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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

Aliases: C8.22SD16, C42.335D4, (C4×C16)⋊7C2, (C2×C4).62D8, (C2×C8).279D4, C8.5Q82C2, C2.Q323C2, C2.D16.1C2, C8.52(C4○D4), C4.16(C2×SD16), C2.17(C4○D16), (C2×C8).540C23, (C4×C8).413C22, (C2×C16).74C22, C8.12D4.3C2, C4.9(C4.4D4), (C2×D8).13C22, C22.126(C2×D8), C2.D8.25C22, C2.11(C4.4D8), (C2×Q16).14C22, (C2×C4).808(C2×D4), SmallGroup(128,974)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C8.22SD16
C1C2C4C8C2×C8C2×C16C4×C16 — C8.22SD16
C1C2C4C2×C8 — C8.22SD16
C1C22C42C4×C8 — C8.22SD16
C1C2C2C2C2C4C4C2×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 2A2B2C2D4A···4F4G4H4I8A···8H16A···16P
order122224···44448···816···16
size1111162···21616162···22···2

38 irreducible representations

dim111111222222
type+++++++++
imageC1C2C2C2C2C2D4D4SD16C4○D4D8C4○D16
kernelC8.22SD16C4×C16C2.D16C2.Q32C8.12D4C8.5Q8C42C2×C8C8C8C2×C4C2
# reps1122111144416

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

01300
13000
00010
001210
,
0100
1000
0092
001611
,
1000
01600
0010
00116
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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