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G = C2×C4×SD16order 128 = 27

Direct product of C2×C4 and SD16

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2×C4×SD16, C42.350D4, C42.684C23, C4(C4×SD16), C86(C22×C4), C4.43(C4×D4), (C4×C8)⋊76C22, Q81(C22×C4), C4.14(C23×C4), (C4×Q8)⋊74C22, D4.1(C22×C4), C4.Q873C22, C4⋊C4.354C23, (C2×C8).590C23, (C2×C4).194C24, (C22×C4).825D4, C22.116(C4×D4), C23.841(C2×D4), C2.4(C22×SD16), (C2×D4).365C23, (C4×D4).288C22, C22.84(C4○D8), (C2×Q8).337C23, C22.81(C2×SD16), Q8⋊C4100C22, (C22×C8).511C22, (C22×SD16).15C2, C22.138(C22×D4), D4⋊C4.214C22, (C2×C42).1112C22, (C22×C4).1510C23, (C2×SD16).176C22, (C22×D4).556C22, (C22×Q8).459C22, (C2×C4×C8)⋊34C2, (C2×C4×Q8)⋊31C2, C42(C2×C4.Q8), C2.54(C2×C4×D4), (C2×C8)⋊31(C2×C4), (C2×C4)(C4×SD16), (C2×C4×D4).73C2, C4.2(C2×C4○D4), C2.4(C2×C4○D8), C43(C2×D4⋊C4), C42(C2×Q8⋊C4), (C2×Q8)⋊27(C2×C4), (C2×C4.Q8)⋊39C2, (C2×C4)3(C4.Q8), (C2×C4)4(D4⋊C4), (C2×C4)3(Q8⋊C4), (C2×Q8⋊C4)⋊60C2, (C2×D4).175(C2×C4), (C2×C4).1574(C2×D4), (C2×D4⋊C4).40C2, (C2×C4).686(C4○D4), (C2×C4⋊C4).907C22, (C2×C4).465(C22×C4), (C2×C4)2(C2×C4.Q8), (C2×C4)3(C2×D4⋊C4), (C2×C4)2(C2×Q8⋊C4), SmallGroup(128,1669)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×C4×SD16
C1C2C22C2×C4C22×C4C2×C42C2×C4×Q8 — C2×C4×SD16
C1C2C4 — C2×C4×SD16
C1C22×C4C2×C42 — C2×C4×SD16
C1C2C2C2×C4 — C2×C4×SD16

Generators and relations for C2×C4×SD16
 G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 476 in 276 conjugacy classes, 156 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×6], C22 [×16], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×12], C2×C4 [×24], D4 [×4], D4 [×6], Q8 [×4], Q8 [×6], C23, C23 [×10], C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×2], SD16 [×16], C22×C4 [×3], C22×C4 [×11], C2×D4 [×6], C2×D4 [×3], C2×Q8 [×6], C2×Q8 [×3], C24, C4×C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×4], C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C4×D4 [×2], C4×Q8 [×4], C4×Q8 [×2], C22×C8 [×2], C2×SD16 [×12], C23×C4, C22×D4, C22×Q8, C2×C4×C8, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.Q8, C4×SD16 [×8], C2×C4×D4, C2×C4×Q8, C22×SD16, C2×C4×SD16
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], SD16 [×4], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C2×SD16 [×6], C4○D8 [×2], C23×C4, C22×D4, C2×C4○D4, C4×SD16 [×4], C2×C4×D4, C22×SD16, C2×C4○D8, C2×C4×SD16

Smallest permutation representation of C2×C4×SD16
On 64 points
Generators in S64
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 49)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)
(1 33 55 9)(2 34 56 10)(3 35 49 11)(4 36 50 12)(5 37 51 13)(6 38 52 14)(7 39 53 15)(8 40 54 16)(17 58 29 46)(18 59 30 47)(19 60 31 48)(20 61 32 41)(21 62 25 42)(22 63 26 43)(23 64 27 44)(24 57 28 45)
(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 30)(2 25)(3 28)(4 31)(5 26)(6 29)(7 32)(8 27)(9 59)(10 62)(11 57)(12 60)(13 63)(14 58)(15 61)(16 64)(17 52)(18 55)(19 50)(20 53)(21 56)(22 51)(23 54)(24 49)(33 47)(34 42)(35 45)(36 48)(37 43)(38 46)(39 41)(40 44)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P4Q···4AB8A···8P
order12···222224···44···44···48···8
size11···144441···12···24···42···2

56 irreducible representations

dim111111111122222
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C4D4D4SD16C4○D4C4○D8
kernelC2×C4×SD16C2×C4×C8C2×D4⋊C4C2×Q8⋊C4C2×C4.Q8C4×SD16C2×C4×D4C2×C4×Q8C22×SD16C2×SD16C42C22×C4C2×C4C2×C4C22
# reps1111181111622848

Matrix representation of C2×C4×SD16 in GL4(𝔽17) generated by

16000
0100
0010
0001
,
1000
01300
00160
00016
,
1000
0100
001212
00512
,
1000
01600
00160
0001
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,12,5,0,0,12,12],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1] >;

C2×C4×SD16 in GAP, Magma, Sage, TeX

C_2\times C_4\times {\rm SD}_{16}
% in TeX

G:=Group("C2xC4xSD16");
// GroupNames label

G:=SmallGroup(128,1669);
// by ID

G=gap.SmallGroup(128,1669);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,184,2804,1411,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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