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G = (C2×D4).301D4  order 128 = 27

54th non-split extension by C2×D4 of D4 acting via D4/C22=C2

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

Aliases: (C2×D4).301D4, C2.15(D4○D8), (C2×Q8).236D4, C4.Q8⋊11C22, C2.D8⋊22C22, C4⋊C4.394C23, C22⋊C8⋊12C22, (C2×C4).294C24, (C2×C8).144C23, (C2×D4).82C23, C23.245(C2×D4), C2.23(D4○SD16), C4⋊D4.23C22, C22.D8⋊13C2, C23.46D4⋊3C2, C23.25D4⋊7C2, M4(2)⋊C4⋊26C2, C42⋊C2⋊15C22, C23.37D4⋊11C2, C23.19D4⋊14C2, (C22×C8).184C22, C22.554(C22×D4), D4⋊C4.180C22, C22.29C24.12C2, C23.33C23⋊9C2, (C22×C4).1010C23, C4.82(C22.D4), (C22×D4).360C22, (C2×M4(2)).76C22, C22.18(C22.D4), (C2×C4⋊C4)⋊49C22, C4.104(C2×C4○D4), (C2×C4).489(C2×D4), (C2×D4⋊C4)⋊30C2, (C22×C8)⋊C2⋊9C2, (C2×C4).296(C4○D4), (C2×C4○D4).139C22, C2.59(C2×C22.D4), SmallGroup(128,1828)

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

Derived series
C1 — C2×C4 — (C2×D4).301D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C23.33C23 — (C2×D4).301D4
Lower central
C1 — C2 — C2×C4 — (C2×D4).301D4
Upper central
C1 — C22 — C2×C4○D4 — (C2×D4).301D4
Jennings
C1 — C2 — C2 — C2×C4 — (C2×D4).301D4

Generators and relations for (C2×D4).301D4
 G = < a,b,c,d,e | a2=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=ebe=ab-1, dcd-1=ece=ab2c, ede=ad3 >

Subgroups: 452 in 209 conjugacy classes, 92 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C22⋊C8, D4⋊C4, C4.Q8, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C22×C8, C2×M4(2), C22×D4, C2×C4○D4, (C22×C8)⋊C2, C2×D4⋊C4, C23.37D4, C23.25D4, M4(2)⋊C4, C22.D8, C23.46D4, C23.19D4, C23.33C23, C22.29C24, (C2×D4).301D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22.D4, C22×D4, C2×C4○D4, C2×C22.D4, D4○D8, D4○SD16, (C2×D4).301D4

Smallest permutation representation of (C2×D4).301D4
►On 32 points
Generators in S32
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4N4O4P8A8B8C8D8E8F
order122222222244444···444888888
size111122448822224···488444488

32 irreducible representations

dim1111111111122244
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○D8D4○SD16
kernel(C2×D4).301D4(C22×C8)⋊C2C2×D4⋊C4C23.37D4C23.25D4M4(2)⋊C4C22.D8C23.46D4C23.19D4C23.33C23C22.29C24C2×D4C2×Q8C2×C4C2C2
# reps1111112241131822

Matrix representation of (C2×D4).301D4 ►in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
0160000
1600000
0010150
000011
0010160
00161610
,
010000
100000
0010150
000011
0000160
000110
,
1300000
040000
006600
0014000
0003314
00141433
,
100000
0160000
001000
00161600
000010
00160016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,0,1,16,0,0,0,0,0,16,0,0,15,1,16,1,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,15,1,16,1,0,0,0,1,0,0],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,6,14,0,14,0,0,6,0,3,14,0,0,0,0,3,3,0,0,0,0,14,3],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,16,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
 

(C2×D4).301D4 in GAP, Magma, Sage, TeX 

(C_2\times D_4)._{301}D_4
 
% in TeX
 
G:=Group("(C2xD4).301D4");
 
// GroupNames label
 
G:=SmallGroup(128,1828);
 
// by ID
 
G=gap.SmallGroup(128,1828);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,100,1018,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=e^2=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e=a*b^-1,d*c*d^-1=e*c*e=a*b^2*c,e*d*e=a*d^3>;
 
// generators/relations
 

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