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G = C2×C2.D16order 128 = 27

Direct product of C2 and C2.D16

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

Aliases: C2×C2.D16, C23.57D8, C22.12D16, C22.10SD32, D87(C2×C4), (C2×D8)⋊9C4, C8.90(C2×D4), C2.1(C2×D16), (C2×C4).77D8, (C22×C16)⋊4C2, (C2×C8).246D4, C2.1(C2×SD32), C4.8(C2×SD16), (C2×C16)⋊16C22, C8.27(C22×C4), (C2×C4).75SD16, (C22×D8).6C2, C22.51(C2×D8), C2.D842C22, C8.24(C22⋊C4), (C2×C8).492C23, (C2×D8).99C22, (C22×C4).582D4, C4.23(D4⋊C4), (C22×C8).526C22, C22.52(D4⋊C4), (C2×C2.D8)⋊15C2, (C2×C8).174(C2×C4), (C2×C4).754(C2×D4), C4.48(C2×C22⋊C4), C2.26(C2×D4⋊C4), (C2×C4).269(C22⋊C4), SmallGroup(128,868)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C2×C2.D16
C1C2C4C2×C4C2×C8C22×C8C22×D8 — C2×C2.D16
C1C2C4C8 — C2×C2.D16
C1C23C22×C4C22×C8 — C2×C2.D16
C1C2C2C2C2C4C4C2×C8 — C2×C2.D16

Generators and relations for C2×C2.D16
 G = < a,b,c,d | a2=b2=c16=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 404 in 136 conjugacy classes, 60 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×16], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], D4 [×10], C23, C23 [×10], C16 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×4], D8 [×4], D8 [×6], C22×C4, C22×C4, C2×D4 [×9], C24, C2.D8 [×2], C2.D8, C2×C16 [×2], C2×C16 [×2], C2×C4⋊C4, C22×C8, C2×D8 [×6], C2×D8 [×3], C22×D4, C2.D16 [×4], C2×C2.D8, C22×C16, C22×D8, C2×C2.D16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], D16 [×2], SD32 [×2], C2×C22⋊C4, C2×D8, C2×SD16, C2.D16 [×4], C2×D4⋊C4, C2×D16, C2×SD32, C2×C2.D16

Smallest permutation representation of C2×C2.D16
On 64 points
Generators in S64
(1 46)(2 47)(3 48)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 49)(29 50)(30 51)(31 52)(32 53)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 17)(16 18)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)
(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 18 19 16)(2 15 20 17)(3 32 21 14)(4 13 22 31)(5 30 23 12)(6 11 24 29)(7 28 25 10)(8 9 26 27)(33 42 59 52)(34 51 60 41)(35 40 61 50)(36 49 62 39)(37 38 63 64)(43 48 53 58)(44 57 54 47)(45 46 55 56)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H8A···8H16A···16P
order12···22222444444448···816···16
size11···18888222288882···22···2

44 irreducible representations

dim1111112222222
type++++++++++
imageC1C2C2C2C2C4D4D4D8SD16D8D16SD32
kernelC2×C2.D16C2.D16C2×C2.D8C22×C16C22×D8C2×D8C2×C8C22×C4C2×C4C2×C4C23C22C22
# reps1411183124288

Matrix representation of C2×C2.D16 in GL4(𝔽17) generated by

16000
0100
0010
0001
,
1000
01600
00160
00016
,
16000
0400
00167
001016
,
1000
01300
00167
0071
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,4,0,0,0,0,16,10,0,0,7,16],[1,0,0,0,0,13,0,0,0,0,16,7,0,0,7,1] >;

C2×C2.D16 in GAP, Magma, Sage, TeX

C_2\times C_2.D_{16}
% in TeX

G:=Group("C2xC2.D16");
// GroupNames label

G:=SmallGroup(128,868);
// by ID

G=gap.SmallGroup(128,868);
# by ID

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

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

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