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G = C2×C22.SD16  order 128 = 27

Direct product of C2 and C22.SD16

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2×C22.SD16, C23.34D8, C24.148D4, C23.40SD16, C4⋊D4⋊1C4, (C22×D4)⋊2C4, C22.9(C2×D8), C22.36C4≀C2, C22⋊C8⋊49C22, C23.488(C2×D4), (C22×C4).209D4, C22.27(C2×SD16), C22.2(D4⋊C4), C4⋊D4.128C22, C22.45(C23⋊C4), (C22×C4).620C23, (C23×C4).203C22, C23.100(C22⋊C4), C2.C42⋊56C22, (C2×C4⋊C4)⋊5C4, C4⋊C4⋊1(C2×C4), (C2×D4)⋊1(C2×C4), C2.15(C2×C4≀C2), (C2×C22⋊C8)⋊5C2, C2.7(C2×D4⋊C4), (C2×C4⋊D4).2C2, C2.12(C2×C23⋊C4), (C2×C4).1144(C2×D4), (C2×C4).85(C22⋊C4), (C2×C4).110(C22×C4), (C22×C4).194(C2×C4), (C2×C2.C42)⋊13C2, C22.174(C2×C22⋊C4), SmallGroup(128,230)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C2×C22.SD16
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4⋊D4 — C2×C22.SD16
Lower central
C1 — C22 — C2×C4 — C2×C22.SD16
Upper central
C1 — C23 — C23×C4 — C2×C22.SD16
Jennings
C1 — C2 — C22 — C22×C4 — C2×C22.SD16

Generators and relations for C2×C22.SD16
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=bcd3 >

Subgroups: 492 in 194 conjugacy classes, 60 normal (28 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2.C42, C2.C42, C22⋊C8, C22⋊C8, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22×C8, C23×C4, C23×C4, C22×D4, C22×D4, C22.SD16, C2×C2.C42, C2×C22⋊C8, C2×C4⋊D4, C2×C22.SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, C23⋊C4, D4⋊C4, C4≀C2, C2×C22⋊C4, C2×D8, C2×SD16, C22.SD16, C2×C23⋊C4, C2×D4⋊C4, C2×C4≀C2, C2×C22.SD16

Smallest permutation representation of C2×C22.SD16
►On 32 points
Generators in S32
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)

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

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

(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)

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4N4O4P8A···8H
order12···222222244444···4448···8
size11···122228822224···4884···4

38 irreducible representations

dim11111111222224
type+++++++++
imageC1C2C2C2C2C4C4C4D4D4D8SD16C4≀C2C23⋊C4
kernelC2×C22.SD16C22.SD16C2×C2.C42C2×C22⋊C8C2×C4⋊D4C2×C4⋊C4C4⋊D4C22×D4C22×C4C24C23C23C22C22
# reps14111242314482

Matrix representation of C2×C22.SD16 ►in GL5(𝔽17)

160000
01000
00100
000160
000016
,
10000
016000
00100
000160
000016
,
10000
016000
001600
00010
00001
,
160000
00400
01000
000211
00009
,
10000
001600
016000
000811
00029

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,11,9],[1,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,8,2,0,0,0,11,9] >;
 

C2×C22.SD16 in GAP, Magma, Sage, TeX 

C_2\times C_2^2.{\rm SD}_{16}
 
% in TeX
 
G:=Group("C2xC2^2.SD16");
 
// GroupNames label
 
G:=SmallGroup(128,230);
 
// by ID
 
G=gap.SmallGroup(128,230);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1123,1018,248,1971]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=b*c*d^3>;
 
// generators/relations
 

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