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## G = C23.21SD16order 128 = 27

### 11st non-split extension by C23 of SD16 acting via SD16/C4=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C23.21SD16
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×C4○D8 — C23.21SD16
 Lower central C1 — C2 — C4 — C8 — C23.21SD16
 Upper central C1 — C4 — C22×C4 — C22×C8 — C23.21SD16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C23.21SD16

Generators and relations for C23.21SD16
G = < a,b,c,d,e | a2=b2=c2=e2=1, d8=c, ab=ba, eae=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=bd3 >

Subgroups: 244 in 110 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×2], C22 [×3], C22 [×5], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×5], D4 [×7], Q8 [×3], C23, C23, C16 [×2], C2×C8 [×6], C2×C8, M4(2) [×3], D8 [×2], D8, SD16 [×4], Q16 [×2], Q16, C22×C4, C22×C4, C2×D4 [×2], C2×Q8, C4○D4 [×6], C8.C4 [×2], C8.C4, C2×C16, M5(2) [×2], M5(2), C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4○D8 [×4], C4○D8 [×2], C2×C4○D4, M5(2)⋊C2 [×2], C8.17D4 [×2], C2×C8.C4, C2×M5(2), C2×C4○D8, C23.21SD16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C2×D4⋊C4, C23.21SD16

Smallest permutation representation of C23.21SD16
On 32 points
Generators in S32
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)
(1 9)(3 11)(5 13)(7 15)(17 25)(19 27)(21 29)(23 31)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(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)
(2 12)(3 15)(4 10)(5 13)(6 8)(7 11)(14 16)(17 29)(18 24)(19 27)(20 22)(21 25)(26 32)(28 30)

G:=sub<Sym(32)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26), (1,9)(3,11)(5,13)(7,15)(17,25)(19,27)(21,29)(23,31), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (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), (2,12)(3,15)(4,10)(5,13)(6,8)(7,11)(14,16)(17,29)(18,24)(19,27)(20,22)(21,25)(26,32)(28,30)>;

G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26), (1,9)(3,11)(5,13)(7,15)(17,25)(19,27)(21,29)(23,31), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (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), (2,12)(3,15)(4,10)(5,13)(6,8)(7,11)(14,16)(17,29)(18,24)(19,27)(20,22)(21,25)(26,32)(28,30) );

G=PermutationGroup([(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26)], [(1,9),(3,11),(5,13),(7,15),(17,25),(19,27),(21,29),(23,31)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(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)], [(2,12),(3,15),(4,10),(5,13),(6,8),(7,11),(14,16),(17,29),(18,24),(19,27),(20,22),(21,25),(26,32),(28,30)])

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D4 D8 SD16 SD16 C23.21SD16 kernel C23.21SD16 M5(2)⋊C2 C8.17D4 C2×C8.C4 C2×M5(2) C2×C4○D8 C4○D8 C2×C8 C22×C4 C2×C4 C2×C4 C23 C1 # reps 1 2 2 1 1 1 8 3 1 4 2 2 4

Matrix representation of C23.21SD16 in GL4(𝔽17) generated by

 0 4 0 0 13 0 0 0 1 4 13 8 11 10 13 4
,
 16 0 0 0 0 16 0 0 8 6 1 0 14 8 0 1
,
 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 10 5 16 2 13 14 16 0 3 9 7 9 11 11 11 3
,
 0 16 0 0 16 0 0 0 2 15 6 11 8 9 3 11
G:=sub<GL(4,GF(17))| [0,13,1,11,4,0,4,10,0,0,13,13,0,0,8,4],[16,0,8,14,0,16,6,8,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[10,13,3,11,5,14,9,11,16,16,7,11,2,0,9,3],[0,16,2,8,16,0,15,9,0,0,6,3,0,0,11,11] >;

C23.21SD16 in GAP, Magma, Sage, TeX

C_2^3._{21}{\rm SD}_{16}
% in TeX

G:=Group("C2^3.21SD16");
// GroupNames label

G:=SmallGroup(128,880);
// by ID

G=gap.SmallGroup(128,880);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,352,1123,1466,136,1411,172,4037,2028,124]);
// Polycyclic

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

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