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G = C2×C23.7D4order 128 = 27

Direct product of C2 and C23.7D4

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

Aliases: C2×C23.7D4, C24.45D4, C23.6C24, C24.470C23, 2+ 1+4.12C22, (C2×D4).148D4, C22⋊C42C23, C23⋊C46C22, (C22×C4)⋊8C23, C23.25(C2×D4), (C23×C4)⋊14C22, (C2×D4).40C23, (C22×C4).114D4, C22.26C22≀C2, C22.40(C22×D4), (C2×2+ 1+4).9C2, (C22×D4).333C22, C22.D428C22, (C2×C4).26(C2×D4), (C2×C23⋊C4)⋊17C2, C2.61(C2×C22≀C2), (C2×C22⋊C4)⋊37C22, (C2×C22.D4)⋊47C2, SmallGroup(128,1756)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C2×C23.7D4
C1C2C22C23C24C22×D4C2×2+ 1+4 — C2×C23.7D4
C1C2C23 — C2×C23.7D4
C1C22C24 — C2×C23.7D4
C1C2C23 — C2×C23.7D4

Generators and relations for C2×C23.7D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf-1=bcd, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef-1=de-1 >

Subgroups: 836 in 397 conjugacy classes, 108 normal (8 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×14], C22, C22 [×6], C22 [×35], C2×C4 [×6], C2×C4 [×40], D4 [×36], Q8 [×4], C23, C23 [×12], C23 [×20], C22⋊C4 [×6], C22⋊C4 [×15], C4⋊C4 [×12], C22×C4 [×5], C22×C4 [×12], C2×D4 [×12], C2×D4 [×39], C2×Q8, C4○D4 [×24], C24, C24 [×3], C24, C23⋊C4 [×12], C2×C22⋊C4 [×3], C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C22.D4 [×12], C22.D4 [×6], C23×C4, C22×D4 [×3], C22×D4 [×3], C2×C4○D4 [×3], 2+ 1+4 [×4], 2+ 1+4 [×6], C2×C23⋊C4 [×3], C23.7D4 [×8], C2×C22.D4 [×3], C2×2+ 1+4, C2×C23.7D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C23.7D4 [×2], C2×C22≀C2, C2×C23.7D4

Smallest permutation representation of C2×C23.7D4
On 32 points
Generators in S32
(1 30)(2 31)(3 32)(4 29)(5 9)(6 10)(7 11)(8 12)(13 23)(14 24)(15 21)(16 22)(17 27)(18 28)(19 25)(20 26)
(2 31)(3 5)(4 10)(6 29)(8 12)(9 32)(14 27)(15 18)(16 22)(17 24)(19 25)(21 28)
(1 11)(2 31)(3 9)(4 29)(5 32)(6 10)(7 30)(8 12)(13 23)(14 27)(15 21)(16 25)(17 24)(18 28)(19 22)(20 26)
(1 7)(2 8)(3 5)(4 6)(9 32)(10 29)(11 30)(12 31)(13 20)(14 17)(15 18)(16 19)(21 28)(22 25)(23 26)(24 27)
(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 25 7 22)(2 21 8 28)(3 27 5 24)(4 23 6 26)(9 14 32 17)(10 20 29 13)(11 16 30 19)(12 18 31 15)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I2J···2O4A···4J4K···4P
order12222···22···24···44···4
size11112···24···44···48···8

32 irreducible representations

dim111112224
type++++++++
imageC1C2C2C2C2D4D4D4C23.7D4
kernelC2×C23.7D4C2×C23⋊C4C23.7D4C2×C22.D4C2×2+ 1+4C22×C4C2×D4C24C2
# reps138313634

Matrix representation of C2×C23.7D4 in GL6(𝔽5)

400000
040000
004000
000400
000040
000004
,
100000
010000
004400
000100
000044
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
040000
100000
000022
000003
003300
004200
,
400000
010000
003000
004200
000030
000003

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4,1,0,0,0,0,0,0,4,0,0,0,0,0,4,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,0,3,4,0,0,0,0,3,2,0,0,2,0,0,0,0,0,2,3,0,0],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,4,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;

C2×C23.7D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._7D_4
% in TeX

G:=Group("C2xC2^3.7D4");
// GroupNames label

G:=SmallGroup(128,1756);
// by ID

G=gap.SmallGroup(128,1756);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,718,2028]);
// Polycyclic

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

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