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G = C23.9D8order 128 = 27

2nd non-split extension by C23 of D8 acting via D8/C4=C22

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

Aliases: C23.9D8, C8.12C42, (C2×C16)⋊6C4, C4.Q83C4, C8.2(C4⋊C4), (C2×C8).3Q8, (C2×C8).82D4, (C2×C4).7Q16, (C2×C4).94SD16, C4.15(C4.Q8), C8.35(C22⋊C4), C4.6(Q8⋊C4), (C22×C4).190D4, (C2×M5(2)).12C2, C22.10(C2.D8), C4.5(C2.C42), (C22×C8).203C22, C23.25D4.8C2, C22.22(D4⋊C4), C2.14(C22.4Q16), (C2×C8).50(C2×C4), (C2×C4).112(C4⋊C4), (C2×C4).63(C22⋊C4), SmallGroup(128,116)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C23.9D8
C1C2C4C8C2×C8C22×C8C2×M5(2) — C23.9D8
C1C2C4C8 — C23.9D8
C1C4C22×C4C22×C8 — C23.9D8
C1C2C2C2C2C4C4C22×C8 — C23.9D8

Generators and relations for C23.9D8
 G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=abc, ab=ba, eae-1=ac=ca, ad=da, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=ad7 >

Subgroups: 136 in 66 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22, C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C16 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×4], C22×C4, C4.Q8 [×4], C2.D8 [×2], C2×C16 [×2], M5(2) [×2], C42⋊C2 [×2], C22×C8, C23.25D4 [×2], C2×M5(2), C23.9D8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, C22.4Q16, C23.9D8

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F···4M8A8B8C8D8E8F16A···16H
order12222444444···488888816···16
size11222112228···82222444···4

32 irreducible representations

dim111112222224
type++++-+-+
imageC1C2C2C4C4D4Q8D4SD16Q16D8C23.9D8
kernelC23.9D8C23.25D4C2×M5(2)C4.Q8C2×C16C2×C8C2×C8C22×C4C2×C4C2×C4C23C1
# reps121842114224

Matrix representation of C23.9D8 in GL4(𝔽17) generated by

0400
13000
0004
00130
,
0400
13000
00013
0040
,
16000
01600
00160
00016
,
0010
0001
14300
141400
,
51200
121200
0001
0010
G:=sub<GL(4,GF(17))| [0,13,0,0,4,0,0,0,0,0,0,13,0,0,4,0],[0,13,0,0,4,0,0,0,0,0,0,4,0,0,13,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,0,14,14,0,0,3,14,1,0,0,0,0,1,0,0],[5,12,0,0,12,12,0,0,0,0,0,1,0,0,1,0] >;

C23.9D8 in GAP, Magma, Sage, TeX

C_2^3._9D_8
% in TeX

G:=Group("C2^3.9D8");
// GroupNames label

G:=SmallGroup(128,116);
// by ID

G=gap.SmallGroup(128,116);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,184,1018,1684,242,4037,124]);
// Polycyclic

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

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