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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

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 2A 2B 2C 2D 2E 2F 2G 2H 4A ··· 4F 4G 4H 4I 4J 4K 8A ··· 8H 8I 8J 8K 8L order 1 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 8 ··· 8 8 8 8 8 size 1 1 1 1 2 2 2 2 4 2 ··· 2 4 8 8 8 8 4 ··· 4 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C4 D4 C8○D4 C23⋊C4 C4.10D4 kernel C24.45(C2×C4) C22.M4(2) C2×C22⋊C8 C24.4C4 C2×C22⋊Q8 C2×C22⋊C4 C2×C4⋊C4 C22×Q8 C22×C4 C22 C4 C22 # reps 1 4 1 1 1 4 2 2 4 8 2 2

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

 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 13 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 15 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 16 0 0

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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