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G = C24.45(C2×C4)  order 128 = 27

10th non-split extension by C24 of C2×C4 acting via C2×C4/C2=C22

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

Aliases: C24.45(C2×C4), (C22×Q8).5C4, C4.10(C23⋊C4), (C22×C4).727D4, C22.13(C8○D4), C22⋊C8.158C22, C23.42(C22⋊C4), C24.4C4.12C2, (C23×C4).202C22, (C22×C4).436C23, C23.173(C22×C4), C22.1(C4.10D4), C22.M4(2)⋊16C2, (C2×C4⋊C4).14C4, (C2×C22⋊C8).9C2, (C2×C22⋊C4).9C4, C2.11(C2×C23⋊C4), (C2×C22⋊Q8).1C2, (C2×C4).1133(C2×D4), (C2×C4⋊C4).11C22, (C22×C4).13(C2×C4), C2.8(C2×C4.10D4), (C2×C4).72(C22⋊C4), C22.154(C2×C22⋊C4), C2.10((C22×C8)⋊C2), SmallGroup(128,204)

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

Derived series
C1 — C23 — C24.45(C2×C4)
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C22⋊Q8 — C24.45(C2×C4)
Lower central
C1 — C2 — C23 — C24.45(C2×C4)
Upper central
C1 — C22 — C23×C4 — C24.45(C2×C4)
Jennings
C1 — C2 — C22 — C22×C4 — C24.45(C2×C4)

Generators and relations for C24.45(C2×C4)
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e2=c, f4=d, ab=ba, ac=ca, ad=da, eae-1=acd, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bce >

Subgroups: 284 in 135 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C23, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×Q8, C24, C22⋊C8, C22⋊C8, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C22×C8, C2×M4(2), C23×C4, C22×Q8, C22.M4(2), C2×C22⋊C8, C24.4C4, C2×C22⋊Q8, C24.45(C2×C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C4.10D4, C2×C22⋊C4, C8○D4, (C22×C8)⋊C2, C2×C23⋊C4, C2×C4.10D4, C24.45(C2×C4)

Smallest permutation representation of C24.45(C2×C4)
►On 32 points
Generators in S32
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4F4G4H4I4J4K8A···8H8I8J8K8L
order1222222224···4444448···88888
size1111222242···2488884···48888

32 irreducible representations

dim111111112244
type+++++++-
imageC1C2C2C2C2C4C4C4D4C8○D4C23⋊C4C4.10D4
kernelC24.45(C2×C4)C22.M4(2)C2×C22⋊C8C24.4C4C2×C22⋊Q8C2×C22⋊C4C2×C4⋊C4C22×Q8C22×C4C22C4C22
# reps141114224822

Matrix representation of C24.45(C2×C4) ►in GL6(𝔽17)

1600000
010000
0016000
000100
0000160
000001
,
100000
010000
001000
000100
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
0130000
400000
000100
0016000
000001
0000160
,
1500000
0150000
000010
000001
001000
0001600

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[15,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C24.45(C2×C4) in GAP, Magma, Sage, TeX 

C_2^4._{45}(C_2\times C_4)
 
% in TeX
 
G:=Group("C2^4.45(C2xC4)");
 
// GroupNames label
 
G:=SmallGroup(128,204);
 
// by ID
 
G=gap.SmallGroup(128,204);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,723,352,1123,851,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^2=c,f^4=d,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,a*f=f*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*c*e>;
 
// generators/relations
 

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