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G = D164C4order 128 = 27

3rd semidirect product of D16 and C4 acting via C4/C2=C2

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

Aliases: D164C4, C42.134D4, C166(C2×C4), D83(C2×C4), (C4×D8)⋊37C2, C164C43C2, C165C42C2, C2.18(C4×D8), C4.30(C4×D4), (C2×D16).6C2, (C2×C4).111D8, (C2×C8).212D4, C2.D1616C2, C4.16(C4○D8), C8.43(C4○D4), C8.40(C22×C4), C22.66(C2×D8), C2.5(C16⋊C22), (C4×C8).220C22, (C2×C16).21C22, (C2×C8).507C23, (C2×D8).106C22, C2.D8.155C22, (C2×C4).773(C2×D4), SmallGroup(128,909)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — D164C4
C1C2C4C2×C4C2×C8C4×C8C4×D8 — D164C4
C1C2C4C8 — D164C4
C1C22C42C4×C8 — D164C4
C1C2C2C2C2C4C4C2×C8 — D164C4

Generators and relations for D164C4
 G = < a,b,c | a16=b2=c4=1, bab=a-1, cac-1=a9, cbc-1=a8b >

Subgroups: 244 in 86 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C16, C16, C42, C22⋊C4, C4⋊C4, C2×C8, D8, D8, C22×C4, C2×D4, C4×C8, D4⋊C4, C2.D8, C2×C16, D16, C4×D4, C2×D8, C165C4, C2.D16, C164C4, C4×D8, C2×D16, D164C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, C4×D4, C2×D8, C4○D8, C4×D8, C16⋊C22, D164C4

Smallest permutation representation of D164C4
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 17)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 64)(45 63)(46 62)(47 61)(48 60)
(1 46 25 62)(2 39 26 55)(3 48 27 64)(4 41 28 57)(5 34 29 50)(6 43 30 59)(7 36 31 52)(8 45 32 61)(9 38 17 54)(10 47 18 63)(11 40 19 56)(12 33 20 49)(13 42 21 58)(14 35 22 51)(15 44 23 60)(16 37 24 53)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F4G4H4I4J8A8B8C8D8E8F16A···16H
order122222224···4444488888816···16
size111188882···288882222444···4

32 irreducible representations

dim1111111222224
type++++++++++
imageC1C2C2C2C2C2C4D4D4C4○D4D8C4○D8C16⋊C22
kernelD164C4C165C4C2.D16C164C4C4×D8C2×D16D16C42C2×C8C8C2×C4C4C2
# reps1121218112444

Matrix representation of D164C4 in GL6(𝔽17)

1620000
1610000
0012111512
0016101416
001010814
001311164
,
1600000
1610000
001000
0001600
0000160
0001211
,
400000
040000
000010
001251615
0016000
005131212

G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,2,1,0,0,0,0,0,0,12,16,10,13,0,0,11,10,10,11,0,0,15,14,8,16,0,0,12,16,14,4],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,12,0,0,0,0,16,1,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,12,16,5,0,0,0,5,0,13,0,0,1,16,0,12,0,0,0,15,0,12] >;

D164C4 in GAP, Magma, Sage, TeX

D_{16}\rtimes_4C_4
% in TeX

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

G:=SmallGroup(128,909);
// by ID

G=gap.SmallGroup(128,909);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,436,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^8*b>;
// generators/relations

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