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G = C42.18C23  order 128 = 27

18th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.18C23, C4○D4⋊7D4, C4⋊C8⋊8C22, D4.9(C2×D4), Q8.9(C2×D4), C4⋊D8⋊20C2, C4⋊C4.339D4, C4⋊SD16⋊4C2, C2.9(D4○D8), (C4×D4)⋊4C22, (C4×Q8)⋊4C22, (C2×D8)⋊41C22, (C2×C8).20C23, C4.76(C22×D4), D4.2D4⋊16C2, C4⋊C4.386C23, (C2×C4).249C24, Q8.D4⋊16C2, C22⋊C4.140D4, (C2×Q16)⋊41C22, (C2×D4).55C23, C23.446(C2×D4), (C2×Q8).42C23, C4.170(C4⋊D4), D4⋊C4⋊17C22, C22.29C24⋊9C2, C2.14(D4○SD16), Q8⋊C4⋊19C22, (C2×SD16)⋊74C22, C4⋊1D4.59C22, C23.36D4⋊7C2, C22.6(C4⋊D4), (C22×C4).979C23, (C22×C8).178C22, C42.6C22⋊7C2, C4.4D4.26C22, C22.509(C22×D4), C23.33C23⋊7C2, (C22×D4).343C22, (C2×M4(2)).56C22, C42⋊C2.104C22, (C2×C4○D8)⋊6C2, (C2×C8⋊C22)⋊17C2, C4.159(C2×C4○D4), (C2×C4).469(C2×D4), C2.67(C2×C4⋊D4), (C2×D4⋊C4)⋊29C2, (C2×C4).280(C4○D4), (C2×C4⋊C4).583C22, (C2×C4○D4).121C22, SmallGroup(128,1777)

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

Derived series
C1 — C2×C4 — C42.18C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C23.33C23 — C42.18C23
Lower central
C1 — C2 — C2×C4 — C42.18C23
Upper central
C1 — C22 — C42⋊C2 — C42.18C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.18C23

Generators and relations for C42.18C23
 G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=a2, ab=ba, cac=a-1, ad=da, eae-1=ab2, cbc=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, de=ed >

Subgroups: 548 in 252 conjugacy classes, 100 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, D4⋊C4, Q8⋊C4, C4⋊C8, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C22×C8, C2×M4(2), C2×D8, C2×D8, C2×SD16, C2×SD16, C2×Q16, C4○D8, C8⋊C22, C22×D4, C2×C4○D4, C2×D4⋊C4, C23.36D4, C42.6C22, C4⋊D8, C4⋊SD16, D4.2D4, Q8.D4, C23.33C23, C22.29C24, C2×C4○D8, C2×C8⋊C22, C42.18C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊D4, D4○D8, D4○SD16, C42.18C23

Smallest permutation representation of C42.18C23
►On 32 points
Generators in S32
(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 12 16 27)(2 9 13 28)(3 10 14 25)(4 11 15 26)(5 20 22 32)(6 17 23 29)(7 18 24 30)(8 19 21 31)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D4E···4N4O8A8B8C8D8E8F
order1222222222244444···44888888
size1111224488822224···48444488

32 irreducible representations

dim111111111111222244
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4C4○D4D4○D8D4○SD16
kernelC42.18C23C2×D4⋊C4C23.36D4C42.6C22C4⋊D8C4⋊SD16D4.2D4Q8.D4C23.33C23C22.29C24C2×C4○D8C2×C8⋊C22C22⋊C4C4⋊C4C4○D4C2×C4C2C2
# reps111122221111224422

Matrix representation of C42.18C23 ►in GL6(𝔽17)

1380000
040000
001001010
0050010
000121212
001251212
,
1600000
0160000
0011500
0011600
00161016
000110
,
490000
4130000
0000710
0050010
00512512
00012512
,
1150000
0160000
00160150
0000161
000010
000110
,
490000
0130000
0000710
0012070
00012512
00512512

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,8,4,0,0,0,0,0,0,10,5,0,12,0,0,0,0,12,5,0,0,10,0,12,12,0,0,10,10,12,12],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,16,0,0,0,15,16,1,1,0,0,0,0,0,1,0,0,0,0,16,0],[4,4,0,0,0,0,9,13,0,0,0,0,0,0,0,5,5,0,0,0,0,0,12,12,0,0,7,0,5,5,0,0,10,10,12,12],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,16,1,1,0,0,0,1,0,0],[4,0,0,0,0,0,9,13,0,0,0,0,0,0,0,12,0,5,0,0,0,0,12,12,0,0,7,7,5,5,0,0,10,0,12,12] >;
 

C42.18C23 in GAP, Magma, Sage, TeX 

C_4^2._{18}C_2^3
 
% in TeX
 
G:=Group("C4^2.18C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1777);
 
// by ID
 
G=gap.SmallGroup(128,1777);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,1018,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=1,e^2=a^2,a*b=b*a,c*a*c=a^-1,a*d=d*a,e*a*e^-1=a*b^2,c*b*c=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*c,d*e=e*d>;
 
// generators/relations
 

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