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## G = (C2×C4)⋊3SD16order 128 = 27

### 1st semidirect product of C2×C4 and SD16 acting via SD16/C4=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C4)⋊3SD16
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×Q8 — C2×C4⋊Q8 — (C2×C4)⋊3SD16
 Lower central C1 — C2 — C22×C4 — (C2×C4)⋊3SD16
 Upper central C1 — C23 — C2×C42 — (C2×C4)⋊3SD16
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C4)⋊3SD16

Generators and relations for (C2×C4)⋊3SD16
G = < a,b,c,d | a2=b4=c8=d2=1, dbd=ab=ba, ac=ca, ad=da, cbc-1=ab-1, dcd=c3 >

Subgroups: 456 in 203 conjugacy classes, 58 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×9], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×15], D4 [×6], Q8 [×12], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], SD16 [×16], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×4], C2×Q8 [×10], C24, Q8⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C4⋊Q8 [×4], C22×C8 [×2], C2×SD16 [×12], C22×D4, C22×Q8 [×2], C22.7C42, C24.3C22, C2×Q8⋊C4 [×2], C2×C4⋊Q8, C22×SD16 [×2], (C2×C4)⋊3SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, SD16 [×4], C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C2×SD16 [×2], C8.C22 [×2], C232D4, Q8⋊D4 [×2], D4.D4 [×2], C85D4, C8.2D4, (C2×C4)⋊3SD16

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

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 8A ··· 8H order 1 2 ··· 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 8 8 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D4 SD16 C4○D4 C8.C22 kernel (C2×C4)⋊3SD16 C22.7C42 C24.3C22 C2×Q8⋊C4 C2×C4⋊Q8 C22×SD16 C2×C8 C22×C4 C2×D4 C2×Q8 C2×C4 C2×C4 C22 # reps 1 1 1 2 1 2 4 2 2 4 8 2 2

Matrix representation of (C2×C4)⋊3SD16 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 16 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 5 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

(C2×C4)⋊3SD16 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_3{\rm SD}_{16}
% in TeX

G:=Group("(C2xC4):3SD16");
// GroupNames label

G:=SmallGroup(128,745);
// by ID

G=gap.SmallGroup(128,745);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,456,422,387,352,2019,1018,521,248,2804,1411,718,172]);
// Polycyclic

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

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