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G = C42.240D4order 128 = 27

222nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.240D4, C42.707C23, (C4×C8)⋊64C22, C4⋊Q863C22, C4.4D840C2, C4⋊C4.91C23, C8⋊C465C22, C4.20(C8⋊C22), (C4×M4(2))⋊40C2, (C2×C4).336C24, (C2×C8).457C23, (C22×C4).460D4, C23.678(C2×D4), D4⋊C453C22, C4.77(C4.4D4), (C2×D4).104C23, C42.C235C22, C23.37D437C2, C41D4.147C22, (C2×C42).847C22, C22.596(C22×D4), (C22×C4).1034C23, C22.31(C4.4D4), (C22×D4).369C22, C23.37C2310C2, C42.29C2218C2, C42⋊C2.141C22, (C2×M4(2)).373C22, C4.45(C2×C4○D4), (C2×C4).514(C2×D4), C2.39(C2×C8⋊C22), (C2×C41D4).24C2, C2.47(C2×C4.4D4), (C2×C4).300(C4○D4), SmallGroup(128,1870)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.240D4
C1C2C4C2×C4C22×C4C2×M4(2)C4×M4(2) — C42.240D4
C1C2C2×C4 — C42.240D4
C1C22C2×C42 — C42.240D4
C1C2C2C2×C4 — C42.240D4

Generators and relations for C42.240D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2c3 >

Subgroups: 580 in 242 conjugacy classes, 96 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×6], C22, C22 [×2], C22 [×22], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×8], D4 [×28], Q8 [×4], C23, C23 [×16], C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×26], C2×Q8 [×2], C24 [×2], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×16], C2×C42, C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C41D4 [×4], C41D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C22×D4 [×2], C22×D4 [×2], C4×M4(2), C23.37D4 [×4], C4.4D8 [×4], C42.29C22 [×4], C2×C41D4, C23.37C23, C42.240D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C8⋊C22 [×4], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, C2×C8⋊C22 [×2], C42.240D4

Smallest permutation representation of C42.240D4
On 32 points
Generators in S32
(1 18 25 11)(2 19 26 12)(3 20 27 13)(4 21 28 14)(5 22 29 15)(6 23 30 16)(7 24 31 9)(8 17 32 10)
(1 31 5 27)(2 28 6 32)(3 25 7 29)(4 30 8 26)(9 22 13 18)(10 19 14 23)(11 24 15 20)(12 21 16 17)
(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 23 29 12)(2 11 30 22)(3 21 31 10)(4 9 32 20)(5 19 25 16)(6 15 26 18)(7 17 27 14)(8 13 28 24)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I4J4K4L4M4N8A···8H
order12222222224···44444448···8
size11112288882···24488884···4

32 irreducible representations

dim11111112224
type++++++++++
imageC1C2C2C2C2C2C2D4D4C4○D4C8⋊C22
kernelC42.240D4C4×M4(2)C23.37D4C4.4D8C42.29C22C2×C41D4C23.37C23C42C22×C4C2×C4C4
# reps11444112284

Matrix representation of C42.240D4 in GL6(𝔽17)

1690000
1310000
0001600
001000
0000016
000010
,
1600000
0160000
0001600
001000
000001
0000160
,
1300000
0130000
0000016
000010
0016000
0001600
,
1320000
040000
000010
0000016
001000
0001600

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

C42.240D4 in GAP, Magma, Sage, TeX

C_4^2._{240}D_4
% in TeX

G:=Group("C4^2.240D4");
// GroupNames label

G:=SmallGroup(128,1870);
// by ID

G=gap.SmallGroup(128,1870);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,100,1018,521,2804,172,4037,124]);
// Polycyclic

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

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