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G = C2×C23.31D4order 128 = 27

Direct product of C2 and C23.31D4

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

Aliases: C2×C23.31D4, C24.149D4, C23.16Q16, C23.31SD16, C22⋊Q81C4, (C22×Q8)⋊2C4, C22.37C4≀C2, C22.7(C2×Q16), C23.489(C2×D4), (C22×C4).210D4, C22.8(C2×SD16), C22⋊C8.160C22, C22.46(C23⋊C4), (C23×C4).204C22, (C22×C4).621C23, C22.6(Q8⋊C4), C22⋊Q8.133C22, C23.101(C22⋊C4), C2.C42.498C22, (C2×C4⋊C4)⋊6C4, C4⋊C42(C2×C4), (C2×Q8)⋊1(C2×C4), C2.16(C2×C4≀C2), C2.13(C2×C23⋊C4), C2.7(C2×Q8⋊C4), (C2×C22⋊Q8).2C2, (C2×C4).1145(C2×D4), (C2×C22⋊C8).10C2, (C2×C4).86(C22⋊C4), (C2×C4).111(C22×C4), (C22×C4).195(C2×C4), C22.175(C2×C22⋊C4), (C2×C2.C42).16C2, SmallGroup(128,231)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C23.31D4
C1C2C22C23C22×C4C23×C4C2×C22⋊Q8 — C2×C23.31D4
C1C22C2×C4 — C2×C23.31D4
C1C23C23×C4 — C2×C23.31D4
C1C2C22C22×C4 — C2×C23.31D4

Generators and relations for C2×C23.31D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bcde3 >

Subgroups: 364 in 168 conjugacy classes, 60 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×12], C22 [×3], C22 [×8], C22 [×12], C8 [×2], C2×C4 [×4], C2×C4 [×32], Q8 [×4], C23 [×3], C23 [×4], C23 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], C22×C4 [×6], C22×C4 [×14], C2×Q8 [×2], C2×Q8 [×3], C24, C2.C42 [×2], C2.C42, C22⋊C8 [×2], C22⋊C8, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8, C23×C4, C23×C4, C22×Q8, C23.31D4 [×4], C2×C2.C42, C2×C22⋊C8, C2×C22⋊Q8, C2×C23.31D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C23⋊C4 [×2], Q8⋊C4 [×4], C4≀C2 [×2], C2×C22⋊C4, C2×SD16, C2×Q16, C23.31D4 [×4], C2×C23⋊C4, C2×Q8⋊C4, C2×C4≀C2, C2×C23.31D4

Smallest permutation representation of C2×C23.31D4
On 32 points
Generators in S32
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)
(2 26)(4 28)(6 30)(8 32)(9 18)(11 20)(13 22)(15 24)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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 5)(2 8 26 32)(3 27)(4 30 28 6)(7 31)(9 15 18 24)(10 19)(11 22 20 13)(12 16)(14 23)(17 21)(25 29)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4N4O4P4Q4R8A···8H
order12···2222244444···444448···8
size11···1222222224···488884···4

38 irreducible representations

dim11111111222224
type+++++++-+
imageC1C2C2C2C2C4C4C4D4D4SD16Q16C4≀C2C23⋊C4
kernelC2×C23.31D4C23.31D4C2×C2.C42C2×C22⋊C8C2×C22⋊Q8C2×C4⋊C4C22⋊Q8C22×Q8C22×C4C24C23C23C22C22
# reps14111242314482

Matrix representation of C2×C23.31D4 in GL6(𝔽17)

100000
010000
0016000
0001600
0000160
0000016
,
100000
010000
001000
000100
000010
0000016
,
1600000
0160000
0016000
0001600
000010
000001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
0100000
12100000
0012500
00121200
000004
0000160
,
1600000
1610000
0016000
000100
0000160
000004

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,12,0,0,0,0,10,10,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,0,16,0,0,0,0,4,0],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,4] >;

C2×C23.31D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{31}D_4
% in TeX

G:=Group("C2xC2^3.31D4");
// GroupNames label

G:=SmallGroup(128,231);
// by ID

G=gap.SmallGroup(128,231);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1123,1018,248,1971]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*c*d*e^3>;
// generators/relations

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