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G = C2×C23.7D4  order 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⋊C4⋊2C23, C23⋊C4⋊6C22, (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.D4⋊28C22, (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

Derived series
C1 — C23 — C2×C23.7D4
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — C2×2+ (1+4) — C2×C23.7D4
Lower central
C1 — C2 — C23 — C2×C23.7D4
Upper central
C1 — C22 — C24 — C2×C23.7D4
Jennings
C1 — C2 — C23 — C2×C23.7D4

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

Generators and relations
 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 >

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

(2 31)(3 26)(4 17)(5 21)(6 16)(8 10)(11 15)(12 22)(14 24)(19 25)(20 32)(27 29)

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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+4)C22×C4C2×D4C24C2
# reps138313634

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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