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G = C165D4order 128 = 27

2nd semidirect product of C16 and D4 acting via D4/C4=C2

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

Aliases: C165D4, C41SD32, C8.17D8, C42.338D4, C4.9(C2×D8), (C4×C16)⋊13C2, (C2×C4).85D8, C8.41(C2×D4), C4⋊Q167C2, (C2×C8).254D4, (C2×SD32)⋊16C2, C84D4.10C2, C4.3(C41D4), C2.16(C2×SD32), C2.14(C84D4), (C4×C8).403C22, (C2×C16).99C22, (C2×C8).546C23, (C2×D8).17C22, C22.132(C2×D8), (C2×Q16).18C22, (C2×C4).814(C2×D4), SmallGroup(128,980)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C165D4
C1C2C4C2×C4C2×C8C4×C8C4×C16 — C165D4
C1C2C4C2×C8 — C165D4
C1C22C42C4×C8 — C165D4
C1C2C2C2C2C4C4C2×C8 — C165D4

Generators and relations for C165D4
 G = < a,b,c | a16=b4=c2=1, ab=ba, cac=a7, cbc=b-1 >

Subgroups: 296 in 98 conjugacy classes, 40 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C4 [×2], C22, C22 [×6], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×8], Q8 [×4], C23 [×2], C16 [×4], C42, C4⋊C4 [×2], C2×C8 [×2], D8 [×6], Q16 [×6], C2×D4 [×4], C2×Q8 [×2], C4×C8, C2×C16 [×2], SD32 [×8], C41D4, C4⋊Q8, C2×D8 [×2], C2×D8, C2×Q16 [×2], C2×Q16, C4×C16, C84D4, C4⋊Q16, C2×SD32 [×4], C165D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×4], C2×D4 [×3], SD32 [×4], C41D4, C2×D8 [×2], C84D4, C2×SD32 [×2], C165D4

Smallest permutation representation of C165D4
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 41 58 25)(2 42 59 26)(3 43 60 27)(4 44 61 28)(5 45 62 29)(6 46 63 30)(7 47 64 31)(8 48 49 32)(9 33 50 17)(10 34 51 18)(11 35 52 19)(12 36 53 20)(13 37 54 21)(14 38 55 22)(15 39 56 23)(16 40 57 24)
(1 25)(2 32)(3 23)(4 30)(5 21)(6 28)(7 19)(8 26)(9 17)(10 24)(11 31)(12 22)(13 29)(14 20)(15 27)(16 18)(33 50)(34 57)(35 64)(36 55)(37 62)(38 53)(39 60)(40 51)(41 58)(42 49)(43 56)(44 63)(45 54)(46 61)(47 52)(48 59)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111222222
type++++++++++
imageC1C2C2C2C2D4D4D4D8D8SD32
kernelC165D4C4×C16C84D4C4⋊Q16C2×SD32C16C42C2×C8C8C2×C4C4
# reps111144114416

Matrix representation of C165D4 in GL4(𝔽17) generated by

141400
31400
00814
001011
,
01600
1000
0012
001616
,
01600
16000
0012
00016
G:=sub<GL(4,GF(17))| [14,3,0,0,14,14,0,0,0,0,8,10,0,0,14,11],[0,1,0,0,16,0,0,0,0,0,1,16,0,0,2,16],[0,16,0,0,16,0,0,0,0,0,1,0,0,0,2,16] >;

C165D4 in GAP, Magma, Sage, TeX

C_{16}\rtimes_5D_4
% in TeX

G:=Group("C16:5D4");
// GroupNames label

G:=SmallGroup(128,980);
// by ID

G=gap.SmallGroup(128,980);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,736,422,100,1123,360,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^4=c^2=1,a*b=b*a,c*a*c=a^7,c*b*c=b^-1>;
// generators/relations

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