11] >;

C4.4D16 in GAP, Magma, Sage, TeX

C_4._4D_{16}
% in TeX

G:=Group("C4.4D16");
// GroupNames label

G:=SmallGroup(128,972);
// by ID

G=gap.SmallGroup(128,972);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,120,422,58,1123,360,3924,102,4037,124]);
// Polycyclic

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

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