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## G = C8.22SD16order 128 = 27

### 8th non-split extension by C8 of SD16 acting via SD16/C8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C8.22SD16
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — C4×C16 — C8.22SD16
 Lower central C1 — C2 — C4 — C2×C8 — C8.22SD16
 Upper central C1 — C22 — C42 — C4×C8 — C8.22SD16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C8.22SD16

Generators and relations for C8.22SD16
G = < a,b,c | a8=c2=1, b8=a4, ab=ba, cac=a3, cbc=a2b3 >

Subgroups: 184 in 69 conjugacy classes, 32 normal (16 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×2], C23, C16 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], D8 [×2], SD16 [×2], Q16 [×2], C2×D4, C2×Q8, C4×C8, C4.Q8, C2.D8 [×2], C2×C16 [×2], C4.4D4, C42.C2, C2×D8, C2×SD16, C2×Q16, C4×C16, C2.D16 [×2], C2.Q32 [×2], C8.12D4, C8.5Q8, C8.22SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], SD16 [×2], C2×D4, C4○D4 [×2], C4.4D4, C2×D8, C2×SD16, C4.4D8, C4○D16 [×2], C8.22SD16

Smallest permutation representation of C8.22SD16
On 64 points
Generators in S64
(1 32 33 49 9 24 41 57)(2 17 34 50 10 25 42 58)(3 18 35 51 11 26 43 59)(4 19 36 52 12 27 44 60)(5 20 37 53 13 28 45 61)(6 21 38 54 14 29 46 62)(7 22 39 55 15 30 47 63)(8 23 40 56 16 31 48 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)
(2 36)(3 15)(4 34)(5 13)(6 48)(7 11)(8 46)(10 44)(12 42)(14 40)(16 38)(17 27)(18 63)(19 25)(20 61)(21 23)(22 59)(24 57)(26 55)(28 53)(29 31)(30 51)(32 49)(33 41)(35 39)(43 47)(50 52)(54 64)(56 62)(58 60)

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 4A ··· 4F 4G 4H 4I 8A ··· 8H 16A ··· 16P order 1 2 2 2 2 4 ··· 4 4 4 4 8 ··· 8 16 ··· 16 size 1 1 1 1 16 2 ··· 2 16 16 16 2 ··· 2 2 ··· 2

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 SD16 C4○D4 D8 C4○D16 kernel C8.22SD16 C4×C16 C2.D16 C2.Q32 C8.12D4 C8.5Q8 C42 C2×C8 C8 C8 C2×C4 C2 # reps 1 1 2 2 1 1 1 1 4 4 4 16

Matrix representation of C8.22SD16 in GL4(𝔽17) generated by

 0 13 0 0 13 0 0 0 0 0 0 10 0 0 12 10
,
 0 1 0 0 1 0 0 0 0 0 9 2 0 0 16 11
,
 1 0 0 0 0 16 0 0 0 0 1 0 0 0 1 16
G:=sub<GL(4,GF(17))| [0,13,0,0,13,0,0,0,0,0,0,12,0,0,10,10],[0,1,0,0,1,0,0,0,0,0,9,16,0,0,2,11],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;

C8.22SD16 in GAP, Magma, Sage, TeX

C_8._{22}{\rm SD}_{16}
% in TeX

G:=Group("C8.22SD16");
// GroupNames label

G:=SmallGroup(128,974);
// by ID

G=gap.SmallGroup(128,974);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,568,422,58,1123,360,3924,102,4037,124]);
// Polycyclic

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

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