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G = (C2×D4)⋊21D4  order 128 = 27

17th semidirect product of C2×D4 and D4 acting via D4/C2=C22

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

Aliases: C4○D4⋊3D4, (C2×D4)⋊21D4, D4.47(C2×D4), C2.7(D4○D8), Q8.47(C2×D4), C22⋊D8⋊14C2, D4⋊D4⋊17C2, C4.84C22≀C2, C22⋊SD16⋊4C2, (C2×D8)⋊39C22, C4⋊C4.14C23, C4⋊D4⋊2C22, (C2×Q8).231D4, C4.49(C22×D4), C22⋊Q8⋊2C22, D4.7D4⋊17C2, C22⋊C8⋊10C22, (C2×C8).302C23, (C2×C4).231C24, (C22×C8)⋊15C22, (C2×Q16)⋊39C22, C22.3C22≀C2, (C2×D4).31C23, C23.235(C2×D4), (C2×Q8).24C23, D4⋊C4⋊14C22, C2.10(D4○SD16), Q8⋊C4⋊16C22, (C2×SD16)⋊71C22, C23.36D4⋊3C2, (C2×M4(2))⋊6C22, (C2×2+ 1+4)⋊2C2, (C22×C4).279C23, C22.491(C22×D4), C22.31C24⋊3C2, (C22×D4).328C22, (C2×C4○D8)⋊3C2, (C2×C4⋊C4)⋊48C22, (C2×C8⋊C22)⋊10C2, (C2×C4).458(C2×D4), (C2×C4○D4)⋊5C22, (C2×D4⋊C4)⋊28C2, (C22×C8)⋊C2⋊7C2, C2.49(C2×C22≀C2), SmallGroup(128,1744)

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

Derived series
C1 — C2×C4 — (C2×D4)⋊21D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2+ 1+4 — (C2×D4)⋊21D4
Lower central
C1 — C2 — C2×C4 — (C2×D4)⋊21D4
Upper central
C1 — C22 — C2×C4○D4 — (C2×D4)⋊21D4
Jennings
C1 — C2 — C2 — C2×C4 — (C2×D4)⋊21D4

Generators and relations for (C2×D4)⋊21D4
 G = < a,b,c,d,e | a2=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, eae=ab2, cbc=dbd-1=ebe=b-1, dcd-1=bc, ece=b-1c, ede=d-1 >

Subgroups: 836 in 382 conjugacy classes, 108 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C22⋊C8, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×D8, C2×SD16, C2×SD16, C2×Q16, C4○D8, C8⋊C22, C22×D4, C22×D4, C22×D4, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, (C22×C8)⋊C2, C2×D4⋊C4, C23.36D4, C22⋊D8, D4⋊D4, C22⋊SD16, D4.7D4, C22.31C24, C2×C4○D8, C2×C8⋊C22, C2×2+ 1+4, (C2×D4)⋊21D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, C2×C22≀C2, D4○D8, D4○SD16, (C2×D4)⋊21D4

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

(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 2)(3 4)(6 8)(9 10)(11 12)(13 14)(15 16)(17 18)(19 20)(22 24)(26 28)(29 31)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F···2M2N4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F
order1222222···2244444444444888888
size1111224···4822224444888444488

32 irreducible representations

dim11111111111122244
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D4○D8D4○SD16
kernel(C2×D4)⋊21D4(C22×C8)⋊C2C2×D4⋊C4C23.36D4C22⋊D8D4⋊D4C22⋊SD16D4.7D4C22.31C24C2×C4○D8C2×C8⋊C22C2×2+ 1+4C2×D4C2×Q8C4○D4C2C2
# reps11112222111171422

Matrix representation of (C2×D4)⋊21D4 ►in GL6(𝔽17)

100000
010000
0016000
0001600
0016010
0001601
,
1600000
0160000
0001600
001000
0000016
000010
,
100000
12160000
0016000
000100
0016010
0001016
,
1040000
1370000
0031400
00141400
0000314
00001414
,
7130000
12100000
00314116
00141466
0000143
000033

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,16,0,16,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,12,0,0,0,0,0,16,0,0,0,0,0,0,16,0,16,0,0,0,0,1,0,1,0,0,0,0,1,0,0,0,0,0,0,16],[10,13,0,0,0,0,4,7,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,0,0,0,3,14,0,0,0,0,14,14],[7,12,0,0,0,0,13,10,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,0,11,6,14,3,0,0,6,6,3,3] >;
 

(C2×D4)⋊21D4 in GAP, Magma, Sage, TeX 

(C_2\times D_4)\rtimes_{21}D_4
 
% in TeX
 
G:=Group("(C2xD4):21D4");
 
// GroupNames label
 
G:=SmallGroup(128,1744);
 
// by ID
 
G=gap.SmallGroup(128,1744);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,521,248,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a*b^2,c*b*c=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b*c,e*c*e=b^-1*c,e*d*e=d^-1>;
 
// generators/relations
 

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