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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C4×SD16
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×Q8 — C2×C4×SD16
 Lower central C1 — C2 — C4 — C2×C4×SD16
 Upper central C1 — C22×C4 — C2×C42 — C2×C4×SD16
 Jennings C1 — C2 — C2 — C2×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, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4×Q8, C22×C8, C2×SD16, C23×C4, C22×D4, C22×Q8, C2×C4×C8, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.Q8, C4×SD16, C2×C4×D4, C2×C4×Q8, C22×SD16, C2×C4×SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, SD16, C22×C4, C2×D4, C4○D4, C24, C4×D4, C2×SD16, C4○D8, C23×C4, C22×D4, C2×C4○D4, C4×SD16, 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 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 56)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(1 33 55 47)(2 34 56 48)(3 35 49 41)(4 36 50 42)(5 37 51 43)(6 38 52 44)(7 39 53 45)(8 40 54 46)(9 24 63 26)(10 17 64 27)(11 18 57 28)(12 19 58 29)(13 20 59 30)(14 21 60 31)(15 22 61 32)(16 23 62 25)
(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 37)(10 40)(11 35)(12 38)(13 33)(14 36)(15 39)(16 34)(17 54)(18 49)(19 52)(20 55)(21 50)(22 53)(23 56)(24 51)(41 57)(42 60)(43 63)(44 58)(45 61)(46 64)(47 59)(48 62)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4P 4Q ··· 4AB 8A ··· 8P order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 SD16 C4○D4 C4○D8 kernel C2×C4×SD16 C2×C4×C8 C2×D4⋊C4 C2×Q8⋊C4 C2×C4.Q8 C4×SD16 C2×C4×D4 C2×C4×Q8 C22×SD16 C2×SD16 C42 C22×C4 C2×C4 C2×C4 C22 # reps 1 1 1 1 1 8 1 1 1 16 2 2 8 4 8

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

 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 12 0 0 5 12
,
 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 1
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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