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G = C42.372D4order 128 = 27

5th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.372D4, C24.4(C2×C4), C23⋊C8.10C2, C42.12C44C2, C42.6C424C2, C22.14(C8○D4), C22⋊C8.159C22, (C22×C4).437C23, C23.174(C22×C4), (C2×C42).153C22, C22.M4(2)⋊17C2, C2.11(C23.C23), C2.9(M4(2).8C22), (C2×C4⋊C4).15C4, (C2×C4).1134(C2×D4), (C2×C4⋊C4).12C22, (C2×C22⋊C4).10C4, (C22×C4).14(C2×C4), (C2×C4).73(C22⋊C4), (C2×C422C2).1C2, (C2×C22⋊C4).89C22, C22.155(C2×C22⋊C4), C2.11((C22×C8)⋊C2), SmallGroup(128,205)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.372D4
C1C2C22C2×C4C22×C4C2×C42C2×C422C2 — C42.372D4
C1C2C23 — C42.372D4
C1C22C2×C42 — C42.372D4
C1C2C22C22×C4 — C42.372D4

Generators and relations for C42.372D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=ab2, ad=da, cbc-1=a2b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 220 in 107 conjugacy classes, 44 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×9], C22, C22 [×2], C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×11], C23, C23 [×5], C42 [×4], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×4], C22×C4 [×6], C24, C4×C8, C8⋊C4, C22⋊C8 [×4], C4⋊C8 [×2], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C422C2 [×4], C23⋊C8 [×2], C22.M4(2) [×2], C42.12C4, C42.6C4, C2×C422C2, C42.372D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C8○D4 [×2], (C22×C8)⋊C2, C23.C23, M4(2).8C22, C42.372D4

Smallest permutation representation of C42.372D4
On 32 points
Generators in S32
(1 15 25 19)(2 12 26 24)(3 9 27 21)(4 14 28 18)(5 11 29 23)(6 16 30 20)(7 13 31 17)(8 10 32 22)
(1 3 5 7)(2 32 6 28)(4 26 8 30)(9 11 13 15)(10 20 14 24)(12 22 16 18)(17 19 21 23)(25 27 29 31)
(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 30 3 4 5 26 7 8)(2 31 32 25 6 27 28 29)(9 14 11 24 13 10 15 20)(12 17 22 19 16 21 18 23)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A···4H4I4J4K4L4M8A···8H8I8J8K8L
order12222224···4444448···88888
size11112282···2448884···48888

32 irreducible representations

dim111111112244
type+++++++
imageC1C2C2C2C2C2C4C4D4C8○D4C23.C23M4(2).8C22
kernelC42.372D4C23⋊C8C22.M4(2)C42.12C4C42.6C4C2×C422C2C2×C22⋊C4C2×C4⋊C4C42C22C2C2
# reps122111444822

Matrix representation of C42.372D4 in GL6(𝔽17)

400000
4130000
001800
0001600
0008013
0001540
,
400000
040000
0041500
0001300
0001501
0009160
,
2130000
2150000
0020150
00801316
0001150
0016090
,
1500000
1520000
0020150
0000131
0001150
000090

G:=sub<GL(6,GF(17))| [4,4,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,8,16,8,15,0,0,0,0,0,4,0,0,0,0,13,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,15,13,15,9,0,0,0,0,0,16,0,0,0,0,1,0],[2,2,0,0,0,0,13,15,0,0,0,0,0,0,2,8,0,16,0,0,0,0,1,0,0,0,15,13,15,9,0,0,0,16,0,0],[15,15,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,15,13,15,9,0,0,0,1,0,0] >;

C42.372D4 in GAP, Magma, Sage, TeX

C_4^2._{372}D_4
% in TeX

G:=Group("C4^2.372D4");
// GroupNames label

G:=SmallGroup(128,205);
// by ID

G=gap.SmallGroup(128,205);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,723,520,1123,851,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

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