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G = C23.20SD16order 128 = 27

10th non-split extension by C23 of SD16 acting via SD16/C4=C22

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

Aliases: C23.20SD16, C4○D8.4C4, C4.89(C2×D8), D8.10(C2×C4), (C2×D8).13C4, (C2×C8).120D4, C8.115(C2×D4), (C2×C4).140D8, (C2×C16)⋊12C22, D8.C45C2, C8.7(C22⋊C4), C8.34(C22×C4), (C2×Q16).13C4, Q16.10(C2×C4), (C2×C4).50SD16, (C2×M5(2))⋊14C2, (C2×C8).577C23, C8.C48C22, C4○D8.13C22, (C22×C4).333D4, C4.38(D4⋊C4), C22.2(C2×SD16), (C22×C8).232C22, C22.31(D4⋊C4), (C2×C8).82(C2×C4), (C2×C4○D8).10C2, (C2×C4).761(C2×D4), C4.55(C2×C22⋊C4), (C2×C8.C4)⋊18C2, C2.33(C2×D4⋊C4), (C2×C4).151(C22⋊C4), SmallGroup(128,875)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C23.20SD16
C1C2C4C2×C4C2×C8C22×C8C2×C4○D8 — C23.20SD16
C1C2C4C8 — C23.20SD16
C1C4C22×C4C22×C8 — C23.20SD16
C1C2C2C2C2C4C4C2×C8 — C23.20SD16

Generators and relations for C23.20SD16
 G = < a,b,c,d,e | a2=b2=c2=e2=1, d8=c, ab=ba, dad-1=eae=ac=ca, ebe=bc=cb, bd=db, 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 [×2], M5(2) [×2], C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4○D8 [×4], C4○D8 [×2], C2×C4○D4, D8.C4 [×4], C2×C8.C4, C2×M5(2), C2×C4○D8, C23.20SD16
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, C23.20SD16

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J16A···16H
order12222224444444888888888816···16
size1122288112228822224488884···4

32 irreducible representations

dim11111111222224
type++++++++
imageC1C2C2C2C2C4C4C4D4D4D8SD16SD16C23.20SD16
kernelC23.20SD16D8.C4C2×C8.C4C2×M5(2)C2×C4○D8C2×D8C2×Q16C4○D8C2×C8C22×C4C2×C4C2×C4C23C1
# reps14111224314224

Matrix representation of C23.20SD16 in GL4(𝔽17) generated by

01300
4000
1516013
16240
,
01300
4000
31604
13130
,
16000
01600
00160
00016
,
30015
913150
151448
26014
,
0100
1000
77143
161633
G:=sub<GL(4,GF(17))| [0,4,15,16,13,0,16,2,0,0,0,4,0,0,13,0],[0,4,3,1,13,0,16,3,0,0,0,13,0,0,4,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[3,9,15,2,0,13,14,6,0,15,4,0,15,0,8,14],[0,1,7,16,1,0,7,16,0,0,14,3,0,0,3,3] >;

C23.20SD16 in GAP, Magma, Sage, TeX

C_2^3._{20}{\rm SD}_{16}
% in TeX

G:=Group("C2^3.20SD16");
// GroupNames label

G:=SmallGroup(128,875);
// by ID

G=gap.SmallGroup(128,875);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,1123,570,360,172,4037,2028,124]);
// Polycyclic

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

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