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G = C23.40D8order 128 = 27

11st non-split extension by C23 of D8 acting via D8/D4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C23.40D8, (C2×D8)⋊14C4, D8.8(C2×C4), C8.94(C2×D4), (C2×C8).118D4, (C2×C4).138D8, C2.D1615C2, (C2×C16)⋊11C22, C8.5(C22⋊C4), C8.31(C22×C4), (C2×C4).48SD16, C4.12(C2×SD16), C22.55(C2×D8), C2.D843C22, C2.2(C16⋊C22), (C2×M5(2))⋊13C2, (C2×C8).496C23, (C22×D8).13C2, (C22×C4).331D4, C4.11(D4⋊C4), (C2×D8).102C22, C23.25D418C2, (C22×C8).230C22, C22.29(D4⋊C4), (C2×C8).80(C2×C4), (C2×C4).758(C2×D4), C4.52(C2×C22⋊C4), C2.30(C2×D4⋊C4), (C2×C4).149(C22⋊C4), SmallGroup(128,872)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C23.40D8
C1C2C4C2×C4C2×C8C22×C8C22×D8 — C23.40D8
C1C2C4C8 — C23.40D8
C1C22C22×C4C22×C8 — C23.40D8
C1C2C2C2C2C4C4C2×C8 — C23.40D8

Generators and relations for C23.40D8
 G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=b, ab=ba, dad-1=eae-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bcd7 >

Subgroups: 388 in 130 conjugacy classes, 52 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×18], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], D4 [×10], C23, C23 [×10], C16 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×4], D8 [×4], D8 [×6], C22×C4, C2×D4 [×9], C24, C4.Q8, C2.D8 [×2], C2×C16 [×2], M5(2) [×2], C42⋊C2, C22×C8, C2×D8 [×6], C2×D8 [×3], C22×D4, C2.D16 [×4], C23.25D4, C2×M5(2), C22×D8, C23.40D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C2×D4⋊C4, C16⋊C22 [×2], C23.40D8

Smallest permutation representation of C23.40D8
On 32 points
Generators in S32
(1 23)(2 32)(3 25)(4 18)(5 27)(6 20)(7 29)(8 22)(9 31)(10 24)(11 17)(12 26)(13 19)(14 28)(15 21)(16 30)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 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 16 23 22)(2 21 24 15)(3 14 25 20)(4 19 26 13)(5 12 27 18)(6 17 28 11)(7 10 29 32)(8 31 30 9)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H8A8B8C8D8E8F16A···16H
order12222222224444444488888816···16
size1111228888222288882222444···4

32 irreducible representations

dim111111222224
type++++++++++
imageC1C2C2C2C2C4D4D4D8SD16D8C16⋊C22
kernelC23.40D8C2.D16C23.25D4C2×M5(2)C22×D8C2×D8C2×C8C22×C4C2×C4C2×C4C23C2
# reps141118312424

Matrix representation of C23.40D8 in GL6(𝔽17)

100000
010000
0016000
0001600
000010
000001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
1090000
670000
0000016
0000160
0031400
00141400
,
780000
15100000
000010
000001
0016000
0001600

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[10,6,0,0,0,0,9,7,0,0,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,16,0,0,0,0,16,0,0,0],[7,15,0,0,0,0,8,10,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

C23.40D8 in GAP, Magma, Sage, TeX

C_2^3._{40}D_8
% in TeX

G:=Group("C2^3.40D8");
// GroupNames label

G:=SmallGroup(128,872);
// by ID

G=gap.SmallGroup(128,872);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,352,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^7>;
// generators/relations

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