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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⋊C8⋊43C22, (C2×C8).389C23, (C2×C4).629C24, (C22×C8)⋊45C22, (C22×D4).34C4, C4.178(C22×D4), (C22×C4).413D4, (C22×Q8).27C4, C24.4C4⋊30C2, C2.6(Q8○M4(2)), C23.96(C22×C4), C23.86(C22⋊C4), (C22×M4(2))⋊15C2, (C2×M4(2))⋊64C22, (C23×C4).506C22, C22.159(C23×C4), (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)⋊C2⋊26C2, (C22×C4).319(C2×C4), (C2×C4).240(C22×C4), (C22×C4○D4).14C2, C2.22(C22×C22⋊C4), C22.18(C2×C22⋊C4), (C2×C4).155(C22⋊C4), (C2×C4○D4).268C22, SmallGroup(128,1611)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C24.73(C2×C4)
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C24.73(C2×C4)
Lower central
C1 — C22 — C24.73(C2×C4)
Upper central
C1 — C2×C4 — C24.73(C2×C4)
Jennings
C1 — C2 — C2 — C2×C4 — C24.73(C2×C4)

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)

Generators and relations
 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 >

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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
 

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