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## G = C24.73(C2×C4)  order 128 = 27

### 38th non-split extension by C24 of C2×C4 acting via C2×C4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C24.73(C2×C4)
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C24.73(C2×C4)
 Lower central C1 — C22 — C24.73(C2×C4)
 Upper central C1 — C2×C4 — C24.73(C2×C4)
 Jennings C1 — C2 — C2 — C2×C4 — C24.73(C2×C4)

Generators and relations for C24.73(C2×C4)
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f4=d, ab=ba, faf-1=ac=ca, eae=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 636 in 382 conjugacy classes, 172 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C22⋊C8, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C24.4C4, (C22×C8)⋊C2, C22×M4(2), C22×C4○D4, C24.73(C2×C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C23×C4, C22×D4, C22×C22⋊C4, Q8○M4(2), C24.73(C2×C4)

Smallest permutation representation of C24.73(C2×C4)
On 32 points
Generators in S32
(1 13)(2 18)(3 15)(4 20)(5 9)(6 22)(7 11)(8 24)(10 28)(12 30)(14 32)(16 26)(17 31)(19 25)(21 27)(23 29)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
(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 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)
(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)| (1,13)(2,18)(3,15)(4,20)(5,9)(6,22)(7,11)(8,24)(10,28)(12,30)(14,32)(16,26)(17,31)(19,25)(21,27)(23,29), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (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), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (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( (1,13)(2,18)(3,15)(4,20)(5,9)(6,22)(7,11)(8,24)(10,28)(12,30)(14,32)(16,26)(17,31)(19,25)(21,27)(23,29), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (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), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (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([[(1,13),(2,18),(3,15),(4,20),(5,9),(6,22),(7,11),(8,24),(10,28),(12,30),(14,32),(16,26),(17,31),(19,25),(21,27),(23,29)], [(1,31),(2,28),(3,25),(4,30),(5,27),(6,32),(7,29),(8,26),(9,21),(10,18),(11,23),(12,20),(13,17),(14,22),(15,19),(16,24)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20)], [(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)], [(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28)], [(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)]])

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E ··· 4J 4K 4L 4M 4N 8A ··· 8P order 1 2 2 2 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 1 1 2 ··· 2 4 4 4 4 1 1 1 1 2 ··· 2 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 4 type + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 Q8○M4(2) kernel C24.73(C2×C4) C24.4C4 (C22×C8)⋊C2 C22×M4(2) C22×C4○D4 C22×D4 C22×Q8 C2×C4○D4 C22×C4 C2 # reps 1 4 8 2 1 6 2 8 8 4

Matrix representation of C24.73(C2×C4) in GL6(𝔽17)

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

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

C24.73(C2×C4) in GAP, Magma, Sage, TeX

C_2^4._{73}(C_2\times C_4)
% in TeX

G:=Group("C2^4.73(C2xC4)");
// GroupNames label

G:=SmallGroup(128,1611);
// by ID

G=gap.SmallGroup(128,1611);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,2019,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=1,f^4=d,a*b=b*a,f*a*f^-1=a*c=c*a,e*a*e=a*d=d*a,b*c=c*b,f*b*f^-1=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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