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G = C42.375D4order 128 = 27

8th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.375D4, (C4×D4)⋊1C4, (C4×Q8)⋊1C4, C4.45C4≀C2, C4.4D47C4, C42.C23C4, C424C44C2, C42.69(C2×C4), C4(C22.SD16), C23.490(C2×D4), (C22×C4).660D4, C4(C23.31D4), C22.16(C4○D8), C22.SD16.7C2, C42.12C415C2, C4⋊D4.129C22, C23.31D423C2, C22⋊C8.161C22, (C22×C4).622C23, (C2×C42).172C22, C22⋊Q8.134C22, C2.8(C23.24D4), C23.36C23.5C2, C2.C42.499C22, C2.12(C23.C23), C4⋊C4.1(C2×C4), C2.17(C2×C4≀C2), (C2×D4).3(C2×C4), (C2×Q8).3(C2×C4), (C2×C4).1146(C2×D4), (C2×C4).112(C22×C4), (C2×C4)(C22.SD16), (C2×C4).170(C22⋊C4), (C2×C4)(C23.31D4), C22.176(C2×C22⋊C4), SmallGroup(128,232)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.375D4
C1C2C22C23C22×C4C2×C42C23.36C23 — C42.375D4
C1C22C2×C4 — C42.375D4
C1C2×C4C2×C42 — C42.375D4
C1C2C22C22×C4 — C42.375D4

Generators and relations for C42.375D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=b-1c-1 >

Subgroups: 244 in 119 conjugacy classes, 46 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×5], C8 [×2], C2×C4 [×6], C2×C4 [×18], D4 [×3], Q8, C23, C23, C42 [×4], C42 [×5], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2.C42 [×2], C2.C42, C4×C8, C22⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C22.SD16 [×2], C23.31D4 [×2], C424C4, C42.12C4, C23.36C23, C42.375D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C4○D8 [×2], C23.C23, C23.24D4, C2×C4≀C2, C42.375D4

Smallest permutation representation of C42.375D4
On 32 points
Generators in S32
(1 10 31 21)(2 11 32 22)(3 12 25 23)(4 13 26 24)(5 14 27 17)(6 15 28 18)(7 16 29 19)(8 9 30 20)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)
(1 17 31 14)(2 20 6 24)(3 16 25 19)(4 11 8 15)(5 21 27 10)(7 12 29 23)(9 28 13 32)(18 26 22 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)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4J4K···4T4U4V4W8A···8H
order122222244444···44···44448···8
size111122811112···24···48884···4

38 irreducible representations

dim111111111122224
type++++++++
imageC1C2C2C2C2C2C4C4C4C4D4D4C4≀C2C4○D8C23.C23
kernelC42.375D4C22.SD16C23.31D4C424C4C42.12C4C23.36C23C4×D4C4×Q8C4.4D4C42.C2C42C22×C4C4C22C2
# reps122111222222882

Matrix representation of C42.375D4 in GL4(𝔽17) generated by

13000
01300
00160
00016
,
01600
1000
0040
0004
,
13000
0400
0010
001213
,
12500
121200
0099
0008
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[0,1,0,0,16,0,0,0,0,0,4,0,0,0,0,4],[13,0,0,0,0,4,0,0,0,0,1,12,0,0,0,13],[12,12,0,0,5,12,0,0,0,0,9,0,0,0,9,8] >;

C42.375D4 in GAP, Magma, Sage, TeX

C_4^2._{375}D_4
% in TeX

G:=Group("C4^2.375D4");
// GroupNames label

G:=SmallGroup(128,232);
// by ID

G=gap.SmallGroup(128,232);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,184,1123,1018,248,1971]);
// Polycyclic

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

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