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G = C2×C23.C8order 128 = 27

Direct product of C2 and C23.C8

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

Aliases: C2×C23.C8, C24.3C8, M5(2)⋊10C22, C8.123(C2×D4), (C2×C8).389D4, C4(C23.C8), (C22×C4).6C8, (C2×M5(2))⋊8C2, C23.17(C2×C8), (C23×C4).22C4, (C22×C8).19C4, C8.30(C22⋊C4), C4.25(C22⋊C8), (C2×C8).384C23, C4.49(C2×M4(2)), (C2×C4).81M4(2), (C2×M4(2)).29C4, C22.12(C22×C8), C22.44(C22⋊C8), (C22×C8).415C22, (C22×M4(2)).23C2, (C2×M4(2)).328C22, (C2×C4).38(C2×C8), (C2×C8).149(C2×C4), C2.24(C2×C22⋊C8), C4.115(C2×C22⋊C4), (C22×C4).486(C2×C4), (C2×C4).558(C22×C4), (C2×C4).362(C22⋊C4), SmallGroup(128,846)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C23.C8
C1C2C4C8C2×C8C22×C8C22×M4(2) — C2×C23.C8
C1C2C22 — C2×C23.C8
C1C2×C4C22×C8 — C2×C23.C8
C1C2C2C2C2C4C4C2×C8 — C2×C23.C8

Generators and relations for C2×C23.C8
 G = < a,b,c,d,e | a2=b2=c2=d2=1, e8=d, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 196 in 118 conjugacy classes, 60 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×2], C22 [×3], C22 [×10], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×2], C2×C4 [×10], C23, C23 [×2], C23 [×4], C16 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×2], M4(2) [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C2×C16 [×2], M5(2) [×4], M5(2) [×2], C22×C8 [×2], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4, C23.C8 [×4], C2×M5(2) [×2], C22×M4(2), C2×C23.C8
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], C2×C22⋊C4, C22×C8, C2×M4(2), C23.C8 [×2], C2×C22⋊C8, C2×C23.C8

Smallest permutation representation of C2×C23.C8
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 27)(2 10)(3 21)(5 31)(6 14)(7 25)(9 19)(11 29)(13 23)(15 17)(20 28)(24 32)
(1 27)(2 20)(3 29)(4 22)(5 31)(6 24)(7 17)(8 26)(9 19)(10 28)(11 21)(12 30)(13 23)(14 32)(15 25)(16 18)
(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)| (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), (1,27)(2,10)(3,21)(5,31)(6,14)(7,25)(9,19)(11,29)(13,23)(15,17)(20,28)(24,32), (1,27)(2,20)(3,29)(4,22)(5,31)(6,24)(7,17)(8,26)(9,19)(10,28)(11,21)(12,30)(13,23)(14,32)(15,25)(16,18), (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( (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), (1,27)(2,10)(3,21)(5,31)(6,14)(7,25)(9,19)(11,29)(13,23)(15,17)(20,28)(24,32), (1,27)(2,20)(3,29)(4,22)(5,31)(6,24)(7,17)(8,26)(9,19)(10,28)(11,21)(12,30)(13,23)(14,32)(15,25)(16,18), (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([(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)], [(1,27),(2,10),(3,21),(5,31),(6,14),(7,25),(9,19),(11,29),(13,23),(15,17),(20,28),(24,32)], [(1,27),(2,20),(3,29),(4,22),(5,31),(6,24),(7,17),(8,26),(9,19),(10,28),(11,21),(12,30),(13,23),(14,32),(15,25),(16,18)], [(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)])

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H8A···8H8I8J8K8L16A···16P
order12222222444444448···8888816···16
size11112244111122442···244444···4

44 irreducible representations

dim111111111224
type+++++
imageC1C2C2C2C4C4C4C8C8D4M4(2)C23.C8
kernelC2×C23.C8C23.C8C2×M5(2)C22×M4(2)C22×C8C2×M4(2)C23×C4C22×C4C24C2×C8C2×C4C2
# reps1421242124444

Matrix representation of C2×C23.C8 in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
010000
001000
0001600
000010
0000016
,
1600000
0160000
001000
000100
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
0160000
1300000
000010
000001
000100
0013000

G:=sub<GL(6,GF(17))| [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],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[0,13,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C2×C23.C8 in GAP, Magma, Sage, TeX

C_2\times C_2^3.C_8
% in TeX

G:=Group("C2xC2^3.C8");
// GroupNames label

G:=SmallGroup(128,846);
// by ID

G=gap.SmallGroup(128,846);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,2019,1411,102,124]);
// Polycyclic

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

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