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## G = (C23×C4).C4order 128 = 27

### 20th non-split extension by C23×C4 of C4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for (C23×C4).C4
G = < a,b,c,d,e | a2=b2=c2=d4=1, e4=d2, ab=ba, ac=ca, ad=da, eae-1=ad2, bc=cb, ede-1=bd=db, be=eb, cd=dc, ece-1=acd2 >

Subgroups: 580 in 292 conjugacy classes, 68 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×6], C22 [×3], C22 [×28], C8 [×4], C2×C4 [×12], C2×C4 [×30], D4 [×24], Q8 [×8], C23, C23 [×6], C23 [×12], C2×C8 [×4], M4(2) [×8], C22×C4 [×2], C22×C4 [×6], C22×C4 [×16], C2×D4 [×4], C2×D4 [×16], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24, C24 [×2], C22⋊C8 [×4], C4.D4 [×4], C4.10D4 [×4], C2×M4(2) [×4], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C22×Q8, C2×C4○D4 [×12], C24.4C4 [×2], C2×C4.D4 [×2], C2×C4.10D4 [×2], C22×C4○D4, (C23×C4).C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, C22⋊C4 [×12], C22×C4, C2×D4 [×6], C2×C22⋊C4 [×3], C22≀C2 [×4], C243C4, M4(2).8C22 [×2], (C23×C4).C4

Smallest permutation representation of (C23×C4).C4
On 32 points
Generators in S32
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)
(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)
(2 6)(3 7)(11 15)(12 16)(17 21)(20 24)(25 29)(28 32)
(1 14 5 10)(2 24 6 20)(3 16 7 12)(4 18 8 22)(9 30 13 26)(11 32 15 28)(17 29 21 25)(19 31 23 27)
(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)| (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32), (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), (2,6)(3,7)(11,15)(12,16)(17,21)(20,24)(25,29)(28,32), (1,14,5,10)(2,24,6,20)(3,16,7,12)(4,18,8,22)(9,30,13,26)(11,32,15,28)(17,29,21,25)(19,31,23,27), (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( (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32), (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), (2,6)(3,7)(11,15)(12,16)(17,21)(20,24)(25,29)(28,32), (1,14,5,10)(2,24,6,20)(3,16,7,12)(4,18,8,22)(9,30,13,26)(11,32,15,28)(17,29,21,25)(19,31,23,27), (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([(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(26,30),(28,32)], [(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)], [(2,6),(3,7),(11,15),(12,16),(17,21),(20,24),(25,29),(28,32)], [(1,14,5,10),(2,24,6,20),(3,16,7,12),(4,18,8,22),(9,30,13,26),(11,32,15,28),(17,29,21,25),(19,31,23,27)], [(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 ··· 2K 4A ··· 4H 4I 4J 4K 4L 8A ··· 8H order 1 2 2 2 2 2 2 ··· 2 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 1 1 2 2 4 ··· 4 2 ··· 2 4 4 4 4 8 ··· 8

32 irreducible representations

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

Matrix representation of (C23×C4).C4 in GL6(𝔽17)

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

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

(C23×C4).C4 in GAP, Magma, Sage, TeX

(C_2^3\times C_4).C_4
% in TeX

G:=Group("(C2^3xC4).C4");
// GroupNames label

G:=SmallGroup(128,517);
// by ID

G=gap.SmallGroup(128,517);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,352,2019,2028,124]);
// Polycyclic

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

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