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G = (C2xC4)wrC2order 128 = 27

Wreath product of C2xC4 by C2

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

Aliases: (C2xC4)wrC2, C42:44D4, C24.162D4, C22:1C4wrC2, C4:D4:14C4, C4.116(C4xD4), C22:Q8:14C4, C42:6C4:25C2, (C22xC42):11C2, C23.566(C2xD4), (C22xC4).552D4, C4.188(C4:D4), C24.4C4:27C2, C22.27C22wrC2, (C23xC4).675C22, C22.19C24.7C2, C23.123(C22:C4), (C22xC4).1372C23, C42:C2.25C22, (C2xC42).1063C22, C2.35(C23.23D4), (C2xM4(2)).185C22, C22.25(C22.D4), C4:C4:8(C2xC4), (C2xD4):9(C2xC4), (C2xC4wrC2):15C2, (C2xQ8):9(C2xC4), C2.42(C2xC4wrC2), (C2xC4).1531(C2xD4), (C2xC4).569(C4oD4), (C22xC4).406(C2xC4), (C2xC4).390(C22xC4), (C2xC4oD4).21C22, (C2xC4).195(C22:C4), C22.271(C2xC22:C4), SmallGroup(128,628)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — (C2xC4)wrC2
C1C2C4C2xC4C22xC4C23xC4C22xC42 — (C2xC4)wrC2
C1C2C2xC4 — (C2xC4)wrC2
C1C2xC4C23xC4 — (C2xC4)wrC2
C1C2C2C22xC4 — (C2xC4)wrC2

Generators and relations for (C2xC4)wrC2
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=a-1b-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 420 in 221 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C23, C42, C42, C22:C4, C4:C4, C4:C4, C2xC8, M4(2), C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C22:C8, C4wrC2, C2xC42, C2xC42, C42:C2, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C2xM4(2), C23xC4, C23xC4, C2xC4oD4, C42:6C4, C24.4C4, C2xC4wrC2, C22xC42, C22.19C24, (C2xC4)wrC2
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C4oD4, C4wrC2, C2xC22:C4, C4xD4, C22wrC2, C4:D4, C22.D4, C23.23D4, C2xC4wrC2, (C2xC4)wrC2

Permutation representations of (C2xC4)wrC2
On 16 points - transitive group 16T211
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 4 8 6)(2 3 7 5)(9 14 11 16)(10 15 12 13)
(1 10 7 14)(2 16 8 12)(3 9 6 13)(4 15 5 11)
(1 12)(2 14)(3 11)(4 13)(5 9)(6 15)(7 16)(8 10)

G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,8,6)(2,3,7,5)(9,14,11,16)(10,15,12,13), (1,10,7,14)(2,16,8,12)(3,9,6,13)(4,15,5,11), (1,12)(2,14)(3,11)(4,13)(5,9)(6,15)(7,16)(8,10)>;

G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,8,6)(2,3,7,5)(9,14,11,16)(10,15,12,13), (1,10,7,14)(2,16,8,12)(3,9,6,13)(4,15,5,11), (1,12)(2,14)(3,11)(4,13)(5,9)(6,15)(7,16)(8,10) );

G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,4,8,6),(2,3,7,5),(9,14,11,16),(10,15,12,13)], [(1,10,7,14),(2,16,8,12),(3,9,6,13),(4,15,5,11)], [(1,12),(2,14),(3,11),(4,13),(5,9),(6,15),(7,16),(8,10)]])

G:=TransitiveGroup(16,211);

44 conjugacy classes

class 1 2A2B2C2D···2I2J4A4B4C4D4E···4Z4AA4AB4AC8A8B8C8D
order12222···2244444···44448888
size11112···2811112···28888888

44 irreducible representations

dim1111111122222
type+++++++++
imageC1C2C2C2C2C2C4C4D4D4D4C4oD4C4wrC2
kernel(C2xC4)wrC2C42:6C4C24.4C4C2xC4wrC2C22xC42C22.19C24C4:D4C22:Q8C42C22xC4C24C2xC4C22
# reps12121144431416

Matrix representation of (C2xC4)wrC2 in GL4(F17) generated by

16000
01600
00130
0001
,
1000
0100
0040
0004
,
01600
1000
0004
00130
,
01600
16000
00013
0040
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,16,0,0,0,0,0,0,13,0,0,4,0],[0,16,0,0,16,0,0,0,0,0,0,4,0,0,13,0] >;

(C2xC4)wrC2 in GAP, Magma, Sage, TeX

(C_2\times C_4)\wr C_2
% in TeX

G:=Group("(C2xC4)wrC2");
// GroupNames label

G:=SmallGroup(128,628);
// by ID

G=gap.SmallGroup(128,628);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,248,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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