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## G = (C22×C4).276D4order 128 = 27

### 161st 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).276D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×C42⋊C2 — (C22×C4).276D4
 Lower central C1 — C2 — C23 — (C22×C4).276D4
 Upper central C1 — C22 — C23×C4 — (C22×C4).276D4
 Jennings C1 — C2 — C2 — C22×C4 — (C22×C4).276D4

Generators and relations for (C22×C4).276D4
G = < a,b,c,d,e | a2=b2=c4=1, d4=c2, e2=bc-1, ebe-1=ab=ba, dcd-1=ece-1=ac=ca, ad=da, ae=ea, bc=cb, dbd-1=abc2, ede-1=abcd3 >

Subgroups: 260 in 136 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×6], C22, C22 [×2], C22 [×12], C8 [×4], C2×C4 [×8], C2×C4 [×18], C23, C23 [×2], C23 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×4], M4(2) [×8], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C22⋊C8 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C2×M4(2) [×4], C23×C4, C22.C42 [×4], C24.4C4 [×2], C2×C42⋊C2, (C22×C4).276D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C23.34D4, M4(2).8C22 [×2], (C22×C4).276D4

Smallest permutation representation of (C22×C4).276D4
On 32 points
Generators in S32
(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)
(2 31)(4 25)(6 27)(8 29)(9 13)(10 24)(11 15)(12 18)(14 20)(16 22)(17 21)(19 23)
(1 3 5 7)(2 29 6 25)(4 31 8 27)(9 15 13 11)(10 22 14 18)(12 24 16 20)(17 23 21 19)(26 28 30 32)
(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)| (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), (2,31)(4,25)(6,27)(8,29)(9,13)(10,24)(11,15)(12,18)(14,20)(16,22)(17,21)(19,23), (1,3,5,7)(2,29,6,25)(4,31,8,27)(9,15,13,11)(10,22,14,18)(12,24,16,20)(17,23,21,19)(26,28,30,32), (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( (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), (2,31)(4,25)(6,27)(8,29)(9,13)(10,24)(11,15)(12,18)(14,20)(16,22)(17,21)(19,23), (1,3,5,7)(2,29,6,25)(4,31,8,27)(9,15,13,11)(10,22,14,18)(12,24,16,20)(17,23,21,19)(26,28,30,32), (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([(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)], [(2,31),(4,25),(6,27),(8,29),(9,13),(10,24),(11,15),(12,18),(14,20),(16,22),(17,21),(19,23)], [(1,3,5,7),(2,29,6,25),(4,31,8,27),(9,15,13,11),(10,22,14,18),(12,24,16,20),(17,23,21,19),(26,28,30,32)], [(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 2E 2F 2G 4A ··· 4H 4I ··· 4P 8A ··· 8H order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 4 4 2 ··· 2 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 type + + + + + image C1 C2 C2 C2 C4 C4 D4 C4○D4 M4(2).8C22 kernel (C22×C4).276D4 C22.C42 C24.4C4 C2×C42⋊C2 C2×C42 C2×C22⋊C4 C22×C4 C2×C4 C2 # reps 1 4 2 1 4 4 4 8 4

Matrix representation of (C22×C4).276D4 in GL6(𝔽17)

 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
,
 16 0 0 0 0 0 13 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 1
,
 16 0 0 0 0 0 13 1 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 15 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 13 0 0 0 0 13 0 0 0
,
 16 9 0 0 0 0 0 1 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 13 0 0

G:=sub<GL(6,GF(17))| [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],[16,13,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,13,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,15,13,0,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,16,0,0,0,0,1,0,0,0],[16,0,0,0,0,0,9,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C22×C4).276D4 in GAP, Magma, Sage, TeX

(C_2^2\times C_4)._{276}D_4
% in TeX

G:=Group("(C2^2xC4).276D4");
// GroupNames label

G:=SmallGroup(128,554);
// by ID

G=gap.SmallGroup(128,554);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,58,2804,1027,124]);
// Polycyclic

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

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