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G = C2×M5(2)⋊C2  order 128 = 27

Direct product of C2 and M5(2)⋊C2

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

Aliases: C2×M5(2)⋊C2, C23.36SD16, M5(2)⋊15C22, C4.64(C2×D8), C8.96(C2×D4), (C2×D8).14C4, D8.12(C2×C4), (C2×C4).142D8, (C2×C8).123D4, C8.8(C22×C4), (C2×C4).53SD16, C8.10(C22⋊C4), (C2×M5(2))⋊17C2, (C2×C8).226C23, C8.C4⋊9C22, (C22×D8).14C2, (C22×C4).336D4, C4.26(D4⋊C4), (C2×D8).153C22, C22.16(C2×SD16), (C22×C8).235C22, C22.33(D4⋊C4), (C2×C8).85(C2×C4), (C2×C4).271(C2×D4), C4.58(C2×C22⋊C4), (C2×C8.C4)⋊19C2, C2.36(C2×D4⋊C4), (C2×C4).274(C22⋊C4), SmallGroup(128,878)

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

Derived series
C1 — C8 — C2×M5(2)⋊C2
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C22×D8 — C2×M5(2)⋊C2
Lower central
C1 — C2 — C4 — C8 — C2×M5(2)⋊C2
Upper central
C1 — C22 — C22×C4 — C22×C8 — C2×M5(2)⋊C2
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×M5(2)⋊C2

Generators and relations for C2×M5(2)⋊C2
 G = < a,b,c,d | a2=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b9, dbd=b3c, cd=dc >

Subgroups: 372 in 128 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C8, C2×C4, D4, C23, C23, C16, C2×C8, C2×C8, M4(2), D8, D8, C22×C4, C2×D4, C24, C8.C4, C8.C4, C2×C16, M5(2), M5(2), C22×C8, C2×M4(2), C2×D8, C2×D8, C22×D4, M5(2)⋊C2, C2×C8.C4, C2×M5(2), C22×D8, C2×M5(2)⋊C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, M5(2)⋊C2, C2×D4⋊C4, C2×M5(2)⋊C2

Smallest permutation representation of C2×M5(2)⋊C2
►On 32 points
Generators in S32
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D8A8B8C8D8E8F8G8H8I8J16A···16H
order12222222224444888888888816···16
size1111228888222222224488884···4

32 irreducible representations

dim111111222224
type+++++++++
imageC1C2C2C2C2C4D4D4D8SD16SD16M5(2)⋊C2
kernelC2×M5(2)⋊C2M5(2)⋊C2C2×C8.C4C2×M5(2)C22×D8C2×D8C2×C8C22×C4C2×C4C2×C4C23C2
# reps141118314224

Matrix representation of C2×M5(2)⋊C2 ►in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
11140000
3130000
005020
00140161
001314120
0014310
,
100000
010000
001000
000100
00120160
0010016
,
1600000
1210000
0016000
001100
0020143
0014033

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],[11,3,0,0,0,0,14,13,0,0,0,0,0,0,5,14,13,14,0,0,0,0,14,3,0,0,2,16,12,1,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,12,1,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,12,0,0,0,0,0,1,0,0,0,0,0,0,16,1,2,14,0,0,0,1,0,0,0,0,0,0,14,3,0,0,0,0,3,3] >;
 

C2×M5(2)⋊C2 in GAP, Magma, Sage, TeX 

C_2\times M_5(2)\rtimes C_2
 
% in TeX
 
G:=Group("C2xM5(2):C2");
 
// GroupNames label
 
G:=SmallGroup(128,878);
 
// by ID
 
G=gap.SmallGroup(128,878);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1123,1466,136,1411,172,4037,2028,124]);
 
// 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^9,d*b*d=b^3*c,c*d=d*c>;
 
// generators/relations
 

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