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G = D4.(C2×D4)  order 128 = 27

8th non-split extension by D4 of C2×D4 acting via C2×D4/C23=C2

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

Aliases: C4○D4.13D4, D4.44(C2×D4), Q8.44(C2×D4), C4.40C22≀C2, C22⋊SD163C2, (C2×D4).146D4, C4⋊C4.11C23, C22⋊C88C22, (C2×Q8).230D4, C4.46(C22×D4), C22⋊Q81C22, D4.7D415C2, (C2×C4).228C24, (C2×C8).300C23, (C22×C8)⋊35C22, C2.8(D4○SD16), (C2×Q16)⋊15C22, C23.232(C2×D4), (C2×Q8).22C23, D4⋊C413C22, C42⋊C28C22, Q8⋊C475C22, C22.20C22≀C2, (C22×SD16)⋊19C2, (C2×SD16)⋊70C22, (C2×D4).382C23, C23.37D45C2, (C2×M4(2))⋊5C22, (C22×Q8)⋊14C22, C23.24D421C2, (C22×C4).276C23, C22.488(C22×D4), (C2×2+ 1+4).7C2, C23.38C233C2, (C22×D4).326C22, (C2×C4).455(C2×D4), (C2×C8.C22)⋊8C2, (C22×C8)⋊C24C2, C2.46(C2×C22≀C2), (C2×C4○D4).101C22, SmallGroup(128,1741)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D4.(C2×D4)
C1C2C22C2×C4C22×C4C2×C4○D4C2×2+ 1+4 — D4.(C2×D4)
C1C2C2×C4 — D4.(C2×D4)
C1C22C2×C4○D4 — D4.(C2×D4)
C1C2C2C2×C4 — D4.(C2×D4)

Generators and relations for D4.(C2×D4)
 G = < a,b,c,d,e | a4=b2=1, c2=d4=e2=a2, bab=eae-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=ebe-1=a-1b, cd=dc, ece-1=a2c, ede-1=d3 >

Subgroups: 796 in 376 conjugacy classes, 108 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×32], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×23], D4 [×6], D4 [×33], Q8 [×2], Q8 [×9], C23, C23 [×2], C23 [×21], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], SD16 [×12], Q16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×8], C2×D4 [×40], C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C4○D4 [×22], C24 [×3], C22⋊C8 [×4], D4⋊C4 [×6], Q8⋊C4 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16 [×6], C2×SD16 [×4], C2×Q16 [×2], C8.C22 [×4], C22×D4, C22×D4 [×2], C22×D4 [×3], C22×Q8, C2×C4○D4 [×2], C2×C4○D4 [×2], C2×C4○D4, 2+ 1+4 [×8], (C22×C8)⋊C2, C23.24D4, C23.37D4, C22⋊SD16 [×4], D4.7D4 [×4], C23.38C23, C22×SD16, C2×C8.C22, C2×2+ 1+4, D4.(C2×D4)
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, D4○SD16 [×2], D4.(C2×D4)

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F···2M4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F
order1222222···2444444444444888888
size1111224···4222244448888444488

32 irreducible representations

dim11111111112224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4D4D4○SD16
kernelD4.(C2×D4)(C22×C8)⋊C2C23.24D4C23.37D4C22⋊SD16D4.7D4C23.38C23C22×SD16C2×C8.C22C2×2+ 1+4C2×D4C2×Q8C4○D4C2
# reps11114411117144

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

1600000
0160000
00161500
001100
000101
001616160
,
100000
7160000
001000
00161600
00160160
000001
,
1600000
0160000
0010015
00160161
000101
0010016
,
1020000
1070000
00101000
0012000
00012512
005555
,
1020000
1070000
00101000
0012700
00012512
00551212

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,0,16,0,0,15,1,1,16,0,0,0,0,0,16,0,0,0,0,1,0],[1,7,0,0,0,0,0,16,0,0,0,0,0,0,1,16,16,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,15,1,1,16],[10,10,0,0,0,0,2,7,0,0,0,0,0,0,10,12,0,5,0,0,10,0,12,5,0,0,0,0,5,5,0,0,0,0,12,5],[10,10,0,0,0,0,2,7,0,0,0,0,0,0,10,12,0,5,0,0,10,7,12,5,0,0,0,0,5,12,0,0,0,0,12,12] >;

D4.(C2×D4) in GAP, Magma, Sage, TeX

D_4.(C_2\times D_4)
% in TeX

G:=Group("D4.(C2xD4)");
// GroupNames label

G:=SmallGroup(128,1741);
// by ID

G=gap.SmallGroup(128,1741);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,521,2804,1411,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=1,c^2=d^4=e^2=a^2,b*a*b=e*a*e^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=e*b*e^-1=a^-1*b,c*d=d*c,e*c*e^-1=a^2*c,e*d*e^-1=d^3>;
// generators/relations

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