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## G = C24.(C2×C4)  order 128 = 27

### 3rd non-split extension by C24 of C2×C4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C24.(C2×C4)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C4⋊D4 — C24.(C2×C4)
 Lower central C1 — C2 — C23 — C24.(C2×C4)
 Upper central C1 — C22 — C23×C4 — C24.(C2×C4)
 Jennings C1 — C2 — C22 — C22×C4 — C24.(C2×C4)

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

Subgroups: 412 in 161 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, C22⋊C8, C22⋊C8, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C22×C8, C2×M4(2), C23×C4, C22×D4, C22×D4, C23⋊C8, C2×C22⋊C8, C24.4C4, C2×C4⋊D4, C24.(C2×C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C4.D4, C2×C22⋊C4, C8○D4, (C22×C8)⋊C2, C2×C23⋊C4, C2×C4.D4, C24.(C2×C4)

Smallest permutation representation of C24.(C2×C4)
On 32 points
Generators in S32
(1 10)(2 22)(3 23)(4 13)(5 14)(6 18)(7 19)(8 9)(11 32)(12 25)(15 28)(16 29)(17 27)(20 30)(21 31)(24 26)
(2 32)(4 26)(6 28)(8 30)(9 20)(11 22)(13 24)(15 18)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(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 14 5 10)(2 15 6 11)(3 16 7 12)(4 9 8 13)(17 27 21 31)(18 28 22 32)(19 29 23 25)(20 30 24 26)
(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)| (1,10)(2,22)(3,23)(4,13)(5,14)(6,18)(7,19)(8,9)(11,32)(12,25)(15,28)(16,29)(17,27)(20,30)(21,31)(24,26), (2,32)(4,26)(6,28)(8,30)(9,20)(11,22)(13,24)(15,18), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (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), (1,14,5,10)(2,15,6,11)(3,16,7,12)(4,9,8,13)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26), (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( (1,10)(2,22)(3,23)(4,13)(5,14)(6,18)(7,19)(8,9)(11,32)(12,25)(15,28)(16,29)(17,27)(20,30)(21,31)(24,26), (2,32)(4,26)(6,28)(8,30)(9,20)(11,22)(13,24)(15,18), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (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), (1,14,5,10)(2,15,6,11)(3,16,7,12)(4,9,8,13)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26), (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([[(1,10),(2,22),(3,23),(4,13),(5,14),(6,18),(7,19),(8,9),(11,32),(12,25),(15,28),(16,29),(17,27),(20,30),(21,31),(24,26)], [(2,32),(4,26),(6,28),(8,30),(9,20),(11,22),(13,24),(15,18)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19)], [(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)], [(1,14,5,10),(2,15,6,11),(3,16,7,12),(4,9,8,13),(17,27,21,31),(18,28,22,32),(19,29,23,25),(20,30,24,26)], [(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 2I 2J 4A ··· 4F 4G 4H 4I 8A ··· 8H 8I 8J 8K 8L order 1 2 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 8 ··· 8 8 8 8 8 size 1 1 1 1 2 2 2 2 4 8 8 2 ··· 2 4 8 8 4 ··· 4 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 D4 C8○D4 C23⋊C4 C4.D4 kernel C24.(C2×C4) C23⋊C8 C2×C22⋊C8 C24.4C4 C2×C4⋊D4 C2×C4⋊C4 C22×D4 C22×C4 C22 C4 C22 # reps 1 4 1 1 1 2 6 4 8 2 2

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

 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 0 10 0 16 2 0 0 5 5 0 1
,
 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 10 7 16 0 0 0 0 7 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 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 16 2 0 0 0 0 16 1
,
 0 2 0 0 0 0 15 0 0 0 0 0 0 0 12 5 1 0 0 0 12 12 1 15 0 0 0 0 0 10 0 0 0 0 12 10

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,10,5,0,0,16,0,0,5,0,0,0,0,16,0,0,0,0,0,2,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,10,0,0,0,0,1,7,7,0,0,0,0,16,0,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[0,15,0,0,0,0,2,0,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,1,1,0,12,0,0,0,15,10,10] >;

C24.(C2×C4) in GAP, Magma, Sage, TeX

C_2^4.(C_2\times C_4)
% in TeX

G:=Group("C2^4.(C2xC4)");
// GroupNames label

G:=SmallGroup(128,203);
// by ID

G=gap.SmallGroup(128,203);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,723,352,1123,851,172]);
// Polycyclic

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

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