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G = C42⋊9(C2×C4)  order 128 = 27

4th semidirect product of C42 and C2×C4 acting via C2×C4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42⋊9(C2×C4)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C23.33C23 — C42⋊9(C2×C4)
 Lower central C1 — C2 — C2×C4 — C42⋊9(C2×C4)
 Upper central C1 — C22 — C22×C4 — C42⋊9(C2×C4)
 Jennings C1 — C2 — C2 — C22×C4 — C42⋊9(C2×C4)

Generators and relations for C429(C2×C4)
G = < a,b,c,d | a4=b4=c2=d4=1, cac=ab=ba, dad-1=a-1b, cbc=b-1, bd=db, cd=dc >

Subgroups: 284 in 143 conjugacy classes, 56 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×9], C22 [×3], C22 [×6], C8 [×3], C2×C4 [×6], C2×C4 [×20], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C42 [×4], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×2], M4(2) [×2], M4(2) [×3], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C2.C42 [×2], D4⋊C4, Q8⋊C4, C4≀C2 [×4], C4.Q8, C2.D8, C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×3], C4×Q8, C2×M4(2) [×2], C2×C4○D4, C426C4, C22.C42, C428C4, C23.36D4, C2×C4≀C2, M4(2)⋊C4, C23.33C23, C429(C2×C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C23.8Q8, D4.8D4, D4.9D4, C429(C2×C4)

Smallest permutation representation of C429(C2×C4)
On 32 points
Generators in S32
(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 4 5 8)(2 3 6 7)(9 15 12 14)(10 16 11 13)(17 26 19 28)(18 27 20 25)(21 32 23 30)(22 29 24 31)
(1 17)(2 27)(3 18)(4 28)(5 19)(6 25)(7 20)(8 26)(9 24)(10 32)(11 30)(12 22)(13 23)(14 31)(15 29)(16 21)
(1 21 6 29)(2 31 5 23)(3 22 8 30)(4 32 7 24)(9 28 10 20)(11 18 12 26)(13 27 14 19)(15 17 16 25)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4R 4S 4T 8A 8B 8C 8D order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 8 8 8 8 size 1 1 1 1 2 2 4 4 2 2 2 2 4 ··· 4 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 D4 D4 Q8 C4○D4 D4.8D4 D4.9D4 kernel C42⋊9(C2×C4) C42⋊6C4 C22.C42 C42⋊8C4 C23.36D4 C2×C4≀C2 M4(2)⋊C4 C23.33C23 C4≀C2 C4⋊C4 C22×C4 C2×D4 C2×Q8 C4○D4 C2×C4 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 2 1 1 2 4 2 2

Matrix representation of C429(C2×C4) in GL6(𝔽17)

 0 16 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 1 0 4 0 0 0 11 0 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 4 4 0 0 0 10 0 0 13
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 0 12 6 0 1 0 0 6 12 1 0
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 12 1 2 0 0 0 11 0 0 15 0 0 7 11 5 1 0 0 15 6 11 0

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,1,11,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,10,0,0,0,13,4,0,0,0,0,0,4,0,0,0,0,0,0,13],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,16,12,6,0,0,16,0,6,12,0,0,0,0,0,1,0,0,0,0,1,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,12,11,7,15,0,0,1,0,11,6,0,0,2,0,5,11,0,0,0,15,1,0] >;

C429(C2×C4) in GAP, Magma, Sage, TeX

C_4^2\rtimes_9(C_2\times C_4)
% in TeX

G:=Group("C4^2:9(C2xC4)");
// GroupNames label

G:=SmallGroup(128,592);
// by ID

G=gap.SmallGroup(128,592);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,352,2019,1018,248,2804,718,172]);
// Polycyclic

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

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