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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

Aliases: (C2×C42).3C4, C24.54(C2×C4), (C22×C4).276D4, C22.C4216C2, C23.75(C22⋊C4), C24.4C4.19C2, (C22×C4).668C23, (C23×C4).242C22, C23.191(C22×C4), C22.31(C42⋊C2), C4.102(C22.D4), C2.10(C23.34D4), (C2×M4(2)).161C22, C2.25(M4(2).8C22), (C2×C4).1323(C2×D4), (C2×C22⋊C4).27C4, (C22×C4).55(C2×C4), (C2×C4).311(C4○D4), (C2×C4⋊C4).756C22, (C2×C4).334(C22⋊C4), (C2×C42⋊C2).16C2, C22.256(C2×C22⋊C4), SmallGroup(128,554)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — (C22×C4).276D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×C42⋊C2 — (C22×C4).276D4
C1C2C23 — (C22×C4).276D4
C1C22C23×C4 — (C22×C4).276D4
C1C2C2C22×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 2A2B2C2D2E2F2G4A···4H4I···4P8A···8H
order122222224···44···48···8
size111122442···24···48···8

32 irreducible representations

dim111111224
type+++++
imageC1C2C2C2C4C4D4C4○D4M4(2).8C22
kernel(C22×C4).276D4C22.C42C24.4C4C2×C42⋊C2C2×C42C2×C22⋊C4C22×C4C2×C4C2
# reps142144484

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

1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
1310000
001000
0001600
0000160
000001
,
1600000
1310000
0013000
0001300
000040
000004
,
4150000
0130000
000001
0000160
0001300
0013000
,
1690000
010000
000010
000001
004000
0001300

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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