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G = C24.5C8order 128 = 27

2nd non-split extension by C24 of C8 acting via C8/C4=C2

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C24.5C8, C222M5(2), (C2×C16)⋊9C22, C8.131(C2×D4), (C2×C8).387D4, C22⋊C1613C2, (C2×M5(2))⋊6C2, (C22×C8).47C4, (C23×C8).23C2, C23.36(C2×C8), (C22×C4).13C8, (C23×C4).37C4, C2.6(C2×M5(2)), C4.24(C22⋊C8), C8.53(C22⋊C4), (C2×C8).624C23, (C2×C4).80M4(2), C4.62(C2×M4(2)), C22.46(C22×C8), C22.30(C22⋊C8), (C22×C8).498C22, (C2×C4).86(C2×C8), (C2×C8).249(C2×C4), C2.22(C2×C22⋊C8), C4.113(C2×C22⋊C4), (C22×C4).446(C2×C4), (C2×C4).609(C22×C4), (C2×C4).361(C22⋊C4), SmallGroup(128,844)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C24.5C8
C1C2C4C8C2×C8C22×C8C23×C8 — C24.5C8
C1C22 — C24.5C8
C1C2×C8 — C24.5C8
C1C2C2C2C2C4C4C2×C8 — C24.5C8

Generators and relations for C24.5C8
 G = < a,b,c,d,e | a2=b2=c2=d2=1, e8=d, ab=ba, eae-1=ac=ca, ad=da, bc=cb, ebe-1=bd=db, cd=dc, ce=ec, de=ed >

Subgroups: 196 in 130 conjugacy classes, 64 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×14], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×10], C23, C23 [×2], C23 [×6], C16 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×10], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C2×C16 [×4], M5(2) [×4], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C23×C4, C22⋊C16 [×4], C2×M5(2) [×2], C23×C8, C24.5C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], M5(2) [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, C2×M5(2) [×2], C24.5C8

Smallest permutation representation of C24.5C8
On 32 points
Generators in S32
(2 27)(4 29)(6 31)(8 17)(10 19)(12 21)(14 23)(16 25)
(1 18)(2 27)(3 20)(4 29)(5 22)(6 31)(7 24)(8 17)(9 26)(10 19)(11 28)(12 21)(13 30)(14 23)(15 32)(16 25)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 17)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)
(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)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J8A···8H8I···8T16A···16P
order12222···244444···48···88···816···16
size11112···211112···21···12···24···4

56 irreducible representations

dim11111111222
type+++++
imageC1C2C2C2C4C4C8C8D4M4(2)M5(2)
kernelC24.5C8C22⋊C16C2×M5(2)C23×C8C22×C8C23×C4C22×C4C24C2×C8C2×C4C22
# reps1421621244416

Matrix representation of C24.5C8 in GL4(𝔽17) generated by

1000
0100
0010
00816
,
16000
0100
00160
00016
,
1000
0100
00160
00016
,
16000
01600
0010
0001
,
0100
9000
0092
00118
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,8,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,9,0,0,1,0,0,0,0,0,9,11,0,0,2,8] >;

C24.5C8 in GAP, Magma, Sage, TeX

C_2^4._5C_8
% in TeX

G:=Group("C2^4.5C8");
// GroupNames label

G:=SmallGroup(128,844);
// by ID

G=gap.SmallGroup(128,844);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,102,124]);
// Polycyclic

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

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