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G = C23⋊C8⋊C2order 128 = 27

15th semidirect product of C23⋊C8 and C2 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C23⋊C815C2, C24.1(C2×C4), (C22×D4).4C4, C22.9(C8○D4), (C22×C4).651D4, C24.4C417C2, C22.5(C23⋊C4), C22⋊C8.124C22, C23.91(C22⋊C4), (C23×C4).200C22, C23.169(C22×C4), (C22×C4).432C23, C22.M4(2)⋊15C2, C2.8(M4(2).8C22), (C2×C4⋊C4).11C4, (C2×C22⋊C8)⋊3C2, C2.9(C2×C23⋊C4), (C2×C4⋊C4).9C22, (C2×C22⋊C4).7C4, (C22×C4).9(C2×C4), (C2×C4).1129(C2×D4), (C2×C4).70(C22⋊C4), C2.6((C22×C8)⋊C2), (C2×C22⋊C4).86C22, C22.150(C2×C22⋊C4), (C2×C22.D4).1C2, SmallGroup(128,200)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23⋊C8⋊C2
C1C2C22C2×C4C22×C4C23×C4C2×C22.D4 — C23⋊C8⋊C2
C1C2C23 — C23⋊C8⋊C2
C1C22C23×C4 — C23⋊C8⋊C2
C1C2C22C22×C4 — C23⋊C8⋊C2

Generators and relations for C23⋊C8⋊C2
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, dad-1=abc, eae=ad4, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 324 in 140 conjugacy classes, 46 normal (34 characteristic)
C1, C2 [×3], C2 [×6], C4 [×7], C22, C22 [×4], C22 [×16], C8 [×4], C2×C4 [×4], C2×C4 [×15], D4 [×4], C23 [×3], C23 [×10], C22⋊C4 [×6], C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4 [×9], C22×C4 [×2], C2×D4 [×4], C24 [×2], C22⋊C8 [×4], C22⋊C8 [×2], C2×C22⋊C4 [×3], C2×C4⋊C4 [×2], C22.D4 [×4], C22×C8, C2×M4(2), C23×C4, C22×D4, C23⋊C8 [×2], C22.M4(2) [×2], C2×C22⋊C8, C24.4C4, C2×C22.D4, C23⋊C8⋊C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C8○D4 [×2], (C22×C8)⋊C2, C2×C23⋊C4, M4(2).8C22, C23⋊C8⋊C2

Smallest permutation representation of C23⋊C8⋊C2
On 32 points
Generators in S32
(1 5)(2 32)(3 29)(6 28)(7 25)(9 13)(11 18)(12 23)(15 22)(16 19)(20 24)(27 31)
(1 5)(2 28)(3 7)(4 30)(6 32)(8 26)(9 24)(10 14)(11 18)(12 16)(13 20)(15 22)(17 21)(19 23)(25 29)(27 31)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(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 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4F4G4H4I4J8A···8H8I8J8K8L
order12222222224···444448···88888
size11112222482···248884···48888

32 irreducible representations

dim1111111112244
type++++++++
imageC1C2C2C2C2C2C4C4C4D4C8○D4C23⋊C4M4(2).8C22
kernelC23⋊C8⋊C2C23⋊C8C22.M4(2)C2×C22⋊C8C24.4C4C2×C22.D4C2×C22⋊C4C2×C4⋊C4C22×D4C22×C4C22C22C2
# reps1221114224822

Matrix representation of C23⋊C8⋊C2 in GL6(𝔽17)

1600000
010000
001000
0001600
000010
0000016
,
100000
010000
001000
000100
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
200000
020000
000010
000001
000100
001000
,
010000
100000
0016000
0001600
0000160
0000016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;

C23⋊C8⋊C2 in GAP, Magma, Sage, TeX

C_2^3\rtimes C_8\rtimes C_2
% in TeX

G:=Group("C2^3:C8:C2");
// GroupNames label

G:=SmallGroup(128,200);
// by ID

G=gap.SmallGroup(128,200);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,723,1123,851,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,e*a*e=a*d^4,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,d*e=e*d>;
// generators/relations

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