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G = C24⋊4Q8  order 128 = 27

3rd semidirect product of C24 and Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C24⋊4Q8, C25.42C22, C24.566C23, C23.337C24, C22.1462+ (1+4), C2.15(D42), C23⋊Q8⋊7C2, C23.61(C2×Q8), C24⋊3C4.7C2, C22⋊C4.125D4, C23.423(C2×D4), C22⋊1(C22⋊Q8), C2.32(D4⋊5D4), C23.4Q8⋊4C2, C23.Q8⋊8C2, (C22×Q8)⋊3C22, (C22×C4).58C23, C23.8Q8⋊37C2, C23.7Q8⋊44C2, C2.3(C23⋊2Q8), C23.303(C4○D4), C22.68(C22×Q8), (C23×C4).350C22, C22.217(C22×D4), C2.C42⋊23C22, C2.14(C22.45C24), (C2×C4⋊C4)⋊17C22, (C2×C4).321(C2×D4), (C2×C22⋊Q8)⋊10C2, C2.16(C2×C22⋊Q8), C22.214(C2×C4○D4), (C22×C22⋊C4).22C2, (C2×C22⋊C4).124C22, SmallGroup(128,1169)

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

Derived series
C1 — C23 — C24⋊4Q8
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C22⋊C4 — C24⋊4Q8
Lower central
C1 — C23 — C24⋊4Q8
Upper central
C1 — C23 — C24⋊4Q8
Jennings
C1 — C23 — C24⋊4Q8

Subgroups: 820 in 390 conjugacy classes, 124 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×10], C4 [×16], C22 [×3], C22 [×12], C22 [×50], C2×C4 [×8], C2×C4 [×48], Q8 [×4], C23, C23 [×14], C23 [×50], C22⋊C4 [×8], C22⋊C4 [×22], C4⋊C4 [×15], C22×C4 [×2], C22×C4 [×10], C22×C4 [×20], C2×Q8 [×5], C24, C24 [×6], C24 [×10], C2.C42 [×2], C2.C42 [×4], C2×C22⋊C4 [×2], C2×C22⋊C4 [×12], C2×C22⋊C4 [×8], C2×C4⋊C4, C2×C4⋊C4 [×8], C22⋊Q8 [×8], C23×C4 [×4], C22×Q8, C25, C24⋊3C4, C23.7Q8 [×2], C23.8Q8 [×4], C23⋊Q8, C23.Q8 [×2], C23.4Q8, C22×C22⋊C4 [×2], C2×C22⋊Q8 [×2], C24⋊4Q8

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], Q8 [×4], C23 [×15], C2×D4 [×12], C2×Q8 [×6], C4○D4 [×4], C24, C22⋊Q8 [×8], C22×D4 [×2], C22×Q8, C2×C4○D4 [×2], 2+ (1+4) [×2], C2×C22⋊Q8 [×2], C23⋊2Q8, D42, D4⋊5D4 [×2], C22.45C24, C24⋊4Q8

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=e2, ab=ba, faf-1=ac=ca, eae-1=ad=da, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
010000
001000
000400
000040
000001
,
100000
010000
001000
000400
000040
000001
,
100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
010000
400000
004000
000400
000020
000003
,
300000
020000
000100
001000
000001
000040

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

38 conjugacy classes

class 1 2A···2G2H···2O2P2Q4A···4P4Q4R4S4T
order12···22···2224···44444
size11···12···2444···48888

38 irreducible representations

dim1111111112224
type++++++++++-+
imageC1C2C2C2C2C2C2C2C2D4Q8C4○D42+ (1+4)
kernelC24⋊4Q8C24⋊3C4C23.7Q8C23.8Q8C23⋊Q8C23.Q8C23.4Q8C22×C22⋊C4C2×C22⋊Q8C22⋊C4C24C23C22
# reps1124121228482

In GAP, Magma, Sage, TeX 

C_2^4\rtimes_4Q_8
 
% in TeX
 
G:=Group("C2^4:4Q8");
 
// GroupNames label
 
G:=SmallGroup(128,1169);
 
// by ID
 
G=gap.SmallGroup(128,1169);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,184,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=e^2,a*b=b*a,f*a*f^-1=a*c=c*a,e*a*e^-1=a*d=d*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
 
// generators/relations
 

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