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G = C2×C24.4C4order 128 = 27

Direct product of C2 and C24.4C4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C24.4C4, C25.8C4, C236M4(2), (C2×C8)⋊9C23, (C23×C4).38C4, (C24×C4).12C2, C22⋊C871C22, C4(C24.4C4), (C2×C4).627C24, C24.125(C2×C4), (C22×C8)⋊44C22, (C22×C4).772D4, C4.176(C22×D4), C223(C2×M4(2)), C2.6(C22×M4(2)), (C22×M4(2))⋊13C2, (C2×M4(2))⋊62C22, (C23×C4).687C22, C22.157(C23×C4), C23.219(C22×C4), C23.206(C22⋊C4), (C22×C4).1491C23, (C2×C22⋊C8)⋊39C2, C4.97(C2×C22⋊C4), (C2×C4).1562(C2×D4), (C2×C4)(C24.4C4), (C22×C4).492(C2×C4), (C2×C4).496(C22×C4), C22.76(C2×C22⋊C4), C2.20(C22×C22⋊C4), (C2×C4).366(C22⋊C4), SmallGroup(128,1609)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C24.4C4
C1C2C4C2×C4C22×C4C23×C4C24×C4 — C2×C24.4C4
C1C22 — C2×C24.4C4
C1C22×C4 — C2×C24.4C4
C1C2C2C2×C4 — C2×C24.4C4

Generators and relations for C2×C24.4C4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f4=e, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 812 in 504 conjugacy classes, 196 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×8], C4 [×4], C22, C22 [×18], C22 [×68], C8 [×8], C2×C4 [×2], C2×C4 [×30], C2×C4 [×44], C23, C23 [×18], C23 [×68], C2×C8 [×8], C2×C8 [×8], M4(2) [×16], C22×C4 [×2], C22×C4 [×34], C22×C4 [×52], C24, C24 [×6], C24 [×12], C22⋊C8 [×16], C22×C8 [×4], C2×M4(2) [×8], C2×M4(2) [×8], C23×C4 [×2], C23×C4 [×12], C23×C4 [×8], C25, C2×C22⋊C8 [×4], C24.4C4 [×8], C22×M4(2) [×2], C24×C4, C2×C24.4C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], M4(2) [×8], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C2×M4(2) [×12], C23×C4, C22×D4 [×2], C24.4C4 [×4], C22×C22⋊C4, C22×M4(2) [×2], C2×C24.4C4

Smallest permutation representation of C2×C24.4C4
On 32 points
Generators in S32
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)
(2 28)(4 30)(6 32)(8 26)(9 22)(11 24)(13 18)(15 20)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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)

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

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

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

56 conjugacy classes

class 1 2A···2G2H···2S4A···4H4I···4T8A···8P
order12···22···24···44···48···8
size11···12···21···12···24···4

56 irreducible representations

dim111111122
type++++++
imageC1C2C2C2C2C4C4D4M4(2)
kernelC2×C24.4C4C2×C22⋊C8C24.4C4C22×M4(2)C24×C4C23×C4C25C22×C4C23
# reps14821142816

Matrix representation of C2×C24.4C4 in GL5(𝔽17)

160000
01000
00100
00010
00001
,
10000
01000
001600
00010
0001416
,
160000
016000
001600
00010
0001416
,
10000
016000
001600
000160
000016
,
10000
01000
00100
000160
000016
,
130000
001600
01000
0001415
000153

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1,14,0,0,0,0,16],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,14,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,14,15,0,0,0,15,3] >;

C2×C24.4C4 in GAP, Magma, Sage, TeX

C_2\times C_2^4._4C_4
% in TeX

G:=Group("C2xC2^4.4C4");
// GroupNames label

G:=SmallGroup(128,1609);
// by ID

G=gap.SmallGroup(128,1609);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,124]);
// Polycyclic

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

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