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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

Aliases: (C23×C4).11C4, C24.118(C2×C4), (C22×C4).275D4, C22.C4215C2, C24.4C4.18C2, (C22×C4).667C23, C23.190(C22×C4), (C23×C4).241C22, C23.198(C22⋊C4), C22.20(C4.D4), C2.9(C23.34D4), C22.30(C42⋊C2), C4.101(C22.D4), C22.12(C4.10D4), (C2×M4(2)).160C22, (C2×C4⋊C4).53C4, (C2×C4).1322(C2×D4), (C22×C4⋊C4).12C2, (C22×C4).54(C2×C4), C2.26(C2×C4.D4), (C2×C4).310(C4○D4), (C2×C4⋊C4).755C22, C2.24(C2×C4.10D4), (C2×C4).125(C22⋊C4), C22.255(C2×C22⋊C4), SmallGroup(128,553)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — (C22×C4).275D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C22×C4⋊C4 — (C22×C4).275D4
C1C2C23 — (C22×C4).275D4
C1C22C23×C4 — (C22×C4).275D4
C1C2C2C22×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 [×3], C2 [×6], C4 [×4], C4 [×6], C22, C22 [×6], C22 [×14], C8 [×4], C2×C4 [×8], C2×C4 [×26], C23 [×3], C23 [×6], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×8], C22×C4 [×2], C22×C4 [×8], C22×C4 [×12], C24, C22⋊C8 [×4], C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C2×M4(2) [×4], C23×C4, C23×C4 [×2], C22.C42 [×4], C24.4C4 [×2], C22×C4⋊C4, (C22×C4).275D4
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], C4.D4 [×2], C4.10D4 [×2], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], 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 2A2B2C2D···2I4A4B4C4D4E···4N8A···8H
order12222···244444···48···8
size11112···222224···48···8

32 irreducible representations

dim1111112244
type++++++-
imageC1C2C2C2C4C4D4C4○D4C4.D4C4.10D4
kernel(C22×C4).275D4C22.C42C24.4C4C22×C4⋊C4C2×C4⋊C4C23×C4C22×C4C2×C4C22C22
# reps1421444822

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

1600000
010000
001000
000100
000010
000001
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0011500
0011600
0001601
00116160
,
0150000
900000
0010059
0060157
006111113
0010131113
,
080000
1500000
0010150
0000161
0001160
0010160

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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