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G = C8×SD16order 128 = 27

Direct product of C8 and SD16

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

Aliases: C8×SD16, C8210C2, C42.633C23, C85(C2×C8), C8(Q8⋊C8), (C8×Q8)⋊1C2, Q81(C2×C8), C8(C82C8), C8(C4.Q8), C2.5(C8×D4), C82(D4⋊C8), Q8⋊C844C2, C82C832C2, (C8×D4).2C2, D4.1(C2×C8), D4⋊C8.17C2, C8(Q8⋊C4), C2.2(C8○D8), C4.8(C8○D4), (C2×C8).218D4, C4.8(C22×C8), C82(D4⋊C4), C4.Q8.12C4, C2.1(C4×SD16), (C2×SD16).8C4, C22.78(C4×D4), D4⋊C4.14C4, C4.126(C4○D8), C4⋊C8.271C22, (C4×C8).367C22, Q8⋊C4.13C4, (C4×SD16).18C2, C4.100(C2×SD16), (C4×D4).269C22, (C4×Q8).255C22, (C2×C8)(Q8⋊C8), (C2×C8)(C4×SD16), (C2×C8)(C82C8), C4⋊C4.127(C2×C4), (C2×C8).155(C2×C4), (C2×D4).149(C2×C4), (C2×C4).1469(C2×D4), (C2×Q8).131(C2×C4), (C2×C4).494(C4○D4), (C2×C4).325(C22×C4), 2-Sylow(CU(3,5)), SmallGroup(128,308)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8×SD16
C1C2C22C2×C4C42C4×C8C8×Q8 — C8×SD16
C1C2C4 — C8×SD16
C1C2×C8C4×C8 — C8×SD16
C1C22C22C42 — C8×SD16

Generators and relations for C8×SD16
 G = < a,b,c | a8=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Subgroups: 152 in 90 conjugacy classes, 52 normal (40 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×5], C22, C22 [×4], C8 [×4], C8 [×6], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], Q8, C23, C42, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×5], SD16 [×4], C22×C4, C2×D4, C2×Q8, C4×C8 [×3], C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8 [×2], C4⋊C8, C4.Q8, C4×D4, C4×Q8, C22×C8, C2×SD16, C82, D4⋊C8, Q8⋊C8, C82C8, C8×D4, C4×SD16, C8×Q8, C8×SD16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C2×C8 [×6], SD16 [×2], C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C2×SD16, C4○D8, C8×D4, C4×SD16, C8○D8, C8×SD16

Smallest permutation representation of C8×SD16
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 21 13 63 31 52 39 43)(2 22 14 64 32 53 40 44)(3 23 15 57 25 54 33 45)(4 24 16 58 26 55 34 46)(5 17 9 59 27 56 35 47)(6 18 10 60 28 49 36 48)(7 19 11 61 29 50 37 41)(8 20 12 62 30 51 38 42)
(1 5)(2 6)(3 7)(4 8)(9 39)(10 40)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 29)(26 30)(27 31)(28 32)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 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), (1,21,13,63,31,52,39,43)(2,22,14,64,32,53,40,44)(3,23,15,57,25,54,33,45)(4,24,16,58,26,55,34,46)(5,17,9,59,27,56,35,47)(6,18,10,60,28,49,36,48)(7,19,11,61,29,50,37,41)(8,20,12,62,30,51,38,42), (1,5)(2,6)(3,7)(4,8)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,29)(26,30)(27,31)(28,32)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53)>;

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4N8A···8H8I···8AB8AC···8AJ
order122222444444444···48···88···88···8
size111144111122224···41···12···24···4

56 irreducible representations

dim1111111111111222222
type+++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4C8D4SD16C4○D4C8○D4C4○D8C8○D8
kernelC8×SD16C82D4⋊C8Q8⋊C8C82C8C8×D4C4×SD16C8×Q8D4⋊C4Q8⋊C4C4.Q8C2×SD16SD16C2×C8C8C2×C4C4C4C2
# reps11111111222216242448

Matrix representation of C8×SD16 in GL3(𝔽17) generated by

200
040
004
,
100
055
0125
,
100
010
0016
G:=sub<GL(3,GF(17))| [2,0,0,0,4,0,0,0,4],[1,0,0,0,5,12,0,5,5],[1,0,0,0,1,0,0,0,16] >;

C8×SD16 in GAP, Magma, Sage, TeX

C_8\times {\rm SD}_{16}
% in TeX

G:=Group("C8xSD16");
// GroupNames label

G:=SmallGroup(128,308);
// by ID

G=gap.SmallGroup(128,308);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,100,1123,570,136,172]);
// Polycyclic

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

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