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G = C42.115D4  order 128 = 27

97th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.115D4, M4(2).29D4, C24.12(C2×C4), C22.60(C4×D4), C4.80(C4⋊D4), C4.19(C4⋊1D4), (C4×M4(2))⋊25C2, C4⋊M4(2)⋊29C2, C4.65(C4.4D4), (C2×C42).321C22, C23.199(C22×C4), (C22×C4).701C23, (C22×D4).47C22, (C22×Q8).37C22, (C2×M4(2)).210C22, C2.19(C24.3C22), C2.28(M4(2).8C22), (C2×C4).32(C2×D4), (C2×C4).66(C4○D4), (C2×C22⋊C4).13C4, (C22×C4).25(C2×C4), (C2×C4.D4).9C2, (C2×C4.4D4).9C2, (C2×C4.10D4)⋊22C2, (C2×C4).202(C22⋊C4), C22.289(C2×C22⋊C4), SmallGroup(128,699)

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

Derived series
C1 — C23 — C42.115D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C4×M4(2) — C42.115D4
Lower central
C1 — C2 — C23 — C42.115D4
Upper central
C1 — C22 — C2×C42 — C42.115D4
Jennings
C1 — C2 — C2 — C22×C4 — C42.115D4

Generators and relations for C42.115D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1b2, cbc-1=dbd=b-1, dcd=bc3 >

Subgroups: 340 in 156 conjugacy classes, 56 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C24, C4×C8, C8⋊C4, C4.D4, C4.10D4, C4⋊C8, C2×C42, C2×C22⋊C4, C4.4D4, C2×M4(2), C22×D4, C22×Q8, C4×M4(2), C2×C4.D4, C2×C4.10D4, C4⋊M4(2), C2×C4.4D4, C42.115D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C24.3C22, M4(2).8C22, C42.115D4

Smallest permutation representation of C42.115D4
►On 32 points
Generators in S32
(1 23 27 14)(2 24 28 15)(3 17 29 16)(4 18 30 9)(5 19 31 10)(6 20 32 11)(7 21 25 12)(8 22 26 13)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I4J4K4L8A···8H8I8J8K8L
order122222224···444448···88888
size111122882···244884···48888

32 irreducible representations

dim11111112224
type++++++++
imageC1C2C2C2C2C2C4D4D4C4○D4M4(2).8C22
kernelC42.115D4C4×M4(2)C2×C4.D4C2×C4.10D4C4⋊M4(2)C2×C4.4D4C2×C22⋊C4C42M4(2)C2×C4C2
# reps11221184444

Matrix representation of C42.115D4 ►in GL6(𝔽17)

0160000
100000
0001300
004000
0000013
000040
,
100000
010000
000100
0016000
0000016
000010
,
010000
1600000
000010
000001
000100
0016000
,
0160000
1600000
001000
0001600
000010
0000016

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

C42.115D4 in GAP, Magma, Sage, TeX 

C_4^2._{115}D_4
 
% in TeX
 
G:=Group("C4^2.115D4");
 
// GroupNames label
 
G:=SmallGroup(128,699);
 
// by ID
 
G=gap.SmallGroup(128,699);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,436,2019,1018,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,d*a*d=a^-1*b^2,c*b*c^-1=d*b*d=b^-1,d*c*d=b*c^3>;
 
// generators/relations
 

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