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G = C42.115D4order 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(C41D4), (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

C1C23 — C42.115D4
C1C2C4C2×C4C22×C4C2×C42C4×M4(2) — C42.115D4
C1C2C23 — C42.115D4
C1C22C2×C42 — C42.115D4
C1C2C2C22×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 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×12], C8 [×6], C2×C4 [×2], C2×C4 [×8], C2×C4 [×8], D4 [×4], Q8 [×4], C23, C23 [×10], C42 [×2], C42 [×2], C22⋊C4 [×8], C2×C8 [×4], M4(2) [×4], M4(2) [×6], C22×C4, C22×C4 [×4], C2×D4 [×6], C2×Q8 [×6], C24 [×2], C4×C8, C8⋊C4, C4.D4 [×4], C4.10D4 [×4], C4⋊C8 [×2], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×4], C2×M4(2) [×4], C22×D4, C22×Q8, C4×M4(2), C2×C4.D4 [×2], C2×C4.10D4 [×2], C4⋊M4(2), C2×C4.4D4, C42.115D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C24.3C22, M4(2).8C22 [×2], C42.115D4

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

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

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

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