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G = (C2×C8).103D4order 128 = 27

71st non-split extension by C2×C8 of D4 acting via D4/C2=C22

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

Aliases: (C2×C8).103D4, (C2×D4).25Q8, (C2×D4).201D4, C23.2(C2×Q8), (C22×C8).10C4, (C2×Q8).159D4, C8.46(C22⋊C4), C23.10(C4⋊C4), C4.10C427C2, C4.122(C4⋊D4), (C2×M4(2)).13C4, C4.84(C42⋊C2), (C22×C8).215C22, (C22×C4).663C23, C22.12(C22⋊Q8), C2.22(C23.7Q8), (C2×M4(2)).156C22, M4(2).8C22.5C2, (C2×C8).12(C2×C4), (C2×C8○D4).2C2, (C2×C8.C4)⋊3C2, (C2×C4).12(C4⋊C4), (C2×C4).230(C2×D4), C22.21(C2×C4⋊C4), C4.92(C2×C22⋊C4), (C22×C4).75(C2×C4), (C2×C4).738(C4○D4), (C2×C4).533(C22×C4), (C2×C4○D4).257C22, SmallGroup(128,545)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — (C2×C8).103D4
C1C2C4C2×C4C22×C4C2×C4○D4C2×C8○D4 — (C2×C8).103D4
C1C2C2×C4 — (C2×C8).103D4
C1C4C2×M4(2) — (C2×C8).103D4
C1C2C2C22×C4 — (C2×C8).103D4

Generators and relations for (C2×C8).103D4
 G = < a,b,c,d | a2=b8=1, c4=b4, d2=ab2, ab=ba, ac=ca, dad-1=ab4, cbc-1=dbd-1=b-1, dcd-1=ab6c3 >

Subgroups: 212 in 126 conjugacy classes, 58 normal (20 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×5], C8 [×2], C8 [×2], C8 [×6], C2×C4 [×2], C2×C4 [×6], C2×C4 [×4], D4 [×6], Q8 [×2], C23, C23 [×2], C2×C8 [×2], C2×C8 [×6], C2×C8 [×8], M4(2) [×14], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C4.D4 [×2], C4.10D4 [×2], C8.C4 [×4], C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×6], C8○D4 [×4], C2×C4○D4, C4.10C42 [×2], M4(2).8C22 [×2], C2×C8.C4 [×2], C2×C8○D4, (C2×C8).103D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C23.7Q8, (C2×C8).103D4

Smallest permutation representation of (C2×C8).103D4
On 32 points
Generators in S32
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)
(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 15 22 28 5 11 18 32)(2 14 23 27 6 10 19 31)(3 13 24 26 7 9 20 30)(4 12 17 25 8 16 21 29)
(1 11 22 28 5 15 18 32)(2 10 23 27 6 14 19 31)(3 9 24 26 7 13 20 30)(4 16 17 25 8 12 21 29)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G8A8B8C8D8E···8J8K···8R
order1222222444444488888···88···8
size1122244112224422224···48···8

32 irreducible representations

dim1111111222224
type+++++++-+
imageC1C2C2C2C2C4C4D4D4Q8D4C4○D4(C2×C8).103D4
kernel(C2×C8).103D4C4.10C42M4(2).8C22C2×C8.C4C2×C8○D4C22×C8C2×M4(2)C2×C8C2×D4C2×D4C2×Q8C2×C4C1
# reps1222144412144

Matrix representation of (C2×C8).103D4 in GL4(𝔽17) generated by

01300
4000
9804
99130
,
2000
0200
14090
3009
,
2010
20016
131150
5020
,
2010
15001
131150
3020
G:=sub<GL(4,GF(17))| [0,4,9,9,13,0,8,9,0,0,0,13,0,0,4,0],[2,0,14,3,0,2,0,0,0,0,9,0,0,0,0,9],[2,2,13,5,0,0,1,0,1,0,15,2,0,16,0,0],[2,15,13,3,0,0,1,0,1,0,15,2,0,1,0,0] >;

(C2×C8).103D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{103}D_4
% in TeX

G:=Group("(C2xC8).103D4");
// GroupNames label

G:=SmallGroup(128,545);
// by ID

G=gap.SmallGroup(128,545);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2019,248,2804,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=a*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^4,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a*b^6*c^3>;
// generators/relations

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