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G = C42.62D4  order 128 = 27

44th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.62D4, C4⋊Q8⋊13C4, (C4×Q8)⋊7C4, (C2×C4).11Q16, C42.79(C2×C4), (C2×C4).25SD16, C22.9(C2×Q16), C42⋊8C4.4C2, (C22×C4).219D4, C23.508(C2×D4), C4.25(Q8⋊C4), C22.10(C2×SD16), C22⋊C8.167C22, (C22×C4).640C23, (C2×C42).182C22, C23.31D4.1C2, C22⋊Q8.146C22, C42.12C4.21C2, C2.C42.8C22, C2.23(C42⋊C22), C23.37C23.8C2, C2.22(C23.C23), C4⋊C4.18(C2×C4), (C2×Q8).14(C2×C4), C2.9(C2×Q8⋊C4), (C2×C4).1164(C2×D4), (C2×C4).130(C22×C4), (C2×C4).240(C22⋊C4), C22.194(C2×C22⋊C4), SmallGroup(128,250)

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

Derived series
C1 — C2×C4 — C42.62D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.37C23 — C42.62D4
Lower central
C1 — C22 — C2×C4 — C42.62D4
Upper central
C1 — C22 — C2×C42 — C42.62D4
Jennings
C1 — C2 — C22 — C22×C4 — C42.62D4

Generators and relations for C42.62D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2b-1, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b, bd=db, dcd-1=b-1c-1 >

Subgroups: 220 in 110 conjugacy classes, 52 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2.C42, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C23.31D4, C42⋊8C4, C42.12C4, C23.37C23, C42.62D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C23.C23, C2×Q8⋊C4, C42⋊C22, C42.62D4

Smallest permutation representation of C42.62D4
►On 32 points
Generators in S32
(1 15 32 17)(2 22 25 12)(3 9 26 19)(4 24 27 14)(5 11 28 21)(6 18 29 16)(7 13 30 23)(8 20 31 10)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J4K···4R8A···8H
order1222224···4444···48···8
size1111222···2448···84···4

32 irreducible representations

dim1111111222244
type+++++++-
imageC1C2C2C2C2C4C4D4D4SD16Q16C23.C23C42⋊C22
kernelC42.62D4C23.31D4C42⋊8C4C42.12C4C23.37C23C4×Q8C4⋊Q8C42C22×C4C2×C4C2×C4C2C2
# reps1411144224422

Matrix representation of C42.62D4 ►in GL6(𝔽17)

13150000
040000
000100
001000
0016161615
000001
,
420000
0130000
004000
000400
000040
000004
,
190000
13160000
0001600
001000
004448
00671313
,
2110000
090000
004448
000040
0001600
00671313

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,15,4,0,0,0,0,0,0,0,1,16,0,0,0,1,0,16,0,0,0,0,0,16,0,0,0,0,0,15,1],[4,0,0,0,0,0,2,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,13,0,0,0,0,9,16,0,0,0,0,0,0,0,1,4,6,0,0,16,0,4,7,0,0,0,0,4,13,0,0,0,0,8,13],[2,0,0,0,0,0,11,9,0,0,0,0,0,0,4,0,0,6,0,0,4,0,16,7,0,0,4,4,0,13,0,0,8,0,0,13] >;
 

C42.62D4 in GAP, Magma, Sage, TeX 

C_4^2._{62}D_4
 
% in TeX
 
G:=Group("C4^2.62D4");
 
// GroupNames label
 
G:=SmallGroup(128,250);
 
// by ID
 
G=gap.SmallGroup(128,250);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,387,184,1123,1018,248,1971]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b^-1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,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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