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G = C4xD16order 128 = 27

Direct product of C4 and D16

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

Aliases: C4xD16, C42.329D4, C16:7(C2xC4), (C4xC16):8C2, D8:2(C2xC4), (C4xD8):1C2, C2.13(C4xD8), C2.3(C2xD16), C4.25(C4xD4), C4o2(C16:3C4), C16:3C4:14C2, (C2xD16).7C2, (C2xC4).172D8, (C2xC8).231D4, C4o2(C2.D16), C2.D16:21C2, C4.11(C4oD8), C2.3(C4oD16), C8.38(C4oD4), C8.35(C22xC4), C22.61(C2xD8), (C4xC8).396C22, (C2xC16).69C22, (C2xC8).502C23, (C2xD8).103C22, C2.D8.150C22, (C2xC4).768(C2xD4), SmallGroup(128,904)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C4xD16
C1C2C4C2xC4C2xC8C4xC8C4xD8 — C4xD16
C1C2C4C8 — C4xD16
C1C2xC4C42C4xC8 — C4xD16
C1C2C2C2C2C4C4C2xC8 — C4xD16

Generators and relations for C4xD16
 G = < a,b,c | a4=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 244 in 87 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, C23, C16, C16, C42, C22:C4, C4:C4, C2xC8, D8, D8, C22xC4, C2xD4, C4xC8, D4:C4, C2.D8, C2xC16, D16, C4xD4, C2xD8, C4xC16, C2.D16, C16:3C4, C4xD8, C2xD16, C4xD16
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D8, C22xC4, C2xD4, C4oD4, D16, C4xD4, C2xD8, C4oD8, C4xD8, C2xD16, C4oD16, C4xD16

Smallest permutation representation of C4xD16
On 64 points
Generators in S64
(1 35 23 49)(2 36 24 50)(3 37 25 51)(4 38 26 52)(5 39 27 53)(6 40 28 54)(7 41 29 55)(8 42 30 56)(9 43 31 57)(10 44 32 58)(11 45 17 59)(12 46 18 60)(13 47 19 61)(14 48 20 62)(15 33 21 63)(16 34 22 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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(33 51)(34 50)(35 49)(36 64)(37 63)(38 62)(39 61)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L8A···8H16A···16P
order122222224444444444448···816···16
size111188881111222288882···22···2

44 irreducible representations

dim11111112222222
type++++++++++
imageC1C2C2C2C2C2C4D4D4C4oD4D8D16C4oD8C4oD16
kernelC4xD16C4xC16C2.D16C16:3C4C4xD8C2xD16D16C42C2xC8C8C2xC4C4C4C2
# reps11212181124848

Matrix representation of C4xD16 in GL3(F17) generated by

1300
010
001
,
1600
01311
0613
,
100
010
0016
G:=sub<GL(3,GF(17))| [13,0,0,0,1,0,0,0,1],[16,0,0,0,13,6,0,11,13],[1,0,0,0,1,0,0,0,16] >;

C4xD16 in GAP, Magma, Sage, TeX

C_4\times D_{16}
% in TeX

G:=Group("C4xD16");
// GroupNames label

G:=SmallGroup(128,904);
// by ID

G=gap.SmallGroup(128,904);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,100,1123,570,360,4037,2028,124]);
// Polycyclic

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

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