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G = C42.366C23  order 128 = 27

227th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.366C23, (C4×C8)⋊63C22, C4⋊C4.238D4, C4⋊Q8⋊10C22, C2.21(D4○D8), C4.4D8⋊39C2, C4⋊C4.89C23, C22⋊C4.78D4, C8⋊C4⋊64C22, (C2×C8).455C23, (C2×C4).334C24, C23.452(C2×D4), (C2×Q8).90C23, C42.C2⋊4C22, D4⋊C4⋊95C22, C8○2M4(2)⋊37C2, C2.33(D4○SD16), Q8⋊C4⋊54C22, C4.45(C4.4D4), (C2×D4).102C23, C4⋊1D4.62C22, C23.37D4⋊36C2, C23.36D4⋊44C2, C23.24D4⋊43C2, (C22×C8).461C22, C4.4D4.31C22, C22.594(C22×D4), C23.41C23⋊6C2, C22.29C24.13C2, (C22×C4).1032C23, C22.15(C4.4D4), (C22×D4).368C22, C42.29C22⋊17C2, C42⋊C2.139C22, C42.78C22⋊25C2, C42.28C22⋊29C2, (C2×M4(2)).371C22, C4.43(C2×C4○D4), (C2×C4).136(C2×D4), (C2×D4⋊C4)⋊57C2, C2.45(C2×C4.4D4), (C2×C4).489(C4○D4), (C2×C4⋊C4).624C22, (C2×C4○D4).149C22, SmallGroup(128,1868)

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

Derived series
C1 — C2×C4 — C42.366C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C8○2M4(2) — C42.366C23
Lower central
C1 — C2 — C2×C4 — C42.366C23
Upper central
C1 — C22 — C42⋊C2 — C42.366C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.366C23

Generators and relations for C42.366C23
 G = < a,b,c,d,e | a4=b4=c2=1, d2=a2b-1, e2=a2b2, ab=ba, cac=a-1b2, ad=da, eae-1=ab2, cbc=b-1, bd=db, be=eb, dcd-1=bc, ece-1=a2b2c, de=ed >

Subgroups: 452 in 202 conjugacy classes, 92 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C4×C8, C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C2×C4⋊C4, C42⋊C2, C22≀C2, C4⋊D4, C22⋊Q8, C4.4D4, C42.C2, C42.C2, C4⋊1D4, C4⋊Q8, C4⋊Q8, C22×C8, C2×M4(2), C22×D4, C2×C4○D4, C8○2M4(2), C2×D4⋊C4, C23.24D4, C23.36D4, C23.37D4, C4.4D8, C42.78C22, C42.28C22, C42.29C22, C22.29C24, C23.41C23, C42.366C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C22×D4, C2×C4○D4, C2×C4.4D4, D4○D8, D4○SD16, C42.366C23

Smallest permutation representation of C42.366C23
►On 32 points
Generators in S32
(1 25 17 11)(2 26 18 12)(3 27 19 13)(4 28 20 14)(5 29 21 15)(6 30 22 16)(7 31 23 9)(8 32 24 10)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I···4M8A8B8C8D8E···8J
order122222222444444444···488888···8
size111122888222244448···822224···4

32 irreducible representations

dim11111111111122244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○D8D4○SD16
kernelC42.366C23C8○2M4(2)C2×D4⋊C4C23.24D4C23.36D4C23.37D4C4.4D8C42.78C22C42.28C22C42.29C22C22.29C24C23.41C23C22⋊C4C4⋊C4C2×C4C2C2
# reps11111122221122822

Matrix representation of C42.366C23 ►in GL6(𝔽17)

790000
2100000
0000143
00001414
003300
0014300
,
1600000
0160000
000100
0016000
000001
0000160
,
100000
6160000
0016000
000100
000001
000010
,
1120000
860000
00141400
0031400
00001414
0000314
,
790000
2100000
0000143
00001414
00141400
0031400

G:=sub<GL(6,GF(17))| [7,2,0,0,0,0,9,10,0,0,0,0,0,0,0,0,3,14,0,0,0,0,3,3,0,0,14,14,0,0,0,0,3,14,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,6,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,8,0,0,0,0,2,6,0,0,0,0,0,0,14,3,0,0,0,0,14,14,0,0,0,0,0,0,14,3,0,0,0,0,14,14],[7,2,0,0,0,0,9,10,0,0,0,0,0,0,0,0,14,3,0,0,0,0,14,14,0,0,14,14,0,0,0,0,3,14,0,0] >;
 

C42.366C23 in GAP, Magma, Sage, TeX 

C_4^2._{366}C_2^3
 
% in TeX
 
G:=Group("C4^2.366C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1868);
 
// by ID
 
G=gap.SmallGroup(128,1868);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,100,1018,2804,172,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=a^2*b^-1,e^2=a^2*b^2,a*b=b*a,c*a*c=a^-1*b^2,a*d=d*a,e*a*e^-1=a*b^2,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b*c,e*c*e^-1=a^2*b^2*c,d*e=e*d>;
 
// generators/relations
 

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