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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

Aliases: C23.21SD16, M5(2).21C22, C4○D8.5C4, C4.66(C2×D8), C8.98(C2×D4), D8.13(C2×C4), (C2×C8).125D4, (C2×C4).144D8, C4(M5(2)⋊C2), M5(2)⋊C29C2, C4(C8.17D4), C8.17D49C2, C8.10(C22×C4), Q16.13(C2×C4), C8.12(C22⋊C4), (C2×M5(2))⋊18C2, (C2×C8).228C23, (C2×C4).107SD16, (C22×C4).338D4, C4.40(D4⋊C4), (C2×D8).154C22, C22.18(C2×SD16), C22.9(D4⋊C4), C8.C4.11C22, (C22×C8).237C22, (C2×Q16).149C22, (C2×C8).87(C2×C4), (C2×C4○D8).13C2, (C2×C4).273(C2×D4), C4.60(C2×C22⋊C4), (C2×C8.C4)⋊20C2, C2.38(C2×D4⋊C4), (C2×C4).153(C22⋊C4), SmallGroup(128,880)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C23.21SD16
C1C2C4C2×C4C2×C8C22×C8C2×C4○D8 — C23.21SD16
C1C2C4C8 — C23.21SD16
C1C4C22×C4C22×C8 — C23.21SD16
C1C2C2C2C2C4C4C2×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 2A2B2C2D2E2F4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J16A···16H
order12222224444444888888888816···16
size1122288112228822224488884···4

32 irreducible representations

dim1111111222224
type+++++++++
imageC1C2C2C2C2C2C4D4D4D8SD16SD16C23.21SD16
kernelC23.21SD16M5(2)⋊C2C8.17D4C2×C8.C4C2×M5(2)C2×C4○D8C4○D8C2×C8C22×C4C2×C4C2×C4C23C1
# reps1221118314224

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

0400
13000
14138
1110134
,
16000
01600
8610
14801
,
16000
01600
00160
00016
,
105162
1314160
3979
1111113
,
01600
16000
215611
89311
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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