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G = (C2xD4).301D4order 128 = 27

54th non-split extension by C2xD4 of D4 acting via D4/C22=C2

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

Aliases: (C2xD4).301D4, C2.15(D4oD8), (C2xQ8).236D4, C4.Q8:11C22, C2.D8:22C22, C4:C4.394C23, C22:C8:12C22, (C2xC4).294C24, (C2xC8).144C23, (C2xD4).82C23, C23.245(C2xD4), C2.23(D4oSD16), 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, (C22xC8).184C22, C22.554(C22xD4), D4:C4.180C22, C22.29C24.12C2, C23.33C23:9C2, (C22xC4).1010C23, C4.82(C22.D4), (C22xD4).360C22, (C2xM4(2)).76C22, C22.18(C22.D4), (C2xC4:C4):49C22, C4.104(C2xC4oD4), (C2xC4).489(C2xD4), (C2xD4:C4):30C2, (C22xC8):C2:9C2, (C2xC4).296(C4oD4), (C2xC4oD4).139C22, C2.59(C2xC22.D4), SmallGroup(128,1828)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — (C2xD4).301D4
C1C2C4C2xC4C22xC4C2xC4:C4C23.33C23 — (C2xD4).301D4
C1C2C2xC4 — (C2xD4).301D4
C1C22C2xC4oD4 — (C2xD4).301D4
C1C2C2C2xC4 — (C2xD4).301D4

Generators and relations for (C2xD4).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, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C23, C42, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C22:C8, D4:C4, C4.Q8, C4.Q8, C2.D8, C2.D8, C2xC4:C4, C2xC4:C4, C42:C2, C42:C2, C4xD4, C4xQ8, C22wrC2, C4:D4, C4.4D4, C4:1D4, C22xC8, C2xM4(2), C22xD4, C2xC4oD4, (C22xC8):C2, C2xD4:C4, C23.37D4, C23.25D4, M4(2):C4, C22.D8, C23.46D4, C23.19D4, C23.33C23, C22.29C24, (C2xD4).301D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C22.D4, C22xD4, C2xC4oD4, C2xC22.D4, D4oD8, D4oSD16, (C2xD4).301D4

Smallest permutation representation of (C2xD4).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++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2D4D4C4oD4D4oD8D4oSD16
kernel(C2xD4).301D4(C22xC8):C2C2xD4:C4C23.37D4C23.25D4M4(2):C4C22.D8C23.46D4C23.19D4C23.33C23C22.29C24C2xD4C2xQ8C2xC4C2C2
# reps1111112241131822

Matrix representation of (C2xD4).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] >;

(C2xD4).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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