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## G = (C22×C4).275D4order 128 = 27

### 160th non-split extension by C22×C4 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — (C22×C4).275D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C22×C4⋊C4 — (C22×C4).275D4
 Lower central C1 — C2 — C23 — (C22×C4).275D4
 Upper central C1 — C22 — C23×C4 — (C22×C4).275D4
 Jennings C1 — C2 — C2 — C22×C4 — (C22×C4).275D4

Generators and relations for (C22×C4).275D4
G = < a,b,c,d,e | a2=b2=c4=1, d4=c2, e2=c, dad-1=eae-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=bc-1d3 >

Subgroups: 308 in 158 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C22⋊C8, C2×C4⋊C4, C2×C4⋊C4, C2×M4(2), C23×C4, C23×C4, C22.C42, C24.4C4, C22×C4⋊C4, (C22×C4).275D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C4.D4, C4.10D4, C2×C22⋊C4, C42⋊C2, C22.D4, C23.34D4, C2×C4.D4, C2×C4.10D4, (C22×C4).275D4

Smallest permutation representation of (C22×C4).275D4
On 32 points
Generators in S32
(2 27)(4 29)(6 31)(8 25)(10 24)(12 18)(14 20)(16 22)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)(17 23 21 19)(18 20 22 24)(25 27 29 31)(26 32 30 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)
(1 14 7 16 5 10 3 12)(2 17 4 23 6 21 8 19)(9 31 15 25 13 27 11 29)(18 26 20 32 22 30 24 28)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4N 8A ··· 8H order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 2 2 2 2 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 type + + + + + + - image C1 C2 C2 C2 C4 C4 D4 C4○D4 C4.D4 C4.10D4 kernel (C22×C4).275D4 C22.C42 C24.4C4 C22×C4⋊C4 C2×C4⋊C4 C23×C4 C22×C4 C2×C4 C22 C22 # reps 1 4 2 1 4 4 4 8 2 2

Matrix representation of (C22×C4).275D4 in GL6(𝔽17)

 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 0 0 0 0 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 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 15 0 0 0 0 1 16 0 0 0 0 0 16 0 1 0 0 1 16 16 0
,
 0 15 0 0 0 0 9 0 0 0 0 0 0 0 10 0 5 9 0 0 6 0 15 7 0 0 6 11 11 13 0 0 10 13 11 13
,
 0 8 0 0 0 0 15 0 0 0 0 0 0 0 1 0 15 0 0 0 0 0 16 1 0 0 0 1 16 0 0 0 1 0 16 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,1,0,0,15,16,16,16,0,0,0,0,0,16,0,0,0,0,1,0],[0,9,0,0,0,0,15,0,0,0,0,0,0,0,10,6,6,10,0,0,0,0,11,13,0,0,5,15,11,11,0,0,9,7,13,13],[0,15,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,15,16,16,16,0,0,0,1,0,0] >;

(C22×C4).275D4 in GAP, Magma, Sage, TeX

(C_2^2\times C_4)._{275}D_4
% in TeX

G:=Group("(C2^2xC4).275D4");
// GroupNames label

G:=SmallGroup(128,553);
// by ID

G=gap.SmallGroup(128,553);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,58,2804,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=1,d^4=c^2,e^2=c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=b*c^-1*d^3>;
// generators/relations

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