Copied to
clipboard

G = C24.73(C2×C4)  order 128 = 27

38th non-split extension by C24 of C2×C4 acting via C2×C4/C2=C22

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

Aliases: C24.73(C2×C4), C22⋊C843C22, (C2×C4).629C24, (C2×C8).389C23, (C22×C8)⋊45C22, (C22×D4).34C4, C4.178(C22×D4), (C22×C4).413D4, (C22×Q8).27C4, C24.4C430C2, C2.6(Q8○M4(2)), C23.96(C22×C4), C23.86(C22⋊C4), (C2×M4(2))⋊64C22, (C22×M4(2))⋊15C2, C22.159(C23×C4), (C23×C4).506C22, (C22×C4).1493C23, (C2×C4○D4).22C4, (C2×C4).839(C2×D4), C4.67(C2×C22⋊C4), (C2×D4).219(C2×C4), (C2×Q8).198(C2×C4), (C22×C8)⋊C226C2, (C22×C4).319(C2×C4), (C2×C4).240(C22×C4), (C22×C4○D4).14C2, C22.18(C2×C22⋊C4), C2.22(C22×C22⋊C4), (C2×C4).155(C22⋊C4), (C2×C4○D4).268C22, SmallGroup(128,1611)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C24.73(C2×C4)
C1C2C4C2×C4C22×C4C23×C4C22×C4○D4 — C24.73(C2×C4)
C1C22 — C24.73(C2×C4)
C1C2×C4 — C24.73(C2×C4)
C1C2C2C2×C4 — C24.73(C2×C4)

Generators and relations for C24.73(C2×C4)
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f4=d, ab=ba, faf-1=ac=ca, eae=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 636 in 382 conjugacy classes, 172 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C4 [×4], C22, C22 [×6], C22 [×30], C8 [×8], C2×C4 [×2], C2×C4 [×30], C2×C4 [×20], D4 [×24], Q8 [×8], C23, C23 [×10], C23 [×10], C2×C8 [×8], C2×C8 [×8], M4(2) [×16], C22×C4 [×2], C22×C4 [×22], C22×C4 [×8], C2×D4 [×12], C2×D4 [×12], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24, C24 [×2], C22⋊C8 [×16], C22×C8 [×4], C2×M4(2) [×8], C2×M4(2) [×8], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C22×Q8, C2×C4○D4 [×8], C2×C4○D4 [×8], C24.4C4 [×4], (C22×C8)⋊C2 [×8], C22×M4(2) [×2], C22×C4○D4, C24.73(C2×C4)
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C22×C22⋊C4, Q8○M4(2) [×2], C24.73(C2×C4)

Smallest permutation representation of C24.73(C2×C4)
On 32 points
Generators in S32
(1 12)(2 18)(3 14)(4 20)(5 16)(6 22)(7 10)(8 24)(9 28)(11 30)(13 32)(15 26)(17 31)(19 25)(21 27)(23 29)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(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 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)
(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)| (1,12)(2,18)(3,14)(4,20)(5,16)(6,22)(7,10)(8,24)(9,28)(11,30)(13,32)(15,26)(17,31)(19,25)(21,27)(23,29), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(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), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (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( (1,12)(2,18)(3,14)(4,20)(5,16)(6,22)(7,10)(8,24)(9,28)(11,30)(13,32)(15,26)(17,31)(19,25)(21,27)(23,29), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(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), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (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([(1,12),(2,18),(3,14),(4,20),(5,16),(6,22),(7,10),(8,24),(9,28),(11,30),(13,32),(15,26),(17,31),(19,25),(21,27),(23,29)], [(1,31),(2,28),(3,25),(4,30),(5,27),(6,32),(7,29),(8,26),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(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)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28)], [(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 2A2B2C2D···2I2J2K2L2M4A4B4C4D4E···4J4K4L4M4N8A···8P
order12222···2222244444···444448···8
size11112···2444411112···244444···4

44 irreducible representations

dim1111111124
type++++++
imageC1C2C2C2C2C4C4C4D4Q8○M4(2)
kernelC24.73(C2×C4)C24.4C4(C22×C8)⋊C2C22×M4(2)C22×C4○D4C22×D4C22×Q8C2×C4○D4C22×C4C2
# reps1482162884

Matrix representation of C24.73(C2×C4) in GL6(𝔽17)

1600000
010000
000400
0013000
000004
0000130
,
100000
010000
001000
000100
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
100000
010000
000100
001000
000001
000010
,
0160000
100000
000010
000001
004000
000400

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0] >;

C24.73(C2×C4) in GAP, Magma, Sage, TeX

C_2^4._{73}(C_2\times C_4)
% in TeX

G:=Group("C2^4.73(C2xC4)");
// GroupNames label

G:=SmallGroup(128,1611);
// by ID

G=gap.SmallGroup(128,1611);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,2019,124]);
// Polycyclic

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

׿
×
𝔽