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G = C2×C16⋊C22  order 128 = 27

Direct product of C2 and C16⋊C22

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

Aliases: C2×C16⋊C22, C16⋊C23, D8⋊2C23, D16⋊3C22, C8.13C24, Q16⋊2C23, C23.53D8, SD32⋊1C22, M5(2)⋊5C22, (C2×C4).54D8, C4.75(C2×D8), C8.37(C2×D4), (C2×D16)⋊12C2, (C2×C16)⋊3C22, (C2×SD32)⋊4C2, (C2×C8).147D4, C4○D8⋊7C22, (C22×D8)⋊21C2, (C2×D8)⋊52C22, (C2×M5(2))⋊3C2, C4.19(C22×D4), C2.28(C22×D8), C22.78(C2×D8), (C2×C8).291C23, (C2×Q16)⋊50C22, (C22×C4).532D4, (C22×C8).294C22, (C2×C4○D8)⋊27C2, (C2×C4).876(C2×D4), SmallGroup(128,2144)

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

Derived series
C1 — C8 — C2×C16⋊C22
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C22×D8 — C2×C16⋊C22
Lower central
C1 — C2 — C4 — C8 — C2×C16⋊C22
Upper central
C1 — C22 — C22×C4 — C22×C8 — C2×C16⋊C22
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C2×C16⋊C22

Generators and relations for C2×C16⋊C22
 G = < a,b,c,d | a2=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b7, dbd=b9, cd=dc >

Subgroups: 532 in 200 conjugacy classes, 92 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C2×C8, C2×C8, D8, D8, SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C16, M5(2), D16, SD32, C22×C8, C2×D8, C2×D8, C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C22×D4, C2×C4○D4, C2×M5(2), C2×D16, C2×SD32, C16⋊C22, C22×D8, C2×C4○D8, C2×C16⋊C22
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C22×D4, C16⋊C22, C22×D8, C2×C16⋊C22

Smallest permutation representation of C2×C16⋊C22
►On 32 points
Generators in S32
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A4B4C4D4E4F8A8B8C8D8E8F16A···16H
order1222222···244444488888816···16
size1111228···82222882222444···4

32 irreducible representations

dim111111122224
type++++++++++++
imageC1C2C2C2C2C2C2D4D4D8D8C16⋊C22
kernelC2×C16⋊C22C2×M5(2)C2×D16C2×SD32C16⋊C22C22×D8C2×C4○D8C2×C8C22×C4C2×C4C23C2
# reps112281131624

Matrix representation of C2×C16⋊C22 ►in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
5120000
610000
0016161615
000010
0014300
009601
,
100000
11160000
000100
001000
003366
00551411
,
1600000
0160000
001000
000100
0000160
001616016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[5,6,0,0,0,0,12,1,0,0,0,0,0,0,16,0,14,9,0,0,16,0,3,6,0,0,16,1,0,0,0,0,15,0,0,1],[1,11,0,0,0,0,0,16,0,0,0,0,0,0,0,1,3,5,0,0,1,0,3,5,0,0,0,0,6,14,0,0,0,0,6,11],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,16,0,0,0,1,0,16,0,0,0,0,16,0,0,0,0,0,0,16] >;
 

C2×C16⋊C22 in GAP, Magma, Sage, TeX 

C_2\times C_{16}\rtimes C_2^2
 
% in TeX
 
G:=Group("C2xC16:C2^2");
 
// GroupNames label
 
G:=SmallGroup(128,2144);
 
// by ID
 
G=gap.SmallGroup(128,2144);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,1430,1684,851,242,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^16=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^7,d*b*d=b^9,c*d=d*c>;
 
// generators/relations
 

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